Theory and Practice of Codes with Locality
Introduction
July 10, 2016
Theory and Practice of Codes with Locality
July 10, 2016
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Information Era
Theory and Practice of Codes with Locality
July 10, 2016
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Information Era
• We live in Information Era
Theory and Practice of Codes with Locality
July 10, 2016
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Information Era
• We live in Information Era
• Big Data players: Facebook, Google, MSFT, Amazon, Alibaba, Dropbox, etc.
Theory and Practice of Codes with Locality
July 10, 2016
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Information Era
• We live in Information Era
• Big Data players: Facebook, Google, MSFT, Amazon, Alibaba, Dropbox, etc.
Theory and Practice of Codes with Locality
July 10, 2016
2 / 22
Information Era
• We live in Information Era
• Big Data players: Facebook, Google, MSFT, Amazon, Alibaba, Dropbox, etc.
• Node failures are the norm
Theory and Practice of Codes with Locality
July 10, 2016
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State-of-the-Art Coding technique
Theory and Practice of Codes with Locality
July 10, 2016
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State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
Theory and Practice of Codes with Locality
July 10, 2016
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State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
Theory and Practice of Codes with Locality
July 10, 2016
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State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
Theory and Practice of Codes with Locality
July 10, 2016
3 / 22
State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
RAID 6
Theory and Practice of Codes with Locality
July 10, 2016
3 / 22
State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
RAID 6
• MDS codes with two parities
Theory and Practice of Codes with Locality
July 10, 2016
3 / 22
State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
RAID 6
• MDS codes with two parities
[n, k] MDS codes
Theory and Practice of Codes with Locality
July 10, 2016
3 / 22
State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
RAID 6
• MDS codes with two parities
[n, k] MDS codes
• Can tolerate any n − k disk failures
Theory and Practice of Codes with Locality
July 10, 2016
3 / 22
State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
RAID 6
• MDS codes with two parities
[n, k] MDS codes
• Can tolerate any n − k disk failures
• FB uses (14, 10) RS codes
Theory and Practice of Codes with Locality
July 10, 2016
3 / 22
State-of-the-Art Coding technique
RAID: Redundant Array of Independent Disks
RAID 1 – Replication (typically 3x)
• Simple implementation!
• High availability of the information
• Can tolerate any 2 disk failures
• Widely used in Hadoop and many other systems
• Storage overhead of 200%!!!!!
RAID 6
• MDS codes with two parities
[n, k] MDS codes
• Can tolerate any n − k disk failures
• FB uses (14, 10) RS codes
• Poor handling of single disk failures (The Repair Problem)
Theory and Practice of Codes with Locality
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Limitations of Reed-Solomon codes
Theory and Practice of Codes with Locality
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
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Theory and Practice of Codes with Locality
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
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Theory and Practice of Codes with Locality
P1
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P4
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
• A disk is lost - Repair job starts to repair it
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Theory and Practice of Codes with Locality
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P3
P4
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
• A disk is lost - Repair job starts to repair it
• Transmit information from 10 disks to recover one lost disk
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Theory and Practice of Codes with Locality
P1
P2
P3
P4
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
• A disk is lost - Repair job starts to repair it
• Transmit information from 10 disks to recover one lost disk
• Generates 10x more traffic compared to replication for recovery of one disk
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Theory and Practice of Codes with Locality
P1
P2
P3
P4
July 10, 2016
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
• A disk is lost - Repair job starts to repair it
• Transmit information from 10 disks to recover one lost disk
• Generates 10x more traffic compared to replication for recovery of one disk
• If a large portion the data is RS-coded =⇒ saturation of the network
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Theory and Practice of Codes with Locality
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P2
P3
P4
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Limitations of Reed-Solomon codes
Example: [14, 10] RS code
• A disk is lost - Repair job starts to repair it
• Transmit information from 10 disks to recover one lost disk
• Generates 10x more traffic compared to replication for recovery of one disk
• If a large portion the data is RS-coded =⇒ saturation of the network
• Goal: Construct efficient codes with ”good“ repair process
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Theory and Practice of Codes with Locality
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P4
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How “good” is the repair process?
Theory and Practice of Codes with Locality
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How “good” is the repair process?
• Total bandwidth during the repair
Theory and Practice of Codes with Locality
July 10, 2016
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How “good” is the repair process?
• Total bandwidth during the repair
• Regenerating Codes - Seminal paper by Dimakis, Godfrey, Wu, Wainwright and
Ramchandran
Theory and Practice of Codes with Locality
July 10, 2016
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How “good” is the repair process?
• Total bandwidth during the repair
• Regenerating Codes - Seminal paper by Dimakis, Godfrey, Wu, Wainwright and
Ramchandran
• Model allows connecting “many” nodes
Theory and Practice of Codes with Locality
July 10, 2016
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How “good” is the repair process?
• Total bandwidth during the repair
• Regenerating Codes - Seminal paper by Dimakis, Godfrey, Wu, Wainwright and
Ramchandran
• Model allows connecting “many” nodes
• Total number of disks participating in the repair
Theory and Practice of Codes with Locality
July 10, 2016
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How “good” is the repair process?
• Total bandwidth during the repair
• Regenerating Codes - Seminal paper by Dimakis, Godfrey, Wu, Wainwright and
Ramchandran
• Model allows connecting “many” nodes
• Total number of disks participating in the repair
• The struggler
Theory and Practice of Codes with Locality
July 10, 2016
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How “good” is the repair process?
• Total bandwidth during the repair
• Regenerating Codes - Seminal paper by Dimakis, Godfrey, Wu, Wainwright and
Ramchandran
• Model allows connecting “many” nodes
• Total number of disks participating in the repair
• The struggler
• Locally Recoverable (LRC) Codes
Theory and Practice of Codes with Locality
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Codes with locality: Plan
Theory and Practice of Codes with Locality
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
Theory and Practice of Codes with Locality
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
Theory and Practice of Codes with Locality
July 10, 2016
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
• LRC codes in practice (Barg)
Theory and Practice of Codes with Locality
July 10, 2016
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
• LRC codes in practice (Barg)
• Break ,
Theory and Practice of Codes with Locality
July 10, 2016
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
• LRC codes in practice (Barg)
• Break ,
• LRC codes - Bounds (Barg)
Theory and Practice of Codes with Locality
July 10, 2016
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
• LRC codes in practice (Barg)
• Break ,
• LRC codes - Bounds (Barg)
• Maximally Recoverable Codes (Tamo)
Theory and Practice of Codes with Locality
July 10, 2016
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Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
• LRC codes in practice (Barg)
• Break ,
• LRC codes - Bounds (Barg)
• Maximally Recoverable Codes (Tamo)
• LRC codes on graphs (Tamo)
Theory and Practice of Codes with Locality
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
Theory and Practice of Codes with Locality
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
Theory and Practice of Codes with Locality
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
1
k−1
k
Theory and Practice of Codes with Locality
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
1
k−1
k
k+1
Theory and Practice of Codes with Locality
k+2
n
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
• An erasure has occurred
1
k−1
k
k+1
Theory and Practice of Codes with Locality
k+2
n
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
• An erasure has occurred
• Any symbol i has a recovery set Ri of r other symbols, r k
1
k−1
k
k+1
Theory and Practice of Codes with Locality
k+2
n
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
• An erasure has occurred
• Any symbol i has a recovery set Ri of r other symbols, r k
1
k−1
k
k+1
Theory and Practice of Codes with Locality
k+2
n
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
• An erasure has occurred
• Any symbol i has a recovery set Ri of r other symbols, r k
1
k−1
k
k+1
k+2
n
r
Theory and Practice of Codes with Locality
July 10, 2016
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Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
• An erasure has occurred
• Any symbol i has a recovery set Ri of r other symbols, r k
1
k−1
k
r
k
k+1
k+2
n
Locally Recoverable Codes - Definition
(n, k, r) LRC Code
• Takes k blocks (symbols) → produces n blocks
• An erasure has occurred
• Any symbol i has a recovery set Ri of r other symbols, r k
• Clearly 1 ≤ r ≤ k
1
k−1
k
k+1
k+2
n
r
k
Theory and Practice of Codes with Locality
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Early references on LRC codes
Theory and Practice of Codes with Locality
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Early references on LRC codes
• J. Han and L. Lastras-Montano, ISIT 2007;
Theory and Practice of Codes with Locality
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Early references on LRC codes
• J. Han and L. Lastras-Montano, ISIT 2007;
• C. Huang, M. Chen, and J. Li, Symp. Networks App. 2007;
Theory and Practice of Codes with Locality
July 10, 2016
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Early references on LRC codes
• J. Han and L. Lastras-Montano, ISIT 2007;
• C. Huang, M. Chen, and J. Li, Symp. Networks App. 2007;
• F. Oggier and A. Datta 2013 - Survey on codes for distributed storage systems;
Theory and Practice of Codes with Locality
July 10, 2016
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Early references on LRC codes
• J. Han and L. Lastras-Montano, ISIT 2007;
• C. Huang, M. Chen, and J. Li, Symp. Networks App. 2007;
• F. Oggier and A. Datta 2013 - Survey on codes for distributed storage systems;
• P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, IEEE Trans. Inf. Theory,
Nov. 2012.
Theory and Practice of Codes with Locality
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Parameters of LRC codes
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• Rate?
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• Rate?
• Minimum distance?
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Proof:
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Proof:
• There exist at most
nr
r+1
coordinates that determine the exact codeword
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Proof:
• There exist at most
nr
r+1
coordinates that determine the exact codeword
• This follows since iteratively:
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Proof:
• There exist at most
nr
r+1
coordinates that determine the exact codeword
• This follows since iteratively:
1. Cost: expose the values of the coordinates in a recovery set Ri , |Ri | ≤ r
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Proof:
• There exist at most
nr
r+1
coordinates that determine the exact codeword
• This follows since iteratively:
1. Cost: expose the values of the coordinates in a recovery set Ri , |Ri | ≤ r
2. Free: the value of the i-th coordinate
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Proof:
• There exist at most
nr
r+1
coordinates that determine the exact codeword
• This follows since iteratively:
1. Cost: expose the values of the coordinates in a recovery set Ri , |Ri | ≤ r
2. Free: the value of the i-th coordinate
3. By exposing at most
nr
r+1
coordinates, the exact codeword is determined
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
• The bound is tight (even over F2 )
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
• The bound is tight (even over F2 )
• Partition the k bits into k/r sets of size r
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The rate is bounded by
k
r
≤
.
n
r+1
• The bound is tight (even over F2 )
• Partition the k bits into k/r sets of size r
• Add parity check bit to each set
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The minimum distance is bounded by
d ≤n−k−
k
+2
r
[Gopalan, Huang, Simitci, Yekhanin 12]
[Papailiopoulos, Dimakis 12]
Theory and Practice of Codes with Locality
July 10, 2016
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Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The minimum distance is bounded by
d ≤n−k−
k
+2
r
[Gopalan, Huang, Simitci, Yekhanin 12]
[Papailiopoulos, Dimakis 12]
Remarks:
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The minimum distance is bounded by
d ≤n−k−
k
+2
r
[Gopalan, Huang, Simitci, Yekhanin 12]
[Papailiopoulos, Dimakis 12]
Remarks:
• Smaller locality =⇒ lower failure resilience
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The minimum distance is bounded by
d ≤n−k−
k
+2
r
[Gopalan, Huang, Simitci, Yekhanin 12]
[Papailiopoulos, Dimakis 12]
Remarks:
• Smaller locality =⇒ lower failure resilience
• Generalization of the Singleton bound (r = k)
Theory and Practice of Codes with Locality
July 10, 2016
9 / 22
Parameters of LRC codes
Let C be an (n, k, r) LRC code
• Assume r|k and r + 1|n
• The minimum distance is bounded by
d ≤n−k−
k
+2
r
[Gopalan, Huang, Simitci, Yekhanin 12]
[Papailiopoulos, Dimakis 12]
Remarks:
• Smaller locality =⇒ lower failure resilience
• Generalization of the Singleton bound (r = k)
• Optimal (n, k, r) LRC code achieves the bound with equality
Theory and Practice of Codes with Locality
July 10, 2016
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Constructing optimal LRC codes
Theory and Practice of Codes with Locality
July 10, 2016
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Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
Theory and Practice of Codes with Locality
July 10, 2016
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Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
1. Non explicit /
Theory and Practice of Codes with Locality
July 10, 2016
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Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
1. Non explicit /
2. Field size is superpolynomial in the length / (is it truly necessary?)
Theory and Practice of Codes with Locality
July 10, 2016
10 / 22
Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
1. Non explicit /
2. Field size is superpolynomial in the length / (is it truly necessary?)
• Optimal ((r + 1)d kr e, k, r) LRC code [Prasanth, Kamath, Lalitha, and Kumar 12]
Theory and Practice of Codes with Locality
July 10, 2016
10 / 22
Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
1. Non explicit /
2. Field size is superpolynomial in the length / (is it truly necessary?)
• Optimal ((r + 1)d kr e, k, r) LRC code [Prasanth, Kamath, Lalitha, and Kumar 12]
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
Theory and Practice of Codes with Locality
July 10, 2016
10 / 22
Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
1. Non explicit /
2. Field size is superpolynomial in the length / (is it truly necessary?)
• Optimal ((r + 1)d kr e, k, r) LRC code [Prasanth, Kamath, Lalitha, and Kumar 12]
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
Theory and Practice of Codes with Locality
July 10, 2016
10 / 22
Constructing optimal LRC codes
• A carefully constructed random generating matrix gives an optimal LRC
1. Non explicit /
2. Field size is superpolynomial in the length / (is it truly necessary?)
• Optimal ((r + 1)d kr e, k, r) LRC code [Prasanth, Kamath, Lalitha, and Kumar 12]
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
2. Field size is superpolynomial
Theory and Practice of Codes with Locality
July 10, 2016
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Optimal LRC codes - Easy cases
Theory and Practice of Codes with Locality
July 10, 2016
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Optimal LRC codes - Easy cases
• r=k
Theory and Practice of Codes with Locality
July 10, 2016
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Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
Theory and Practice of Codes with Locality
July 10, 2016
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Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
Theory and Practice of Codes with Locality
July 10, 2016
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Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
• r=1
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
• r=1
1. d ≤ 2( 2n − k + 1)
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
• r=1
1. d ≤ 2( 2n − k + 1)
2. Duplication of an (n/2, k) RS is an (n, k, 1) optimal LRC code
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
• r=1
1. d ≤ 2( 2n − k + 1)
2. Duplication of an (n/2, k) RS is an (n, k, 1) optimal LRC code
3. |F| = O(n)
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
• r=1
1. d ≤ 2( 2n − k + 1)
2. Duplication of an (n/2, k) RS is an (n, k, 1) optimal LRC code
3. |F| = O(n)
• Q: What happens for 1 < r < k?
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Optimal LRC codes - Easy cases
• r=k
1. d ≤ n − k + 1
2. An (n, k) RS is an (n, k, k) optimal LRC code
3. |F| = O(n)
• r=1
1. d ≤ 2( 2n − k + 1)
2. Duplication of an (n/2, k) RS is an (n, k, 1) optimal LRC code
3. |F| = O(n)
• Q: What happens for 1 < r < k?
• Q: Generalize the optimal codes for r = 1, k to codes with arbitrary r?
Theory and Practice of Codes with Locality
July 10, 2016
11 / 22
Intuition behind the LRC construction
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 )
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
f (x1 )
x1
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
f (x1 )
f (x2 )
x1
x2
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
f (x5 )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
xn
Intuition behind the LRC construction
(a0 , a1 , a2 , a3 ) → f (x) = a0 + a1 x + a2 x2 + a3 x3 → f (x) = (f (x1 ), f (x2 ), ..., f (xn ))
Only two points suffice to recover the lost point
f (x5 )
f (xn )
f (x4 )
f (x1 )
f (x2 ) f (x3 )
x1
x2
x3
x4
x5
Theory and Practice of Codes with Locality
xn
July 10, 2016
12 / 22
Optimal (n, k, r) LRC code construction
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
2.2 g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
2.2 g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Encoding:
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
2.2 g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Encoding: Given k information symbols ai,j , i = 0, ..., r − 1, j = 0, ..., kr − 1
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
2.2 g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Encoding: Given k information symbols ai,j , i = 0, ..., r − 1, j = 0, ..., kr − 1
1. Define the polynomial
k
−1
r−1 X
r
X
i
f (x) =
x
ai,j g(x)j
i=0
j=0
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
2.2 g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Encoding: Given k information symbols ai,j , i = 0, ..., r − 1, j = 0, ..., kr − 1
1. Define the polynomial
k
−1
r−1 X
r
X
i
f (x) =
x
ai,j g(x)j
i=0
j=0
2. Store the length−n vector (f (α) : α ∈ ∪i Ri )
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. g(x) ∈ F[x] is a polynomial s.t.
2.1 deg (g(x)) = r + 1
2.2 g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Encoding: Given k information symbols ai,j , i = 0, ..., r − 1, j = 0, ..., kr − 1
1. Define the polynomial
k
−1
r−1 X
r
X
i
f (x) =
x
ai,j g(x)j
i=0
j=0
2. Store the length−n vector (f (α) : α ∈ ∪i Ri )
Theorem: This is an optimal (n, k, r) LRC code over F
T. and Alexander Barg, “A Family of Optimal Locally Recoverable Codes”, IEEE Trans. on Information Theory
Theory and Practice of Codes with Locality
July 10, 2016
13 / 22
Optimal (n, k, r) LRC code construction - Cont’d
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Locality: Recover f (α) =? for α ∈ R1
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim:
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
Proof:
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
Proof: f (β)
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
P
i
Proof: f (β) = r−1
i=0 β fi (β)
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
P
Pr−1 i
i
Proof: f (β) = r−1
i=0 β fi (β) =
i=0 β fi (R1 )
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
P
Pr−1 i
i
Proof: f (β) = r−1
i=0 β fi (β) =
i=0 β fi (R1 ) = δ(β)
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
P
Pr−1 i
i
Proof: f (β) = r−1
i=0 β fi (β) =
i=0 β fi (R1 ) = δ(β)
1. deg(δ(x)) ≤ r − 1
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
P
Pr−1 i
i
Proof: f (β) = r−1
i=0 β fi (β) =
i=0 β fi (R1 ) = δ(β)
1. deg(δ(x)) ≤ r − 1
2. r points on δ(x) will suffice to recover δ(x)
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
Pr−1 i
j
j=0 ai,j g(x) =
i=0 x
i=0 x fi (x)
Locality: Recover f (α) =? for α ∈ R1
P kr −1
1. Define fi (x) = j=0
ai,j g(x)j
2. fi (x) is constant on the sets Rj
P
i
3. Define δ(x) = r−1
i=0 x fi (R1 )
Claim: f (β) = δ(β) for all β ∈ R1 (In particular f (α) = δ(α))
P
Pr−1 i
i
Proof: f (β) = r−1
i=0 β fi (β) =
i=0 β fi (R1 ) = δ(β)
1. deg(δ(x)) ≤ r − 1
2. r points on δ(x) will suffice to recover δ(x)
3. Read the r values {δ(β) = f (β) : β ∈ R1 \α}
Theory and Practice of Codes with Locality
July 10, 2016
14 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Minimum Distance:
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Minimum Distance:
1. Minimum distance = Minimum weight
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Minimum Distance:
1. Minimum distance = Minimum weight
2. deg(f (x)) ≤ r − 1 + (r + 1)( kr − 1)
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Minimum Distance:
1. Minimum distance = Minimum weight
2. deg(f (x)) ≤ r − 1 + (r + 1)( kr − 1) = k +
k
r
−2
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Minimum Distance:
1. Minimum distance = Minimum weight
2. deg(f (x)) ≤ r − 1 + (r + 1)( kr − 1) = k +
3. =⇒ d ≥ n − (k +
k
r
k
r
−2
− 2)
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Optimal (n, k, r) LRC code construction - Cont’d
• g(x) is constant on the sets Ri , |Ri | = r + 1
• f (x) =
Pr−1 i P kr −1
j
j=0 ai,j g(x)
i=0 x
Minimum Distance:
1. Minimum distance = Minimum weight
2. deg(f (x)) ≤ r − 1 + (r + 1)( kr − 1) = k +
3. =⇒ d ≥ n − (k +
k
r
k
r
−2
− 2)
4. Equality follows since the code is an (n, k, r) LRC code
Theory and Practice of Codes with Locality
July 10, 2016
15 / 22
Example: (9, 4, 2) LRC over F13
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1)
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
Theory and Practice of Codes with Locality
j=0
ai,j (x3 )j
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
Theory and Practice of Codes with Locality
j=0
ai,j (x3 )j = 1 + x + x3 + x4
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
= (4,8,7,1,11,2,0,0,0)
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
• deg 𝛿 𝑥
≤1
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
• deg 𝛿 𝑥
≤1
𝑓 3 =𝛿 3 =8
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
• deg 𝛿 𝑥
≤1
𝑓 3 =𝛿 3 =8
𝑓 9 =𝛿 9 =7
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
• deg 𝛿 𝑥
≤1
𝑓 3 =𝛿 3 =8
𝑓 9 =𝛿 9 =7
• 𝛿 𝑥 = 2𝑥 + 2
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
• deg 𝛿 𝑥
≤1
𝑓 3 =𝛿 3 =8
• 𝛿 𝑥 = 2𝑥 + 2
𝑓 1 =𝛿 1 =4
𝑓 9 =𝛿 9 =7
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Example: (9, 4, 2) LRC over F13
• R1 = {1, 3, 9}, R2 = {2, 6, 5}, R3 = {4, 12, 10}
• The polynomial g(x) = x3 is constant on each set Ri
• (a0,0 , a0,1 , a1,0 , a1,1 ) = (1, 1, 1, 1) → f (x) =
P1
i=0
xi
P1
j=0
ai,j (x3 )j = 1 + x + x3 + x4
• Store the evaluation of f (x) at ∪i Ri
𝑓 1 , 𝑓 3 , 𝑓 9 , 𝑓 2 , 𝑓 6 , 𝑓 5 , 𝑓 4 , 𝑓 12 , 𝑓 10
•
= (4,8,7,1,11,2,0,0,0)
𝑓 1 =?
Local Recovery:
• deg 𝛿 𝑥
≤1
𝑓 3 =𝛿 3 =8
• 𝛿 𝑥 = 2𝑥 + 2
𝑓 1 =𝛿 1 =4
𝑓 9 =𝛿 9 =7
Theory and Practice of Codes with Locality
July 10, 2016
16 / 22
Optimal (n, k, r) LRC code construction
Theory and Practice of Codes with Locality
July 10, 2016
17 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
Theory and Practice of Codes with Locality
July 10, 2016
17 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. A polynomial g(x) of degree r + 1
3. g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Theory and Practice of Codes with Locality
July 10, 2016
17 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. A polynomial g(x) of degree r + 1
3. g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Claim:
Theory and Practice of Codes with Locality
July 10, 2016
17 / 22
Optimal (n, k, r) LRC code construction
Ingredients:
n
1. R1 , ..., R r+1
are disjoint subsets of the field F, s.t. |Ri | = r + 1
2. A polynomial g(x) of degree r + 1
3. g(x) is constant on each subset Ri : g(α) = g(β) for α, β ∈ Ri
Claim: Let H be a subgroup of F∗ or F+ , then the annihilator polynomial of H
Y
g(x) =
(x − h)
h∈H
is constant on each coset of H
Theory and Practice of Codes with Locality
July 10, 2016
17 / 22
Availability problem
Theory and Practice of Codes with Locality
July 10, 2016
18 / 22
Availability problem
“Hot data” accessed simultaneously by a very large number of users
Theory and Practice of Codes with Locality
July 10, 2016
18 / 22
Availability problem
“Hot data” accessed simultaneously by a very large number of users
Theory and Practice of Codes with Locality
July 10, 2016
18 / 22
Availability problem - Cont’d
Theory and Practice of Codes with Locality
July 10, 2016
19 / 22
Availability problem - Cont’d
• Main advantage of replication - High availability for hot data
Theory and Practice of Codes with Locality
July 10, 2016
19 / 22
Availability problem - Cont’d
• Main advantage of replication - High availability for hot data
• Goal: A code with high availability and small overhead
Theory and Practice of Codes with Locality
July 10, 2016
19 / 22
Availability problem - Cont’d
• Main advantage of replication - High availability for hot data
• Goal: A code with high availability and small overhead
• Solution: LRC code with multiple disjoint recovery sets (codes with availability)
Theory and Practice of Codes with Locality
July 10, 2016
19 / 22
Idea behind constructing codes with multiple recovery sets
Theory and Practice of Codes with Locality
July 10, 2016
20 / 22
Idea behind constructing codes with multiple recovery sets
Subspace of
polynomials that
provide locality 𝑟1
42
Theory and Practice of Codes with Locality
July 10, 2016
20 / 22
Idea behind constructing codes with multiple recovery sets
Subspace of
polynomials that
provide locality 𝑟1
Ex: The space of polynomials {𝑎0 + 𝑎1 𝑥 + 𝑎3 𝑥 3 + 𝑎4 𝑥 4 : 𝑎𝑖 ∈ 𝐹13 }
43
Theory and Practice of Codes with Locality
July 10, 2016
20 / 22
Idea behind constructing codes with multiple recovery sets
Subspace of
polynomials that
provide locality 𝑟2
Subspace of
polynomials that
provide locality 𝑟1
Ex: The space of polynomials {𝑎0 + 𝑎1 𝑥 + 𝑎3 𝑥 3 + 𝑎4 𝑥 4 : 𝑎𝑖 ∈ 𝐹13 }
44
Theory and Practice of Codes with Locality
July 10, 2016
20 / 22
Idea behind constructing codes with multiple recovery sets
Subspace of
polynomials that
provide locality 𝑟2
Subspace of
polynomials that
provide locality 𝑟1
Ex: The space of polynomials {𝑎0 + 𝑎1 𝑥 + 𝑎3 𝑥 3 + 𝑎4 𝑥 4 : 𝑎𝑖 ∈ 𝐹13 }
45
Theory and Practice of Codes with Locality
July 10, 2016
20 / 22
LRC code with 2 recovery sets - Example
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
12,6, {2,3} 𝐿𝑅𝐶
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
12,6, {2,3} 𝐿𝑅𝐶
• Overhead of 200%
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
• Overhead of 200%
12,6, {2,3} 𝐿𝑅𝐶
• Overhead of 100%
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
• Overhead of 200%
• Can tolerate any 2 disk failures
12,6, {2,3} 𝐿𝑅𝐶
• Overhead of 100%
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
• Overhead of 200%
• Can tolerate any 2 disk failures
12,6, {2,3} 𝐿𝑅𝐶
• Overhead of 100%
• Can tolerate any 3 disk failures
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
• Overhead of 200%
• Can tolerate any 2 disk failures
• Recovery sets { , , }
12,6, {2,3} 𝐿𝑅𝐶
• Overhead of 100%
• Can tolerate any 3 disk failures
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
LRC code with 2 recovery sets - Example
• Example: (12, 6, {2, 3}) over F13
1. Encoding polynomial:
(a0 , a1 , a6 , a4 , a9 , a10 ) 7→ f (x) = a0 + a1 x + a4 x4 + a6 x6 + a9 x9 + a10 x10
2. Evaluated points: F∗13
•
18,6, 1,1
3𝑥 𝑅𝑒𝑝𝑙𝑖𝑐𝑎𝑡𝑖𝑜𝑛
• Overhead of 200%
• Can tolerate any 2 disk failures
• Recovery sets { , , }
12,6, {2,3} 𝐿𝑅𝐶
• Overhead of 100%
• Can tolerate any 3 disk failures
,
}
• Recovery sets { ,
Theory and Practice of Codes with Locality
July 10, 2016
21 / 22
Codes with locality: Plan
• LRC codes - Basics (Tamo)
• LRC codes - Constructions (Barg)
• LRC codes in practice (Barg)
• Break ,
• LRC codes - Bounds (Barg)
• Maximally Recoverable Codes (Tamo)
• LRC codes on graphs (Tamo)
Theory and Practice of Codes with Locality
July 10, 2016
22 / 22
Constructions of LRC codes
Constructions of LRC codes
1 / 25
From RS codes to other related families
In this part our goal is to construct new families of LRC codes
Constructions of LRC codes
2 / 25
From RS codes to other related families
In this part our goal is to construct new families of LRC codes
Advantages of the LRC RS construction
• small alphabet q « n
• large distance
• easy processing
Constructions of LRC codes
2 / 25
From RS codes to other related families
In this part our goal is to construct new families of LRC codes
Advantages of the LRC RS construction
• small alphabet q « n
• large distance
• easy processing
Limitations
• nďq
Constructions of LRC codes
2 / 25
From RS codes to other related families
In this part our goal is to construct new families of LRC codes
Advantages of the LRC RS construction
• small alphabet q « n
• large distance
• easy processing
Limitations
• nďq
Fixed q, large n?
Constructions of LRC codes
2 / 25
From RS codes to other related families
In this part our goal is to construct new families of LRC codes
Advantages of the LRC RS construction
• small alphabet q « n
• large distance
• easy processing
Limitations
• nďq
Fixed q, large n?
Possibilities: Reduce q or Increase n
Constructions of LRC codes
2 / 25
From RS codes to other related families
In this part our goal is to construct new families of LRC codes
Advantages of the LRC RS construction
• small alphabet q « n
• large distance
• easy processing
Limitations
• nďq
Fixed q, large n?
Possibilities: Reduce q or Increase n
• Take a subfield subcode of an LRC RS code (e.g., a binary code)
• The analysis is simplified if we take a cyclic LRC RS code
• Take a large subset of sampling points (points on an algebraic curve)
Constructions of LRC codes
2 / 25
Cyclic LRC codes
Cyclic codes form a classic topic in coding theory: BCH codes, RM codes, many other well-studied code
families are cyclic.
Cyclic q-ary LRC codes
Recall the LRC RS construction:
Data: subset of points P “ pP1 , . . . , Pn q Ă Fq ; linear space of polynomials
V “ xxi bpxqj y, dim V “ k, bpxq constant on pr ` 1q-subblocks of the set P;
VÑC
fa ÞÑ evP pfa q “ pfa pP1 q, . . . , fa pPn qq
Constructions of LRC codes
3 / 25
Cyclic LRC codes
Cyclic codes form a classic topic in coding theory: BCH codes, RM codes, many other well-studied code
families are cyclic.
Cyclic q-ary LRC codes
Recall the LRC RS construction:
Data: subset of points P “ pP1 , . . . , Pn q Ă Fq ; linear space of polynomials
V “ xxi bpxqj y, dim V “ k, bpxq constant on pr ` 1q-subblocks of the set P;
VÑC
fa ÞÑ evP pfa q “ pfa pP1 q, . . . , fa pPn qq
Additional assumptions: n|pq ´ 1q; pr ` 1q|n; r|k
P “ t1, α, . . . , α
n´1
B
u;
V“
k pr`1q´2
r
ÿ
ai xi , ai P Fq
F
(1)
i“0
i‰r modpr`1q
(α - nth root of unity)
Constructions of LRC codes
3 / 25
Cyclic LRC codes
Cyclic codes form a classic topic in coding theory: BCH codes, RM codes, many other well-studied code
families are cyclic.
Cyclic q-ary LRC codes
Recall the LRC RS construction:
Data: subset of points P “ pP1 , . . . , Pn q Ă Fq ; linear space of polynomials
V “ xxi bpxqj y, dim V “ k, bpxq constant on pr ` 1q-subblocks of the set P;
VÑC
fa ÞÑ evP pfa q “ pfa pP1 q, . . . , fa pPn qq
Additional assumptions: n|pq ´ 1q; pr ` 1q|n; r|k
P “ t1, α, . . . , α
n´1
B
u;
V“
k pr`1q´2
r
ÿ
ai xi , ai P Fq
F
(1)
i“0
i‰r modpr`1q
(α - nth root of unity)
We obtain a cyclic code C:
pc0 , c1 , c2 , . . . , cn´1 q P C
ñ
pcn´1 , c0 , c1 , . . . , cn´2 q P C
Constructions of LRC codes
3 / 25
Zeros of a cyclic LRC code
Let C be a q-ary cyclic code
pco , c1 , . . . , cn´1 q P C
Ñ
cpxq “
n´1
ÿ
ci xi
i“0
C is an ideal in the ring Fq rxs{px ´ 1q; C “ xgpxqy, where gpxq is the generator polynomial
n
of C
Definition: Zeros of the code C “ zeros of gpxq
Constructions of LRC codes
4 / 25
Zeros of a cyclic LRC code
Let C be a q-ary cyclic code
pco , c1 , . . . , cn´1 q P C
Ñ
cpxq “
n´1
ÿ
ci xi
i“0
C is an ideal in the ring Fq rxs{px ´ 1q; C “ xgpxqy, where gpxq is the generator polynomial
n
of C
Definition: Zeros of the code C “ zeros of gpxq
Generator matrix
¨
˚
˚
G“˚
˚
˝
Parity-check matrix
1 1
1
...
1
1 α
α2
...
αn´1
.
.
.
.
.
.
.
.
.
1 αk´1 α2pk´1q . . . αpk´1qpn´1q
˛
¨
‹
‹
‹
‹
‚
˚
˚
H“˚
˚
˝
1 α
α2
...
αn´1
1 α2
α2¨2 . . .
α2pn´1q
.
.
.
.
.
.
.
.
.
1 αn´k α2pn´kq . . . αpn´kqpn´1q
˛
‹
‹
‹
‹
‚
C is a cyclic code with zeros α, α2 , . . . , αn´k
Constructions of LRC codes
4 / 25
Cyclic codes: Example
• RS code C of length n “ 15, k “ 8, d “ 8, q “ 24
Zeros of C: α, α2 , α3 , α4 , α5 , α6 , α7
Generator polynomial gpxq “
ś7
i“1 px
´ αi q, dimpCq “ n ´ degpgq “ 8
BCH bound: dpCq ě number of consecutive 0’s ` 1
Constructions of LRC codes
5 / 25
Cyclic codes: Example
• RS code C of length n “ 15, k “ 8, d “ 8, q “ 24
Zeros of C: α, α2 , α3 , α4 , α5 , α6 , α7
Generator polynomial gpxq “
ś7
i“1 px
´ αi q, dimpCq “ n ´ degpgq “ 8
BCH bound: dpCq ě number of consecutive 0’s ` 1
• Now suppose that C has zeros
tα, α2 , α3 , α4 , α5 , α6 , α7 u Y tα, α4 , α7 , α10 , α13 u.
The distance is still d “ 8, and we get locality r “ 2
Constructions of LRC codes
5 / 25
Cyclic codes: Example
• RS code C of length n “ 15, k “ 8, d “ 8, q “ 24
Zeros of C: α, α2 , α3 , α4 , α5 , α6 , α7
Generator polynomial gpxq “
ś7
i“1 px
´ αi q, dimpCq “ n ´ degpgq “ 8
BCH bound: dpCq ě number of consecutive 0’s ` 1
• Now suppose that C has zeros
tα, α2 , α3 , α4 , α5 , α6 , α7 u Y tα, α4 , α7 , α10 , α13 u.
The distance is still d “ 8, and we get locality r “ 2
Indeed,
fa pxq “ a1 ` a2 x ` a3 x3 ` a4 x4 ` a5 x6 ` a6 x7
Constructions of LRC codes
5 / 25
Cyclic codes: Example
• RS code C of length n “ 15, k “ 8, d “ 8, q “ 24
Zeros of C: α, α2 , α3 , α4 , α5 , α6 , α7
Generator polynomial gpxq “
ś7
i“1 px
´ αi q, dimpCq “ n ´ degpgq “ 8
BCH bound: dpCq ě number of consecutive 0’s ` 1
• Now suppose that C has zeros
tα, α2 , α3 , α4 , α5 , α6 , α7 u Y tα, α4 , α7 , α10 , α13 u.
The distance is still d “ 8, and we get locality r “ 2
Indeed,
fa pxq “ a1 ` a2 x ` a3 x3 ` a4 x4 ` a5 x6 ` a6 x7
The rows of G are 1, α, α3 , α4 , α6 , α7
Constructions of LRC codes
5 / 25
Cyclic codes: Example
• RS code C of length n “ 15, k “ 8, d “ 8, q “ 24
Zeros of C: α, α2 , α3 , α4 , α5 , α6 , α7
Generator polynomial gpxq “
ś7
i“1 px
´ αi q, dimpCq “ n ´ degpgq “ 8
BCH bound: dpCq ě number of consecutive 0’s ` 1
• Now suppose that C has zeros
tα, α2 , α3 , α4 , α5 , α6 , α7 u Y tα, α4 , α7 , α10 , α13 u.
The distance is still d “ 8, and we get locality r “ 2
Indeed,
fa pxq “ a1 ` a2 x ` a3 x3 ` a4 x4 ` a5 x6 ` a6 x7
The rows of G are 1, α, α3 , α4 , α6 , α7
k “ 6; d “ 8 “ n ´ k
Constructions of LRC codes
r`1
`2
r
5 / 25
Cyclic LRC codes
• Consider a cyclic code of length n|pq ´ 1q given by evaluations evalpfa pxqq of the
k pr`1q´2
r
ř
polynomials of the form fa pxq “
ai xi , ai P Fq
i“0
i‰r modpr`1q
• The zeros of the code are a union of two (overlapping) subsets:
subset D gives the distance, |D| “ n ´ kr pr ` 1q ` 1
subset L supports locality, |LzD| “
k
r
´1
• The code is Singleton-optimal by the BCH bound
Constructions of LRC codes
6 / 25
Cyclic LRC codes
• Consider a cyclic code of length n|pq ´ 1q given by evaluations evalpfa pxqq of the
k pr`1q´2
r
ř
polynomials of the form fa pxq “
ai xi , ai P Fq
i“0
i‰r modpr`1q
• The zeros of the code are a union of two (overlapping) subsets:
subset D gives the distance, |D| “ n ´ kr pr ` 1q ` 1
subset L supports locality, |LzD| “
k
r
´1
• The code is Singleton-optimal by the BCH bound
The zeros of C are of the form αj ,
j P t1, 2, . . . , n ´ kr pr ` 1q ` 1u; tn ´ p kr ´ 1qpr ` 1q ` 1, n ´ p kr ´ 2qpr ` 1q ` 1, . . . , n ´ ru
Constructions of LRC codes
6 / 25
Main ideas I
Let 0 ď l ď r, ν “ n{pr ` 1q. Consider the ν ˆ n matrix (the zeros from L)
¨
1
˚1
˚
H1 “ ˚
˚ ..
˝.
1
α2ppr`1q`lq
...
...
αpn´1qpr`1q`l
αpn´1qppr`1q`l
..
.
α2ppν´1qpr`1q`lq
...
αpn´1qppν´1qpr`1q`lq
αl
α2l
αpr`1q`l
αpν´1qpr`1q`l
Constructions of LRC codes
˛
‹
‹
‹
‹
‚
7 / 25
Main ideas I
Let 0 ď l ď r, ν “ n{pr ` 1q. Consider the ν ˆ n matrix (the zeros from L)
¨
1
˚1
˚
H1 “ ˚
˚ ..
˝.
1
α2ppr`1q`lq
...
...
αpn´1qpr`1q`l
αpn´1qppr`1q`l
..
.
α2ppν´1qpr`1q`lq
...
αpn´1qppν´1qpr`1q`lq
αl
α2l
αpr`1q`l
αpν´1qpr`1q`l
˛
‹
‹
‹
‹
‚
The row space of H1 contains all the cyclic shifts of the n-dimensional vector of weight r ` 1
v “ p1 lo
0o.mo
. .o0n αlν lo
0o.mo
. .o0n α2lν lo
0o.mo
. .o0n . . . αrlν lo
0o.mo
. .o0nq, ν “ npr ` 1q
ν´1
ν´1
ν´1
Constructions of LRC codes
ν´1
7 / 25
Main ideas I
Let 0 ď l ď r, ν “ n{pr ` 1q. Consider the ν ˆ n matrix (the zeros from L)
¨
1
˚1
˚
H1 “ ˚
˚ ..
˝.
1
α2ppr`1q`lq
...
...
αpn´1qpr`1q`l
αpn´1qppr`1q`l
..
.
α2ppν´1qpr`1q`lq
...
αpn´1qppν´1qpr`1q`lq
αl
α2l
αpr`1q`l
αpν´1qpr`1q`l
˛
‹
‹
‹
‹
‚
The row space of H1 contains all the cyclic shifts of the n-dimensional vector of weight r ` 1
v “ p1 lo
0o.mo
. .o0n αlν lo
0o.mo
. .o0n α2lν lo
0o.mo
. .o0n . . . αrlν lo
0o.mo
. .o0nq, ν “ npr ` 1q
ν´1
ν´1
ν´1
ν´1
The code C has the parity-check matrix H “ H Y H1 , where
• H is formed by the rows in D
• H1 is formed by the rows in LzD
Constructions of LRC codes
7 / 25
Main ideas II
To obtain a code over a small field (e.g., Fp ) take a subfield subcode of the code C, i.e.,
D “ C X pFp qn
Constructions of LRC codes
8 / 25
Main ideas II
To obtain a code over a small field (e.g., Fp ) take a subfield subcode of the code C, i.e.,
D “ C X pFp qn
Analysis is based on Delsarte’s theorem: C P Fnq , q “ pm
C
Ş
ÐÑ
CK
pFp qn
D
Tm
ÐÑ
DK “ Tm pCK q
(Delsarte ’75; Sidelnikov ’72)
Trace Tm : Fpm Ñ Fp : Tm paq “ a ` ap ` ¨ ¨ ¨ ` ap
m´1
• Since D is cyclic, r “ dK ´ 1
• We would like small r, i.e., upper estimates of dK
• This is in contrast to classical coding where one seeks large dK
Constructions of LRC codes
8 / 25
Main ideas II
To obtain a code over a small field (e.g., Fp ) take a subfield subcode of the code C, i.e.,
D “ C X pFp qn
Analysis is based on Delsarte’s theorem: C P Fnq , q “ pm
C
Ş
CK
ÐÑ
pFp qn
D
Tm
ÐÑ
DK “ Tm pCK q
(Delsarte ’75; Sidelnikov ’72)
Trace Tm : Fpm Ñ Fp : Tm paq “ a ` ap ` ¨ ¨ ¨ ` ap
m´1
Further ideas involve
• Theory of irreducible (minimal) cyclic codes
• Results on upper bounds on the distance of a cyclic code in terms of its zeros
Constructions of LRC codes
8 / 25
A sampling of results
In a number of cases it is possible to estimate the locality and the number of recovery sets
of codes.
Let α be an nth root of unity, let p2t ´ 1q|n for some t.
Let D be an rn, ks binary linear cyclic code whose defining set of zeros contains the group
t
xα2 ´1 y. Then the locality of D satisfies
r ď 2t´1 ´ 1,
and each symbol has ě 2t´1 recovery sets.
For instance, we have the following binary LRC codes:
n
35
45
27
63
k
20
33
7
36
d
3
3
6
3
ZpDq
t1, 15u
t1u
t1, 9u
t1, 9, 11, 15, 23u
t
3
4
2
3
r
r
r
r
r
ď3
ď7
“1
ď3
Constructions of LRC codes
ZpDK q
t0, 1, 7, 15u
t0, 1, 3, 5, 9, 15, 21u
t0, 3u
t0, 1, 7, 9, 11, 15, 21, 23u
dK
4
8
2
4
9 / 25
Some open questions
• In several examples the bounds on locality (dual distance) give tight results.
Is it possible to characterize the cases in which the bounds are tight?
• Find the number of recovery sets per symbol for families of cyclic codes.
Constructions of LRC codes
10 / 25
Codes on curves
In this part we discuss another approach to constructing long codes over small
alphabets.
RS codes can be viewed as codes on the (affine) line, and can be extended to
codes on algebraic curves.
The constructions becomes more technical, and we proceed by example.
Constructions of LRC codes
11 / 25
LRC codes on curves
Consider the set of pairs px, yq P F9 that satisfy the equation x3 ` x “ y4
α7
‚
α6 ‚
α5
‚
α4 ‚
x α3 ‚
α2 ‚
α
‚
1
‚
0 ‚
0 1 α
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
α 2 α3 α4 α5 α6 α7
y
Constructions of LRC codes
12 / 25
LRC codes on curves
Consider the set of pairs px, yq P F9 that satisfy the equation x3 ` x “ y4
α7
‚
α6 ‚
α5
‚
α4 ‚
x α3 ‚
α2 ‚
α
‚
1
‚
0 ‚
0 1 α
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
α 2 α3 α4 α5 α6 α7
y
Affine points of the Hermitian curve X over F9 ; α2 “ α ` 1
Constructions of LRC codes
12 / 25
Hermitian codes
g: X
px, yq
Ñ
P1
ÞÑ
y
Space of functions V :“ x1, y, y2 , x, xy, xy2 y
A={Affine points of the Hermitian curve over F9 }; n “ 27, k “ 6
C : V Ñ Fn9
Constructions of LRC codes
13 / 25
Hermitian codes
g: X
px, yq
Ñ
P1
ÞÑ
y
Space of functions V :“ x1, y, y2 , x, xy, xy2 y
A={Affine points of the Hermitian curve over F9 }; n “ 27, k “ 6
C : V Ñ Fn9
E.g., message p1, α, α2 , α3 , α4 , α5 q
Fpx, yq “ 1 ` αy ` α2 y2 ` α3 x ` α4 xy ` α5 xy2
Fp0, 0q “ 1 etc.
Constructions of LRC codes
13 / 25
LRC codes on curves
α7
α
α7
α5
0
6
2
α α
α5
α6
α4
α2
0
4
7
3
5
α
α
α
α
α5
x α3
α3
α7
α
α
2
3
α α
α
0
0
0
0
1
1
α6
α4
0
0 1
0 1 α α2 α3 α4 α5 α6 α7
y
Constructions of LRC codes
14 / 25
Hermitian LRC codes
α7
α
α7
α5
0
6
2
α α
α5
α6
α4
α2
0
4
7
3
5
α
α
α
α
α5
x α3
α3
α7
α
α
2
3
α α
α
X
0
0
0
0
1
1
α6
α4
0
0 1
0 1 α α2 α3 α4 α5 α6 α7
y
Let P “ pα, 1q be the erased location.
Constructions of LRC codes
15 / 25
Local recovery with Hermitian codes
α7
α
α7
α5
0
6
2
α α
α5
α6
α4
α2
0
4
7
3
5
5
α
α
α
α
α
x α3
α3
α7
α
α
α2 α3
α
?
0
0
0
6
4
1
1
α
α
0
0 1
0 1 α α2 α3 α4 α5 α6 α7
y
Let P “ pα, 1q be the erased location. Recovery set IP “ tpα4 , 1q, pα3 , 1qu
Find f pxq : f pα4 q “ α7 , f pα3 q “ α3
ñ f pxq “ αx ´ α2
Constructions of LRC codes
16 / 25
Local recovery with Hermitian codes
α7
α
α7
α5
0
6
2
α α
α5
α6
α4
α2
0
4
7
3
5
5
α
α
α
α
α
x α3
α3
α7
α
α
α2 α3
α
?
0
0
0
6
4
1
1
α
α
0
0 1
0 1 α α2 α3 α4 α5 α6 α7
y
Let P “ pα, 1q be the erased location. Recovery set IP “ tpα4 , 1q, pα3 , 1qu
Find f pxq : f pα4 q “ α7 , f pα3 q “ α3
ñ f pxq “ αx ´ α2
f pαq “ 0 “ FpPq
Constructions of LRC codes
16 / 25
Local recovery with Hermitian codes
α7
α
α7
α5
0
6
2
α α
α5
α6
α4
α2
0
4
7
3
5
5
α
α
α
α
α
x α3
α3
α7
α
α
α2 α3
α
?
0
0
0
6
4
1
1
α
α
0
0 1
0 1 α α2 α3 α4 α5 α6 α7
y
Let P “ pα, 1q be the erased location. Recovery set IP “ tpα4 , 1q, pα3 , 1qu
Find f pxq : f pα4 q “ α7 , f pα3 q “ α3
ñ f pxq “ αx ´ α2
f pαq “ 0 “ FpPq
Computations can be done using GAP or Magma
Constructions of LRC codes
16 / 25
Hermitian codes
q “ q20 , q0 prime power
Constructions of LRC codes
17 / 25
Hermitian codes
q “ q20 , q0 prime power
X : xq0 ` x “ yq0 `1
Constructions of LRC codes
17 / 25
Hermitian codes
q “ q20 , q0 prime power
X : xq0 ` x “ yq0 `1
X has q30 “ q3{2 points in Fq
Constructions of LRC codes
17 / 25
Hermitian codes
q “ q20 , q0 prime power
X : xq0 ` x “ yq0 `1
X has q30 “ q3{2 points in Fq
We obtain a family of q-ary codes of length n “ q30 ,
k “ pt ` 1qpq0 ´ 1q, d ě n ´ tq0 ´ pq0 ´ 2qpq0 ` 1q
with locality r “ q0 ´ 1.
Constructions of LRC codes
17 / 25
Geometric view of the construction
We take g : X Ñ Y “
α7
‚
α6 ‚
α5
‚
α4 ‚
x α3 ‚
α2 ‚
α
‚
1
‚
0 ‚
0 1 α
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
P1 , gpPq “ gpx, yq :“ y
Projection on y
‚
α2 α3 α4 α5 α6 α7
y
Constructions of LRC codes
18 / 25
Geometric view of the construction
We take g : X Ñ Y “
α7
‚
α6 ‚
α5
‚
α4 ‚
x α3 ‚
α2 ‚
α
‚
1
‚
0 ‚
0 1 α
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
P1 , gpPq “ gpx, yq :“ y
‚
Projection on y
‚
α2 α3 α4 α5 α6 α7
y
Space of functions V :“ x1, y, y2 , x, xy, xy2 y
Since y is constant on the fibers (recovery sets), we get back to the univariate polynomial
interpolation
Constructions of LRC codes
18 / 25
Geometric view of the construction
We take g : X Ñ Y “
α7
‚
α6 ‚
α5
‚
α4 ‚
x α3 ‚
α2 ‚
α
‚
1
‚
0 ‚
0 1 α
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
‚
P1 , gpPq “ gpx, yq :“ y
‚
Projection on y
‚
α2 α3 α4 α5 α6 α7
y
Space of functions V :“ x1, y, y2 , x, xy, xy2 y
Since y is constant on the fibers (recovery sets), we get back to the univariate polynomial
interpolation
It is also possible to take gpPq “ x (projection on x); we obtain LRC codes with locality q0
Constructions of LRC codes
18 / 25
General construction: Covering maps of curves
In the RS-like construction, X “ Y “ P1
Constructions of LRC codes
19 / 25
General construction: Technical details
Map of curves
X, Y smooth projective absolutely irreducible curves over
k
g:XÑY
rational separable map of degree r ` 1
Constructions of LRC codes
20 / 25
General construction: Technical details
Map of curves
X, Y smooth projective absolutely irreducible curves over
k
g:XÑY
rational separable map of degree r ` 1
k
Lift the points of Y
S “ tP1 , . . . , Ps u Ă Yp q. Partition of points:
k
A :“ g´1 pSq “ tPij , i “ 0, . . . , r, j “ 1, . . . , su Ď Xp q
such that gpPij q “ Pj for all i, j
Constructions of LRC codes
20 / 25
General construction: Technical details
Map of curves
X, Y smooth projective absolutely irreducible curves over
k
g:XÑY
rational separable map of degree r ` 1
Lift the points of Y
k
S “ tP1 , . . . , Ps u Ă Yp q. Partition of points:
k
A :“ g´1 pSq “ tPij , i “ 0, . . . , r, j “ 1, . . . , su Ď Xp q
such that gpPij q “ Pj for all i, j
k
k
k
Let x P pXq be such that pXq “ pYqpxq, and let deg x “ h as a projection x : X Ñ P1k
Constructions of LRC codes
20 / 25
General construction, II
Let Q8 Ă π
´1
p8q, deg Q8 “ ℓ ě 1
Let LpQ8 q “ xf1 , . . . , fm y, m ě ℓ ´ gY ` 1
Function space
A
E
V :“ fj xi , i “ 0, . . . , r ´ 1; j “ 1, . . . , m
Constructions of LRC codes
21 / 25
General construction, II
Let Q8 Ă π
´1
p8q, deg Q8 “ ℓ ě 1
Let LpQ8 q “ xf1 , . . . , fm y, m ě ℓ ´ gY ` 1
Function space
A
E
V :“ fj xi , i “ 0, . . . , r ´ 1; j “ 1, . . . , m
The code C is an image of the map
e :“ evA :V ÝÑ
kpr`1qs
F ÞÑ pFpPij q, i “ 0, . . . , r, j “ 1, . . . , sq
Constructions of LRC codes
21 / 25
General construction, II
Let Q8 Ă π
´1
p8q, deg Q8 “ ℓ ě 1
Let LpQ8 q “ xf1 , . . . , fm y, m ě ℓ ´ gY ` 1
Function space
A
E
V :“ fj xi , i “ 0, . . . , r ´ 1; j “ 1, . . . , m
The code C is an image of the map
e :“ evA :V ÝÑ
kpr`1qs
F ÞÑ pFpPij q, i “ 0, . . . , r, j “ 1, . . . , sq
Theorem: The subspace CpD, gq Ă Fq forms an pn, k, rq linear LRC code with the
parameters
n “ pr ` 1qs
k “ rm ě rpℓ ´ gY ` 1q
d ě n ´ ℓpr ` 1q ´ pr ´ 1qh
,
/
/
.
/
/
-
provided that the right-hand side of the inequality for d is a positive integer.
Constructions of LRC codes
21 / 25
Asymptotically good sequences of codes
In classic coding problem, codes on algebraic curves give rise to some excellent code
families, in particular, improving the asymptotic Gilbert-Varshamov bound on the
parameters
(Tsfasman-Vlădutş-Zink ’81)
Constructions of LRC codes
22 / 25
Asymptotically good sequences of codes
Let q “ q20 , where q0 is a prime power. Take Garcia-Stichtenoth towers of curves:
k
k
k
x0 :“ 1; X1 :“ P1 , pX1 q “ px1 q;
zl´1
q
q0 `1
Xl : zl 0 ` zl “ xl´1
, xl´1 :“
P pXl´1 q (if l ě 3q
xl´2
There exist families of q-ary LRC codes with locality r whose rate and relative distance
satisfy
3 ¯
r ´
1´δ´ ?
,
r`1
q`1
?
2 q¯
r ´
1´δ´
,
Rě
r`1
q´1
Rě
Constructions of LRC codes
r“
r“
?
?
q´1
q
22 / 25
Asymptotically good sequences of codes
Let q “ q20 , where q0 is a prime power. Take Garcia-Stichtenoth towers of curves:
k
k
k
x0 :“ 1; X1 :“ P1 , pX1 q “ px1 q;
zl´1
q
q0 `1
Xl : zl 0 ` zl “ xl´1
, xl´1 :“
P pXl´1 q (if l ě 3q
xl´2
There exist families of q-ary LRC codes with locality r whose rate and relative distance
satisfy
3 ¯
r ´
1´δ´ ?
,
r`1
q`1
?
2 q¯
r ´
1´δ´
,
Rě
r`1
q´1
Rě
˚q
r“
r“
?
?
q´1
q
Recall the TVZ ’81 bound without locality: R ě 1 ´ δ ´
Constructions of LRC codes
1
?
q´1
22 / 25
LRC codes on curves better than the GV bound
Constructions of LRC codes
23 / 25
LRC codes on curves better than the GV bound
The asymptotic GV bound can be improved for any given (constant) r for all q greater than
some value.
Constructions of LRC codes
23 / 25
Extensions and open questions
Extensions
• LRC codes that correct locally multiple erasures
• LRC codes on curves with availability
• Adjusting the locality value r
Open questions
• Constructions of LRC codes from known families of good curves
• Parameters of subfield subcodes of AG LRC codes
• Automorphism groups of curves and codes with availability
Constructions of LRC codes
24 / 25
References
S. Goparaju and R. Calderbank, Binary cyclic codes that are locally repairable, Proc. 2014 IEEE Int. Sympos.
Inform. Theory, Honolulu, HI, pp. 676–680.
A. Barg, I. Tamo, and S. Vlăduţ, Locally recoverable codes on algebraic curves, March 2016 (Preprint:
arXiv:1603.08876).
I. Tamo, A. Barg, S. Goparaju, and R. Calderbank, Cyclic LRC codes, binary LRC codes, and upper bounds on
the distance of cyclic codes, International Journal of Information and Coding Theory, to appear (Preprint:
arXiv:1603.08878).
A. Barg and I. Tamo, On codes with the locality constraint, IEEE IT Society Newsletter 65, no. 4 (2015),
pp. 7–15.
Constructions of LRC codes
25 / 25
Erasure (and LRC) codes in practice
A brief overview
LRC codes in practice
1 / 14
Before LRC codes
Replication: Google (x3, x27), FB (x3), HDFS, others
LRC codes in practice
2 / 14
Before LRC codes
Replication: Google (x3, x27), FB (x3), HDFS, others
RS codes, e.g., [6,4] RS codes: RAID; used by FB, others
LRC codes in practice
2 / 14
Before LRC codes
Replication: Google (x3, x27), FB (x3), HDFS, others
RS codes, e.g., [6,4] RS codes: RAID; used by FB, others
Array-type codes: RAID 6 (2 parities per block); IBM (EVENODD, X-Code)
LRC codes in practice
2 / 14
Before LRC codes
Replication: Google (x3, x27), FB (x3), HDFS, others
RS codes, e.g., [6,4] RS codes: RAID; used by FB, others
Array-type codes: RAID 6 (2 parities per block); IBM (EVENODD, X-Code)
“There are two kinds of erasure codes implemented in Raid: XOR code and Reed-Solomon code. [...]
The replication on the source file can be reduced to 1 when using Reed-Solomon without losing data safety. The
downside of having only one replica of a block is that reads of a block have to go to a single machine, reducing
parallelism. Thus Reed-Solomon should be used on data that is not supposed to be used frequently.”
http://wiki.apache.org/hadoop/HDFS-RAID
LRC codes in practice
2 / 14
Applications of erasure codes
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
• HDFS-RAID, including HDFS Xorbas (FB)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
• HDFS-RAID, including HDFS Xorbas (FB)
• Microsoft Azure storage (VLDB Endowment, 2013)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
• HDFS-RAID, including HDFS Xorbas (FB)
• Microsoft Azure storage (VLDB Endowment, 2013)
• HACFS, adaptive coding for HDFS (IBM) (FAST ’15)
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
• HDFS-RAID, including HDFS Xorbas (FB)
• Microsoft Azure storage (VLDB Endowment, 2013)
• HACFS, adaptive coding for HDFS (IBM) (FAST ’15)
• Glacier (NSDI ’05), Petal (ASPLOS ’96), Weaver (FAST ’05), Stair codes (FAST ’14),
CORE (MSST 2013), ...
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
• HDFS-RAID, including HDFS Xorbas (FB)
• Microsoft Azure storage (VLDB Endowment, 2013)
• HACFS, adaptive coding for HDFS (IBM) (FAST ’15)
• Glacier (NSDI ’05), Petal (ASPLOS ’96), Weaver (FAST ’05), Stair codes (FAST ’14),
CORE (MSST 2013), ...
• Proposals based on regenerating codes
LRC codes in practice
3 / 14
Applications of erasure codes
Systems using existing solutions:
• Amazon Web Services (AWS) + Glacier archiving service (previously S3); on the
market
• Dropbox (moved its storage from AWS to own system)
• Google Colossus distributed FS ([9,6] RS codes; OSDI2010)
• Facebook F4 ([14,10] RS code; OSDI 2014)
• Ceph storage platform (ceph.com)
Innovative proposals, erasure codes with some form of locality:
• HDFS-RAID, including HDFS Xorbas (FB)
• Microsoft Azure storage (VLDB Endowment, 2013)
• HACFS, adaptive coding for HDFS (IBM) (FAST ’15)
• Glacier (NSDI ’05), Petal (ASPLOS ’96), Weaver (FAST ’05), Stair codes (FAST ’14),
CORE (MSST 2013), ...
• Proposals based on regenerating codes
(Many more)
LRC codes in practice
3 / 14
Optimization of coding for storage
[Xia et al., HACFS, USENIX FAST’15]
LRC codes in practice
4 / 14
Erasure coding in Ceph
Ceph: Object storage based free
software storage platform for storing
on a single cluster
Erasure coding:
uses a [5,3] MDS code
http://docs.ceph.com
LRC codes in practice
5 / 14
Facebook’s storage system
The FB system stores files as redundant pieces of data distributed across different drives
and data centers.
The system distinguishes between cold, warm, and hot data.
Hot data (Haystack): The data is replicated 3 times, two copies in one center on different
racks, and the third copy in a different center.
LRC codes in practice
6 / 14
Facebook’s storage system
Warm data are written once, accessed many times, and can be deleted but not modified
(e.g., 400B photos).
LRC codes in practice
7 / 14
Facebook’s storage system
Warm data are written once, accessed many times, and can be deleted but not modified
(e.g., 400B photos).
Facebook’s F4 “Warm” BLOB storage system relies on r14, 10s RS codes. The stripe of 14
blocks is placed on 14 different nodes, different racks.
(S. Muralidhar e.a., OSDI ’14)
LRC codes in practice
7 / 14
Facebook’s storage system
Warm data are written once, accessed many times, and can be deleted but not modified
(e.g., 400B photos).
Facebook’s F4 “Warm” BLOB storage system relies on r14, 10s RS codes. The stripe of 14
blocks is placed on 14 different nodes, different racks.
(S. Muralidhar e.a., OSDI ’14)
Main issues: Reliability and load analysis based on the models of failures in the system,
and storage/system architecture
Optimizing storage, input/output, and bandwidth (the amount of data exchanges in the
system).
LRC codes in practice
7 / 14
Data placement
http://www.slideshare.net/ydn/hdfs-raid-facebook
LRC codes in practice
8 / 14
HDFS RAID + LRC (HDFS-XORbas)
LRC codes for Hadoop file system
(M. Sathiamoorthy et al., 2013 VLDB Endowment)
[14,10] RS code with two additional local parities:
X1 , X2 , X3 , X4 , X5
Data
where S1 “
ř5
i“1
ai Xi , S2 “
S1 X6 , X7 , X8 , X9 , X10
S2
Data
ř5
i“1
P1 , P2 , P3 , P4
RS parities
bi Xi`5 are the local parities
LRC codes in practice
9 / 14
HDFS RAID + LRC (HDFS-XORbas)
LRC codes for Hadoop file system
(M. Sathiamoorthy et al., 2013 VLDB Endowment)
[14,10] RS code with two additional local parities:
X1 , X2 , X3 , X4 , X5
Data
where S1 “
ř5
i“1
ai Xi , S2 “
S1 X6 , X7 , X8 , X9 , X10
S2
Data
ř5
i“1
P1 , P2 , P3 , P4
RS parities
bi Xi`5 are the local parities
Reliability analysis in terms of mean time to data loss using a Markov failure model
LRC codes in practice
9 / 14
Microsoft Azure storage
(previously Windows Azure storage)
P1
F16
X1
X2
X3
X4
X5
X6
P2
S1
S2
S1 “ α1,1 X1 ` α1,2 X2 ` α1,3 X3 , S2 “ α2,1 X4 ` α2,2 X5 ` α2,3 X6
LRC codes in practice
10 / 14
Microsoft Azure storage
(previously Windows Azure storage)
P1
F16
X1
X2
X3
X4
X5
X6
P2
S2
S1
S1 “ α1,1 X1 ` α1,2 X2 ` α1,3 X3 , S2 “ α2,1 X4 ` α2,2 X5 ` α2,3 X6
Choose a basis of C so that every 4 erasures other than
tX1 , X2 , X3 , S1 u,
tX4 , X5 , X6 , S2 u
are correctable (Maximally Recoverable Codes, refer to Part 5 “MR codes”).
• C. Huang et al., USENIX 2012: (6,2,2) code, (12,2,2) code (1.33x overhead)
• Installed in Windows 8.1 and Windows Server 2012
LRC codes in practice
10 / 14
Adaptive coding for HDFS
Implements 2 code families:
• 6 x 5 Product codes
• LRC (12,2,2) codes
12 data blocks; 2 local parities, 2
global parities
Optimizing overhead or recovery time
[Xia et al., HACFS, USENIX FAST’15]
LRC codes in practice
11 / 14
Maximally recoverable codes for SSD arrays
RAID 5 or RAID 6 type architecture (r “ 1 to 3 parities per row)
global parities
LRC codes in practice
12 / 14
Maximally recoverable codes for SSD arrays
RAID 5 or RAID 6 type architecture (r “ 1 or 2 parities per row)
Mixed failure model: device failure plus several sector failures
LRC codes in practice
13 / 14
Maximally recoverable codes for SSD arrays
RAID 5 or RAID 6 type architecture (r “ 1 or 2 parities per row)
Mixed failure model: device failure plus several sector failures
Algebraic construction of MR codes similar to rank error codes (Blaum et al., 2013)
LRC codes in practice
13 / 14
A selection of references
C. Huang et al., Erasure coding in Windows Azure storage, Usenix ATC, 2012
M. Blaum et al., Partial MDS codes and applcations in RAID, IT Trans. 2013
˚
f4: Facebook’s Warm BLOB Storage system, 11th USENIX OSDI, 2014, 383–398
˚
M. Xia et al., A tale of two erasure codes in HDFS, 13th USENIX FAST, 2015, 213–226
A. Dimakis, LRC codes and index coding, Talk at DIMACS workshop, Dec. 2015
J. Plank and C. Huang, Tutorial on erasure codes, USENIX FAST 2013
XORing elephants: Erasure codes for Big Data, VLDB Endowment 2013, vol.6, 325–336
K.V. Rashmi et al., Erasure-coded distributed storage systems (FB study), USENIX 2013
˚
˚
K.V. Rashmi et al., Jointly optimal erasure codes for I/O, storage & bandwidth, FAST 2015
Includes overview of other industry-related solutions or proposals
LRC codes in practice
14 / 14
Bounds on LRC codes
Bounds on LRC codes
1 / 22
Problem
C Ă Fnq – LRC code of length n, cardinality qk , distance d, locality r
Def: For any i P rns there exists a recovery (helper) set Ri Ă rnsztiu, |Ri | ď r and a function
ϕi such that for any x P C
xi “ ϕi pxj , j P Ri q
(We do not assume that Ri X Rj “ H, i ‰ j )
Bounds on LRC codes
2 / 22
Problem
C Ă Fnq – LRC code of length n, cardinality qk , distance d, locality r
Def: For any i P rns there exists a recovery (helper) set Ri Ă rnsztiu, |Ri | ď r and a function
ϕi such that for any x P C
xi “ ϕi pxj , j P Ri q
(We do not assume that Ri X Rj “ H, i ‰ j )
Given k define dpn, k, r, qq to be the largest possible distance of C
Bounds on LRC codes
2 / 22
Problem
C Ă Fnq – LRC code of length n, cardinality qk , distance d, locality r
Def: For any i P rns there exists a recovery (helper) set Ri Ă rnsztiu, |Ri | ď r and a function
ϕi such that for any x P C
xi “ ϕi pxj , j P Ri q
(We do not assume that Ri X Rj “ H, i ‰ j )
Given k define dpn, k, r, qq to be the largest possible distance of C
Given d define kpn, d, r, qq to be the largest possible dimension of C
(log-cardinality)
Bounds on LRC codes
2 / 22
Problem
C Ă Fnq – LRC code of length n, cardinality qk , distance d, locality r
Def: For any i P rns there exists a recovery (helper) set Ri Ă rnsztiu, |Ri | ď r and a function
ϕi such that for any x P C
xi “ ϕi pxj , j P Ri q
(We do not assume that Ri X Rj “ H, i ‰ j )
Given k define dpn, k, r, qq to be the largest possible distance of C
Given d define kpn, d, r, qq to be the largest possible dimension of C
(log-cardinality)
Bounds on d, k ?
Bounds on LRC codes
2 / 22
First results
k
r
ď
n
r`1
d ďn´k´
QkU
r
`2
(Gopalan e.a.. IEEE Trans. IT, 2013)
Bounds on LRC codes
3 / 22
First results
k
r
ď
n
r`1
d ďn´k´
QkU
r
`2
(Gopalan e.a.. IEEE Trans. IT, 2013)
The bound on the rate was proved in the first part of the tutorial.
Let us prove the bound on the distance.
Bounds on LRC codes
3 / 22
The Generalized Singleton bound
Main idea. Let C be a q-ary code of length n, size qk . The distance dpCq satisfies
@SĂrns:|CS |ăqk dpCq ď n ´ |S|
pAq
Construct a set S:
Let m “ t k´1
u. Put T :“ H, L :“ H.
r
Bounds on LRC codes
4 / 22
The Generalized Singleton bound
Main idea. Let C be a q-ary code of length n, size qk . The distance dpCq satisfies
@SĂrns:|CS |ăqk dpCq ď n ´ |S|
pAq
Construct a set S:
Let m “ t k´1
u. Put T :“ H, L :“ H.
r
For j “ 1, 2, . . . , m: Take ij P pT Y Lqc , Put T Ð T Y Rij , L Ð L Y tij u (Ri is the recovery set for ij ).
Bounds on LRC codes
4 / 22
The Generalized Singleton bound
Main idea. Let C be a q-ary code of length n, size qk . The distance dpCq satisfies
@SĂrns:|CS |ăqk dpCq ď n ´ |S|
pAq
Construct a set S:
Let m “ t k´1
u. Put T :“ H, L :“ H.
r
For j “ 1, 2, . . . , m: Take ij P pT Y Lqc , Put T Ð T Y Rij , L Ð L Y tij u (Ri is the recovery set for ij ).
Clearly |T| ď k ´ 1, so |CT | ď qk´1 , and |CTYL | ď qk´1
Bounds on LRC codes
4 / 22
The Generalized Singleton bound
Main idea. Let C be a q-ary code of length n, size qk . The distance dpCq satisfies
@SĂrns:|CS |ăqk dpCq ď n ´ |S|
pAq
Construct a set S:
Let m “ t k´1
u. Put T :“ H, L :“ H.
r
For j “ 1, 2, . . . , m: Take ij P pT Y Lqc , Put T Ð T Y Rij , L Ð L Y tij u (Ri is the recovery set for ij ).
Clearly |T| ď k ´ 1, so |CT | ď qk´1 , and |CTYL | ď qk´1
Suppose that |T| “ k ´ 1 (if needed, add some coordinates from rnszT Y L) and put S “ T Y L.
Bounds on LRC codes
4 / 22
The Generalized Singleton bound
Main idea. Let C be a q-ary code of length n, size qk . The distance dpCq satisfies
@SĂrns:|CS |ăqk dpCq ď n ´ |S|
pAq
Construct a set S:
Let m “ t k´1
u. Put T :“ H, L :“ H.
r
For j “ 1, 2, . . . , m: Take ij P pT Y Lqc , Put T Ð T Y Rij , L Ð L Y tij u (Ri is the recovery set for ij ).
Clearly |T| ď k ´ 1, so |CT | ď qk´1 , and |CTYL | ď qk´1
Suppose that |T| “ k ´ 1 (if needed, add some coordinates from rnszT Y L) and put S “ T Y L.
We have |S| “ k ´ 1 ` m “ k ` r kr s ´ 2 and |CS | ă qk . Conclude by pAq.
Bounds on LRC codes
4 / 22
Shortening (alphabet-dependent) bound
Theorem (Cadambe-Mazumdar ’13)
Let kq pm, dq be the maximum dimension of a q-ary code of length m and distance d. Then
kpn, d, r, qq ď min tsr ` kq pn ´ spr ` 1q, dqu.
sě1
Proof.
Let C be a linear LRC code with distance d (the linearity assumption is easily lifted).
• Following the proof of the LRC Singleton bound, construct a subset I Ă rns such that
|I| “ spr ` 1q, dim projI pCq ď sr.
• Consider the shortening CI of C by the coordinates in I. We obtain
length pCI q “ n ´ spr ` 1q,
dpCI q “ d
I
dimpC q “ k ´ dim projI pCq ě k ´ sr
kq pn ´ spr ` 1q, dq ě k ´ sr
Bounds on LRC codes
5 / 22
Asymptotic problem
Bounds on codes are often considered in the asymptotics of n Ñ 8
Let Mpn, r, δnq be the max size of a code of length n, distance d, locality r
Rpr, δq :“ lim sup
nÑ8
1
log Mpn, r, δnq
n
Bounds on LRC codes
6 / 22
Gilbert-Varshamov bound
For binary codes,
Rpr, δq ě 1 ´ min
0ăsď1
! 1
)
log2 pp1 ` sqr`1 ` p1 ´ sqr`1 q ´ δ log2 s .
r`1
Bounds on LRC codes
7 / 22
Gilbert-Varshamov bound
For binary codes,
Rpr, δq ě 1 ´ min
0ăsď1
! 1
)
log2 pp1 ` sqr`1 ` p1 ´ sqr`1 q ´ δ log2 s .
r`1
Proof by random coding: Estimate the average weight enumerator for the ensemble given
by
»
—
—
—
—
—
H“—
—
—
—
–
fi
11 . . . 1
11 . . . 1
..
.
11 . . . 1
HL
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
Bounds on LRC codes
7 / 22
Gilbert-Varshamov bound
For binary codes,
Rpr, δq ě 1 ´ min
0ăsď1
! 1
)
log2 pp1 ` sqr`1 ` p1 ´ sqr`1 q ´ δ log2 s .
r`1
Proof by random coding: Estimate the average weight enumerator for the ensemble given
by
»
—
—
—
—
—
H“—
—
—
—
–
fi
11 . . . 1
11 . . . 1
..
.
11 . . . 1
HL
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
PrptdpCq ă δnuq
n
r
ď δnq´np r`1 ´Rq min
Bounds on LRC codes
0ăsď1
bpsq r`1
sδn
7 / 22
Gilbert-Varshamov bound
For binary codes,
Rpr, δq ě 1 ´ min
0ăsď1
! 1
)
log2 pp1 ` sqr`1 ` p1 ´ sqr`1 q ´ δ log2 s .
r`1
Proof by random coding: Estimate the average weight enumerator for the ensemble given
by
»
—
—
—
—
—
H“—
—
—
—
–
fi
11 . . . 1
11 . . . 1
..
.
11 . . . 1
HL
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
PrptdpCq ă δnuq
n
r
ď δnq´np r`1 ´Rq min
0ăsď1
bpsq “
bpsq r`1
sδn
1
pp1 ` pq ´ 1qsqr`1
q
` pq ´ 1qp1 ´ sqr`1 q
Bounds on LRC codes
7 / 22
Gilbert-Varshamov bound
For binary codes,
Rpr, δq ě 1 ´ min
0ăsď1
! 1
)
log2 pp1 ` sqr`1 ` p1 ´ sqr`1 q ´ δ log2 s .
r`1
Proof by random coding: Estimate the average weight enumerator for the ensemble given
by
»
—
—
—
—
—
H“—
—
—
—
–
fi
11 . . . 1
11 . . . 1
..
.
11 . . . 1
HL
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
PrptdpCq ă δnuq
n
r
ď δnq´np r`1 ´Rq min
0ăsď1
bpsq “
bpsq r`1
sδn
1
pp1 ` pq ´ 1qsqr`1
q
` pq ´ 1qp1 ´ sqr`1 q
(Cadambe-Mazumdar ’15; Tamo-B.-Frolov ’15)
Bounds on LRC codes
7 / 22
Asymptotic upper bounds
Let
Rq pr, δq “ lim sup
nÑ8
1
logq kpn, r, δnq
n
Using the Cadambe-Mazumdar bound together with classic bounds on kq pm, dq (with no
locality)
r
p1 ´ δq, 0 ď δ ď 1
r`1
q ¯
r ´
1´δ
, 0 ď δ ď q{pq ´ 1q
Rq pr, δq ď
r`1
q´1
!
´
¯)
δ
Rq pr, δq ď min τ r ` p1 ´ τ pr ` 1qqfq
1
1 ´ τ pr ` 1q
0ďτ ď r`1
Rq pr, δq ď
where
fq pxq :“ hq p 1q pq ´ 1 ´ xpq ´ 2q ´ 2
b
pq ´ 1qxp1 ´ xqqq,
hq pxq :“ ´x logq px{pq ´ 1qq ´ p1 ´ xq logq p1 ´ xq.
Bounds on LRC codes
8 / 22
Asymptotic bounds
R
0.7
0.6
Si
ng
le
0.5
ton
Plo
n
tki
0.4
bo
un
d
un
bo
0.3
d
0.2
0.1
LP bound
GV bound
0.2
0.4
0.6
0.8
1.0
∆
Binary codes; r “ 3
Bounds on LRC codes
9 / 22
Improving GV bound using LRC codes on curves
(B-Tamo-Vlăduţ, ’15)
(details in part on algebraic constructions)
Bounds on LRC codes
10 / 22
Correcting more than one erasure
Our goal here is to correct locally ě 2 erasures.
Bounds on LRC codes
11 / 22
Correcting more than one erasure
Our goal here is to correct locally ě 2 erasures.
Def. (correcting ρ ´ 1 erasure): C is an pn, k, r, ρq LRC code if each i P rns is contained in a
subset Ii , |Ii | ď r ` ρ ´ 1 such that the projection of C on Ii forms a code with distance ě ρ.
Bounds on LRC codes
11 / 22
Correcting more than one erasure
Our goal here is to correct locally ě 2 erasures.
Def. (correcting ρ ´ 1 erasure): C is an pn, k, r, ρq LRC code if each i P rns is contained in a
subset Ii , |Ii | ď r ` ρ ´ 1 such that the projection of C on Ii forms a code with distance ě ρ.
Singleton-type bound:
d ďn´k`1´
´Q k U
r
¯
´ 1 pρ ´ 1q (Prakash e.a., ISIT ’12)
Bounds on LRC codes
11 / 22
Correcting more than one erasure
Our goal here is to correct locally ě 2 erasures.
Def. (correcting ρ ´ 1 erasure): C is an pn, k, r, ρq LRC code if each i P rns is contained in a
subset Ii , |Ii | ď r ` ρ ´ 1 such that the projection of C on Ii forms a code with distance ě ρ.
Singleton-type bound:
d ďn´k`1´
´Q k U
r
¯
´ 1 pρ ´ 1q (Prakash e.a., ISIT ’12)
GV-type bound:
• Proof by random coding, similar to the LRC GV bound
• For large q the GV bound can be improved using codes on curves (refer to the part
“Algebraic constructions”)
(A.B., I. Tamo, and S. Vlăduţ, ’16).
Bounds on LRC codes
11 / 22
Correcting more than one erasure - recursive bounds
Theorem (Hu, Tamo, B., ISIT’16)
Let Bpn, ρq be an upper bound on the size of a code of length n and distance ρ. If the
function B is log-convex in n, then for any pn, k, r, ρq LRC code,
kď
Q n ´ pd ´ 1q U
r`ρ´1
logq Bpr ` ρ ´ 1, ρq
E.g., locality-dependent Plotkin bound: Let ρ ą pr ` ρ ´ 1q q´1
, then
q
kď
Q n ´ pd ´ 1q U
r`ρ´1
logq
ρ
ρ´
q´1
pr
q
` ρ ´ 1q
It is also possible to derive a locality-dependent Hamming bound.
Bounds on LRC codes
12 / 22
Linear programming bounds
In classical coding theory, all (or most) known upper bounds on the size of codes with a
given distance can be derived via linear programming (Delsarte ’72). In this part we extend
this approach to codes with locality.
Steps:
• Construct an association scheme for codes with locality
• Derive positivity conditions for the distance distribution (Delsarte inequalities)
• Find closed-form expressions for upper bounds
Bounds on LRC codes
13 / 22
Linear programming bounds
C an pn, k, r, ρq LRC code, distance d.
Bounds on LRC codes
14 / 22
Linear programming bounds
C an pn, k, r, ρq LRC code, distance d.
Assume disjoint recovery groups of size r ` ρ ´ 1
Bounds on LRC codes
14 / 22
Linear programming bounds
C an pn, k, r, ρq LRC code, distance d.
Assume disjoint recovery groups of size r ` ρ ´ 1
Product association scheme Hpst, qq “ pHpt, qqqbs , s ě 2
Let C be a q-ary pn, k, r, ρq LRC code with distance d. Define
␣
T :“ i “ pi1 , . . . , is q | i1 ` . . . ` is ě d,
(
ip P t0, ρ, ρ ` 1, . . . , ρ ` r ´ 1u for all p “ 1, . . . , s. .
Bounds on LRC codes
14 / 22
Linear programming bounds
C an pn, k, r, ρq LRC code, distance d.
Assume disjoint recovery groups of size r ` ρ ´ 1
Product association scheme Hpst, qq “ pHpt, qqqbs , s ě 2
Let C be a q-ary pn, k, r, ρq LRC code with distance d. Define
␣
T :“ i “ pi1 , . . . , is q | i1 ` . . . ` is ě d,
(
ip P t0, ρ, ρ ` 1, . . . , ρ ` r ´ 1u for all p “ 1, . . . , s. .
Then |C| ď 1 `
ř
iPT
ai , where pai , i P Tq is given by the following LP problem
ÿ
maximize
ai subject to
iPT
ai ě 0, i P T,
ÿ
pr`ρ´1q
ai Qij ě ´Kj
p0q,
j P rr ` ρ ´ 1ss zt0u
iPT
(Hu et al., Proc. ISIT 2016)
Bounds on LRC codes
14 / 22
Linear programming bounds
Results:
It is possible to
• Derive the Singleton, Hamming, and Plotkin bounds on pn, k, r, ρq LRC codes
• Improve the shortening bound in examples for short length (e.g., n “ 8-20) by solving
the LP problem
Bounds on LRC codes
15 / 22
Availability (Multiple recovery sets)
Def: C Ă F n is an LRC code with availability if every coordinate i Ă rns has t
pjq
pjq
pairwise disjoint recovery sets Ri , and |Ri | ď rj , j “ 1, . . . , t.
For simplicity let rj “ r, j “ 1, . . . , t and use the notation pn, k, r, tq LRC code
Bounds on LRC codes
16 / 22
Availability (Multiple recovery sets)
Def: C Ă F n is an LRC code with availability if every coordinate i Ă rns has t
pjq
pjq
pairwise disjoint recovery sets Ri , and |Ri | ď rj , j “ 1, . . . , t.
For simplicity let rj “ r, j “ 1, . . . , t and use the notation pn, k, r, tq LRC code
Example: p6, 3, 2, 2q LRC code with distance 3, parity-check matrix
fi
0 0 0 1 1 1
—
ffi
H “ – 0 1 1 0 0 1 fl
1 0 1 0 1 0
»
Bounds on LRC codes
16 / 22
Availability (Multiple recovery sets)
Def: C Ă F n is an LRC code with availability if every coordinate i Ă rns has t
pjq
pjq
pairwise disjoint recovery sets Ri , and |Ri | ď rj , j “ 1, . . . , t.
For simplicity let rj “ r, j “ 1, . . . , t and use the notation pn, k, r, tq LRC code
Example: p6, 3, 2, 2q LRC code with distance 3, parity-check matrix
fi
0 0 0 1 1 1
—
ffi
H “ – 0 1 1 0 0 1 fl
1 0 1 0 1 0
»
Constructions:
• Case t “ 2 : Two-level codes give a natural way of constructing LRC codes with t “ 2.
Examples include product codes, codes on bipartite graphs.
• Algebraic constructions of codes with t ě 2 recovery sets are covered in another part
of the tutorial (refer to “Introduction”)
Bounds on LRC codes
16 / 22
Upper bounds on codes with availability
Extension of the Singleton bound:
d ďn´k`2´
Q tpk ´ 1q ` 1 U
tpr ´ 1q ` 1
(Wang-Zhang ’14; Rawat e.a. ’14)
Bounds on LRC codes
17 / 22
Upper bounds on codes with availability
Theorem (Tamo-B-Frolov ’15)
The parameters of a t-LRC code are bounded as follows:
k
1
ď śt
1
n
j“1 p1 ` jr q
t Y
ÿ
k ´ 1]
d ďn´
ri
i“0
Bounds on LRC codes
18 / 22
Upper bounds on codes with availability
Theorem (Tamo-B-Frolov ’15)
The parameters of a t-LRC code are bounded as follows:
k
1
ď śt
1
n
j“1 p1 ` jr q
t Y
ÿ
k ´ 1]
d ďn´
ri
i“0
Proof idea:
• construct a directed graph GpV, Eq, V “ rns, pi, jq P E iff j is a part of some recovery set for i
• study colorings and expansion of the graph, which enables us to trace propagation of dependent vertices
in G
Bounds on LRC codes
18 / 22
Upper bounds on codes with availability
Theorem (Tamo-B-Frolov ’15)
The parameters of a t-LRC code are bounded as follows:
k
1
ď śt
1
n
j“1 p1 ` jr q
t Y
ÿ
k ´ 1]
d ďn´
ri
i“0
Proof idea:
• construct a directed graph GpV, Eq, V “ rns, pi, jq P E iff j is a part of some recovery set for i
• study colorings and expansion of the graph, which enables us to trace propagation of dependent vertices
in G
• The bounds are tight in some examples, but generally are likely to be loose.
• The above bound and the bound on the previous page are incomparable (i.e., there
are examples where each of them is better than the other one).
Bounds on LRC codes
18 / 22
Existence of codes with availability; t “ 2
GV-type bound for codes with two recovery sets (TBF ’16)
Bounds on LRC codes
19 / 22
Existence of codes with availability; t “ 2
GV-type bound for codes with two recovery sets (TBF ’16)
• Let G be a regular graph of degree r ` 1. Each edge is adjacent to two disjoint sets of r edges,
which form 2 recovery sets
Bounds on LRC codes
19 / 22
Existence of codes with availability; t “ 2
GV-type bound for codes with two recovery sets (TBF ’16)
• Let G be a regular graph of degree r ` 1. Each edge is adjacent to two disjoint sets of r edges,
which form 2 recovery sets (bipartite graphs support recovery sets of different sizes)
Bounds on LRC codes
19 / 22
Existence of codes with availability; t “ 2
GV-type bound for codes with two recovery sets (TBF ’16)
• Let G be a regular graph of degree r ` 1. Each edge is adjacent to two disjoint sets of r edges,
which form 2 recovery sets (bipartite graphs support recovery sets of different sizes)
• Consider the ensemble of linear codes with parity-check matrices
»
Ar`2
—
—
Ar`2
—
—
—
..
—
.
H“—
—
—
Ar`2
—
—
–
HL
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
where Ar`2 is E-V incidence matrix of the complete graph Kr`2 (or of Kr1 ,r2 )
HL is a random matrix with independent entries
Bounds on LRC codes
19 / 22
Existence of codes with availability; t “ 2
GV-type bound for codes with two recovery sets (TBF ’16)
• Let G be a regular graph of degree r ` 1. Each edge is adjacent to two disjoint sets of r edges,
which form 2 recovery sets (bipartite graphs support recovery sets of different sizes)
• Consider the ensemble of linear codes with parity-check matrices
»
Ar`2
—
—
Ar`2
—
—
—
..
—
.
H“—
—
—
Ar`2
—
—
–
HL
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
where Ar`2 is E-V incidence matrix of the complete graph Kr`2 (or of Kr1 ,r2 )
HL is a random matrix with independent entries
• Analyze the ensemble-average distance of codes
Bounds on LRC codes
19 / 22
Many recovery sets
Theorem
There exist asymptotically good sequences of q-ary LRC codes (fixed q, n Ñ 8) with t
disjoint recovery sets of size r for any constant t, 1 ď t ď r
Ideas: Existence of bipartite expanding graphs; large finite field alphabets
Bounds on LRC codes
20 / 22
Many recovery sets
Theorem
There exist asymptotically good sequences of q-ary LRC codes (fixed q, n Ñ 8) with t
disjoint recovery sets of size r for any constant t, 1 ď t ď r
Ideas: Existence of bipartite expanding graphs; large finite field alphabets
R
0.8
0.6
t “ 3; r “ 6
δ “ 0, R “ 1 ´
Si
ng
l
et
0.4
Lower bound
on
bo
un
d
t
r`1
Singleton bound
Rptq pr, δq ď
0.2
0.2
0.4
0.6
0.8
1.0
rt pr ´ 1q
p1 ´ δq
rt`1 ´ 1
∆
Bounds on LRC codes
20 / 22
Open questions
• Alphabet-dependent bounds make little use of locality
• Find a good way to derive lower bounds for t ě 3, upper bounds for t ě 2
• Settle the corner point for δ “ 0 of the asymptotic bound for t ě 2
• Remove the structure assumption in the LP bound
• Derive an LP (closed-form asymptotic) bound for LRC codes that correct locally one
erasure
Bounds on LRC codes
21 / 22
Selected references
P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, On the locality of codeword symbols, IEEE Trans. Inform.
Theory, vol. 58, no. 11, pp. 6925–6934, 2011.
N. Prakash, G. M. Kamath, V. Lalitha, and P. V. Kumar, Optimal linear codes with a local-error-correction
property, Proc. 2012 IEEE Internat. Sympos. Inform. Theory, IEEE, 2012, pp. 2776–2780.
V. Cadambe and A. Mazumdar, Bounds on the size of locally recoverable codes, IEEE Trans. Inform. Theory,
vol. 61, no. 11, pp. 5787–5794, 2015.
A. Wang and Z. Zhang, Repair locality with multiple erasure tolerance, IEEE Trans. Inform. Theory, vol. 60,
no. 11, 6979–6987, 2014.
A. Rawat, D. Papailiopoulos, A. Dimakis, and S. Vishwanath, Locality and availability in distributed storage, ISIT
2014.
I. Tamo, A. Barg, and A. Frolov, Bounds on the parameters of locally recoverable codes, IEEE Trans. Inform.
Theory, vol. 62, no. 6, 3070–3083, 2016.
S. Hu, I. Tamo, and A. Barg, Combinatorial bounds for LRC codes, Proc. ISIT 2016.
A. Barg, I. Tamo, and S. Vlăduţ, Locally recoverable codes on algebraic curves, arXiv:1603.08876; also ISIT ’15.
Bounds on LRC codes
22 / 22
Maximally Recoverable Codes
Combining locality and storage efficiency
Maximally Recoverable Codes
1 / 17
Maximally Recoverable (MR) codes
Motivation
Maximally Recoverable Codes
2 / 17
Maximally Recoverable (MR) codes
Motivation
• An (n, k) code is MDS ⇐⇒ Any k symbols of the codeword suffice for decoding
Maximally Recoverable Codes
2 / 17
Maximally Recoverable (MR) codes
Motivation
• An (n, k) code is MDS ⇐⇒ Any k symbols of the codeword suffice for decoding
• MDS property is very important in storage
Maximally Recoverable Codes
2 / 17
Maximally Recoverable (MR) codes
Motivation
• An (n, k) code is MDS ⇐⇒ Any k symbols of the codeword suffice for decoding
• MDS property is very important in storage
• Recovery set = a set of r + 1 symbols s.t. any symbol of the set can be recovered
by the remaining r symbols
Maximally Recoverable Codes
2 / 17
Maximally Recoverable (MR) codes
Motivation
• An (n, k) code is MDS ⇐⇒ Any k symbols of the codeword suffice for decoding
• MDS property is very important in storage
• Recovery set = a set of r + 1 symbols s.t. any symbol of the set can be recovered
by the remaining r symbols
• A set of k symbols of an (n, k, r) LRC code, that contains a recovery set does not
suffice for decoding
Maximally Recoverable Codes
2 / 17
Maximally Recoverable (MR) codes
Motivation
• An (n, k) code is MDS ⇐⇒ Any k symbols of the codeword suffice for decoding
• MDS property is very important in storage
• Recovery set = a set of r + 1 symbols s.t. any symbol of the set can be recovered
by the remaining r symbols
• A set of k symbols of an (n, k, r) LRC code, that contains a recovery set does not
suffice for decoding
• =⇒ LRC codes are not MDS
Maximally Recoverable Codes
2 / 17
Maximally Recoverable (MR) codes
Motivation
• An (n, k) code is MDS ⇐⇒ Any k symbols of the codeword suffice for decoding
• MDS property is very important in storage
• Recovery set = a set of r + 1 symbols s.t. any symbol of the set can be recovered
by the remaining r symbols
• A set of k symbols of an (n, k, r) LRC code, that contains a recovery set does not
suffice for decoding
• =⇒ LRC codes are not MDS
• Goal: LRC codes that are “close” to be MDS
Maximally Recoverable Codes
2 / 17
(n, k, r) MR codes - Definition
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
• Disjoint recovery sets Ri , |Ri | = r + 1 for i = 1, ...,
Maximally Recoverable Codes
n
r+1
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
• Disjoint recovery sets Ri , |Ri | = r + 1 for i = 1, ...,
n
r+1
• Any set S of k symbols s.t. Ri * S suffices for decoding
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
• Disjoint recovery sets Ri , |Ri | = r + 1 for i = 1, ...,
n
r+1
• Any set S of k symbols s.t. Ri * S suffices for decoding
Remarks:
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
• Disjoint recovery sets Ri , |Ri | = r + 1 for i = 1, ...,
n
r+1
• Any set S of k symbols s.t. Ri * S suffices for decoding
Remarks:
• If ∃i, Ri ⊆ S, |S| = k then decoding of the codeword is impossible
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
• Disjoint recovery sets Ri , |Ri | = r + 1 for i = 1, ...,
n
r+1
• Any set S of k symbols s.t. Ri * S suffices for decoding
Remarks:
• If ∃i, Ri ⊆ S, |S| = k then decoding of the codeword is impossible
• Terminology: MR codes = Partially MDS codes
Maximally Recoverable Codes
3 / 17
(n, k, r) MR codes - Definition
An (n, k, r) MR codes has the following properties:
• An (n, k, r) LRC code
• Disjoint recovery sets Ri , |Ri | = r + 1 for i = 1, ...,
n
r+1
• Any set S of k symbols s.t. Ri * S suffices for decoding
Remarks:
• If ∃i, Ri ⊆ S, |S| = k then decoding of the codeword is impossible
• Terminology: MR codes = Partially MDS codes
• Equivalent definition using Matroid Theory language
Maximally Recoverable Codes
3 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
⇔ any d − 1 = n − k −
k
r
+ 1 erasures are recoverable
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
⇔ any d − 1 = n − k −
k
r
⇔ any n − (d − 1) = k +
+ 1 erasures are recoverable
k
r
− 1 coordinates suffice for decoding
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
⇔ any d − 1 = n − k −
k
r
⇔ any n − (d − 1) = k +
• Any k +
k
r
+ 1 erasures are recoverable
k
r
− 1 coordinates suffice for decoding
− 1 coordinates contain a subset S s.t.
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
⇔ any d − 1 = n − k −
k
r
⇔ any n − (d − 1) = k +
• Any k +
k
r
+ 1 erasures are recoverable
k
r
− 1 coordinates suffice for decoding
− 1 coordinates contain a subset S s.t.
1. |S| = k
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
⇔ any d − 1 = n − k −
k
r
⇔ any n − (d − 1) = k +
• Any k +
k
r
+ 1 erasures are recoverable
k
r
− 1 coordinates suffice for decoding
− 1 coordinates contain a subset S s.t.
1. |S| = k
2. ∀i, Ri * S
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Proof:
• Assume r divides k
• MR code is an Optimal LRC code
⇔d =n−k−
k
r
+2
⇔ any d − 1 = n − k −
k
r
⇔ any n − (d − 1) = k +
• Any k +
k
r
+ 1 erasures are recoverable
k
r
− 1 coordinates suffice for decoding
− 1 coordinates contain a subset S s.t.
1. |S| = k
2. ∀i, Ri * S
• By the MR property, S suffices for decoding
Maximally Recoverable Codes
4 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
Maximally Recoverable Codes
5 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
LRC Codes
Maximally Recoverable Codes
5 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
LRC Codes
Optimal LRC
Codes
Maximally Recoverable Codes
5 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
LRC Codes
Optimal LRC
Codes
MR Codes
Maximally Recoverable Codes
5 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
LRC Codes
Optimal LRC
Codes
MR Codes
Q: MR codes = Optimal LRC codes ?
Maximally Recoverable Codes
5 / 17
MR codes - Basics
Lemma
MR codes are optimal LRC codes
LRC Codes
Optimal LRC
Codes
MR Codes
Q: MR codes = Optimal LRC codes ? Ans: No
Maximally Recoverable Codes
5 / 17
Constructing MR codes
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
2. Field size is superpolynomial
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
2. Field size is superpolynomial
• Construction of Partially MDS codes [Blaum, Lee Hafner, Hetzler 13, Blaum, Plank,
Schwartz, Yaakobi 14]
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
2. Field size is superpolynomial
• Construction of Partially MDS codes [Blaum, Lee Hafner, Hetzler 13, Blaum, Plank,
Schwartz, Yaakobi 14]
1. Restricted parameters
Maximally Recoverable Codes
6 / 17
Constructing MR codes
• Probabilistic construction of optimal LRC code results an MR code
1. Non explicit
2. Field size is superpolynomial in the length (is this truly necessary?)
• Explicit constructions [Rawat, Koyluoglu, Silberstein, Vishwanath 14, Gopalan, Huang,
Jenkins, Yekhanin 14, Tamo, Papailiopoulos, Dimakis 14]
1. Any n, k, r
2. Field size is superpolynomial
• Construction of Partially MDS codes [Blaum, Lee Hafner, Hetzler 13, Blaum, Plank,
Schwartz, Yaakobi 14]
1. Restricted parameters
2. Field size |F| = O(n)
Maximally Recoverable Codes
6 / 17
Constructing MR codes - Easy cases
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
• r=1
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
• r=1
1. Duplication of an (n/2, k) RS is an (n, k, 1) MR code
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
• r=1
1. Duplication of an (n/2, k) RS is an (n, k, 1) MR code
2. |F| = O(n)
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
• r=1
1. Duplication of an (n/2, k) RS is an (n, k, 1) MR code
2. |F| = O(n)
• Q: Is the RS-like construction an MR code?
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
• r=1
1. Duplication of an (n/2, k) RS is an (n, k, 1) MR code
2. |F| = O(n)
• Q: Is the RS-like construction an MR code?
1. No, in general
Maximally Recoverable Codes
7 / 17
Constructing MR codes - Easy cases
• r=k
1. An (n, k) RS is an (n, k, k) MR code
2. |F| = O(n)
• r=1
1. Duplication of an (n/2, k) RS is an (n, k, 1) MR code
2. |F| = O(n)
• Q: Is the RS-like construction an MR code?
1. No, in general
2. Yes in some cases (other cases?)
Maximally Recoverable Codes
7 / 17
MR codes through linearized polynomials
Maximally Recoverable Codes
8 / 17
MR codes through linearized polynomials
Basic facts:
• Fqm is a vector space of dimension m over Fq
Maximally Recoverable Codes
8 / 17
MR codes through linearized polynomials
Basic facts:
• Fqm is a vector space of dimension m over Fq
• Linearized polynomial has the form
f (x) =
m−1
X
i
ai xq , where ai ∈ Fqm
i=0
Maximally Recoverable Codes
8 / 17
MR codes through linearized polynomials
Basic facts:
• Fqm is a vector space of dimension m over Fq
• Linearized polynomial has the form
f (x) =
m−1
X
i
ai xq , where ai ∈ Fqm
i=0
• For x, y ∈ Fqm and α1 , α2 ∈ Fq
f (α1 x + α2 y) = α1 f (x) + α2 f (y)
Maximally Recoverable Codes
8 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi }
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
Maximally Recoverable Codes
i=r+1
αi }
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
Maximally Recoverable Codes
i=r+1
n
αi } , ..., R r+1
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
1. | ∪i Ri | = n
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
1. | ∪i Ri | = n
P
2.
α∈Ri α = 0
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
1. | ∪i Ri | = n
P
2.
α∈Ri α = 0
Encoding:
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
1. | ∪i Ri | = n
P
2.
α∈Ri α = 0
Encoding:
(v0 , ..., vk−1 )
1. Given k information symbols of F2m
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
1. | ∪i Ri | = n
P
2.
α∈Ri α = 0
Encoding:
(v0 , ..., vk−1 ) 7→ f (x) =
Pk−1
i=0
vi x2
i
1. Given k information symbols of F2m
2. Form the linearized polynomial
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code [Rawat, Koyluoglu, Silberstein, Vishwanath 14]
• Set m =
nr
r+1
• α1 , ..., αm ∈ F2m linearly independent elements over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• Observations:
1. | ∪i Ri | = n
P
2.
α∈Ri α = 0
Encoding:
(v0 , ..., vk−1 ) 7→ f (x) =
Pk−1
i=0
vi x2
i
7→ (f (α) : α ∈ ∪i Ri )
1. Given k information symbols of F2m
2. Form the linearized polynomial
3. Output the length−n codeword
Maximally Recoverable Codes
9 / 17
Constructing an (n, k, r) MR code - Cont’d
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
Locality: Recover f β =? for β ∈ Ri
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
Locality: Recover f β =? for β ∈ Ri
X
α=0
α∈Ri
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
Locality: Recover f β =? for β ∈ Ri
X
α∈Ri
α=0
=⇒ β =
X
α
α∈Ri \β
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
Locality: Recover f β =? for β ∈ Ri
X
α∈Ri
α=0
=⇒ β =
X
α
=⇒ f β = f
α∈Ri \β
Maximally Recoverable Codes
X
α
α∈Ri \β
10 / 17
Constructing an (n, k, r) MR code - Cont’d
Locality: Recover f β =? for β ∈ Ri
X
α∈Ri
α=0
=⇒ β =
X
α
=⇒ f β = f
α∈Ri \β
Maximally Recoverable Codes
X
α∈Ri \β
α =
X
f α
α∈Ri \β
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi }
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
Maximally Recoverable Codes
i=r+1
αi }
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
Maximally Recoverable Codes
i=r+1
n
αi } , ..., R r+1
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• =⇒ the only linear dependencies in ∪i Ri are the Ri ’s
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• =⇒ the only linear dependencies in ∪i Ri are the Ri ’s
• =⇒ If S ⊆ ∪i Ri s.t. |S| = k, Ri * S then the elements of S are linearly
independent over F2
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• =⇒ the only linear dependencies in ∪i Ri are the Ri ’s
• =⇒ If S ⊆ ∪i Ri s.t. |S| = k, Ri * S then the elements of S are linearly
independent over F2
• =⇒ all 2k possible sums
P
s ∈{0,1} s s
s∈S
are distinct
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• =⇒ the only linear dependencies in ∪i Ri are the Ri ’s
• =⇒ If S ⊆ ∪i Ri s.t. |S| = k, Ri * S then the elements of S are linearly
independent over F2
• =⇒ all 2k possible sums
P
s ∈{0,1} s s
s∈S
are distinct
• =⇒ 2k evaluations of f x :
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• =⇒ the only linear dependencies in ∪i Ri are the Ri ’s
• =⇒ If S ⊆ ∪i Ri s.t. |S| = k, Ri * S then the elements of S are linearly
independent over F2
• =⇒ all 2k possible sums
P
s ∈{0,1} s s
s∈S
are distinct
• =⇒ 2k evaluations of f x :
f
X
s s
s ∈{0,1}
s∈S
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• α1 , ..., αm are linearly independent over F2
• R1 = {α1 , ..., αr ,
Pr
i=1
αi } , R2 = {αr+1 , ..., α2r ,
P2r
i=r+1
n
αi } , ..., R r+1
• =⇒ the only linear dependencies in ∪i Ri are the Ri ’s
• =⇒ If S ⊆ ∪i Ri s.t. |S| = k, Ri * S then the elements of S are linearly
independent over F2
• =⇒ all 2k possible sums
P
s ∈{0,1} s s
s∈S
are distinct
• =⇒ 2k evaluations of f x :
f
X
s ∈{0,1}
s∈S
s s =
X
s f s
s ∈{0,1}
s∈S
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• =⇒ 2k evaluations of f x :
f
X
s ∈{0,1}
s∈S
s s =
X
s f s
s ∈{0,1}
s∈S
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• =⇒ 2k evaluations of f x :
f
X
s ∈{0,1}
s∈S
s s =
X
s f s
s ∈{0,1}
s∈S
• deg(f ) ≤ 2k−1
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• =⇒ 2k evaluations of f x :
f
X
s s =
s ∈{0,1}
s∈S
• deg(f ) ≤ 2k−1 , since f x =
Pk−1
i=0
vi x2
X
s f s
s ∈{0,1}
s∈S
i
Maximally Recoverable Codes
10 / 17
Constructing an (n, k, r) MR code - Cont’d
MR Property:
• =⇒ 2k evaluations of f x :
f
X
s s =
s ∈{0,1}
s∈S
• deg(f ) ≤ 2k−1 , since f x =
Pk−1
i=0
vi x2
X
s f s
s ∈{0,1}
s∈S
i
• f x can be interpolated
Maximally Recoverable Codes
10 / 17
Properties of the MR code construction
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r X
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r X
• Need m =
nr
r+1
linearly independent elements over F2
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r X
• Need m =
nr
r+1
nr
linearly independent elements over F2 =⇒ |F| = 2 r+1
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r X
• Need m =
nr
r+1
nr
linearly independent elements over F2 =⇒ |F| = 2 r+1
• Field size is exponential in n
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r X
• Need m =
nr
r+1
nr
linearly independent elements over F2 =⇒ |F| = 2 r+1
• Field size is exponential in n
×
Maximally Recoverable Codes
11 / 17
Properties of the MR code construction
• Flexible set of parameters n, k, r X
• Need m =
nr
r+1
nr
linearly independent elements over F2 =⇒ |F| = 2 r+1
• Field size is exponential in n
×
• Can we do better?
Maximally Recoverable Codes
11 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
nr
r+1
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
The code is a
n
r+1
0
2
independent parity check codes of length r + 1
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
3
O(n3/2 )
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
3
O(n3/2 )
Maximally Recoverable Codes
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
3
O(n3/2 )
Maximally Recoverable Codes
4
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
3
O(n3/2 )
Maximally Recoverable Codes
4
O(n7/3 )
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
3
O(n3/2 )
Maximally Recoverable Codes
4
O(n7/3 )
≥5
12 / 17
High-rate MR codes [Gopalan, Huang, Jenkins, Yekhanin 14]
• In any (n, k, r) LRC code k ≤
• High rate means
nr
r+1
nr
r+1
− k = O( logn n )
• F(n, k, r) = minimum field size needed for an (n, k, r) MR code
• Known bounds on F(n, k, r):
nr
r+1
−k
F(n, k, r)
0
2
1
r+1
2
θ(n)
3
O(n3/2 )
Maximally Recoverable Codes
4
O(n7/3 )
≥5
O(n
nr −k−1
r+1
)
12 / 17
Summary of best known constructions of MR codes
High-Rate
Constant-Rate
Vanishing Rate
Dimension
Field size
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
High-Rate
Constant-Rate
Vanishing Rate
Dimension
Field size
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
nr
r+1
High-Rate
− k = O( logn n )
Constant-Rate
Vanishing Rate
Field size
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
Constant-Rate
Vanishing Rate
nr
n r+1 −k−1
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
Constant-Rate
Vanishing Rate
nr
n r+1 −k−1
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
• Constant-Rate Regime: Rawat, Koyluoglu, Silberstein, Vishwanath 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
Constant-Rate
r
k
< r+1
n
Vanishing Rate
nr
n r+1 −k−1
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
• Constant-Rate Regime: Rawat, Koyluoglu, Silberstein, Vishwanath 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
nr
n r+1 −k−1
Constant-Rate
r
k
< r+1
n
Vanishing Rate
nr
2 r+1
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
• Constant-Rate Regime: Rawat, Koyluoglu, Silberstein, Vishwanath 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
nr
n r+1 −k−1
Constant-Rate
r
k
< r+1
n
Vanishing Rate
nr
2 r+1
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
• Constant-Rate Regime: Rawat, Koyluoglu, Silberstein, Vishwanath 14
• Vanishing Rate Regime: T., Papailiopoulos, Dimakis 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
nr
n r+1 −k−1
Constant-Rate
r
k
< r+1
n
Vanishing Rate
k = O( logn n )
nr
2 r+1
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
• Constant-Rate Regime: Rawat, Koyluoglu, Silberstein, Vishwanath 14
• Vanishing Rate Regime: T., Papailiopoulos, Dimakis 14
Maximally Recoverable Codes
13 / 17
Summary of best known constructions of MR codes
Dimension
Field size
nr
r+1
High-Rate
− k = O( logn n )
nr
n r+1 −k−1
Constant-Rate
r
k
< r+1
n
Vanishing Rate
k = O( logn n )
nr
2 r+1
nk
• High-Rate Regime: Gopalan, Huang, Jenkins, Yekhanin 14
• Constant-Rate Regime: Rawat, Koyluoglu, Silberstein, Vishwanath 14
• Vanishing Rate Regime: T., Papailiopoulos, Dimakis 14
Maximally Recoverable Codes
13 / 17
Lower bounds on the field size
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
• By the bound on the field size of an MDS code
Maximally Recoverable Codes
nr
r+1
nr
≤ |F| + ( r+1
− k) − 1
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
• By the bound on the field size of an MDS code
nr
r+1
nr
≤ |F| + ( r+1
− k) − 1
Questions:
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
• By the bound on the field size of an MDS code
nr
r+1
nr
≤ |F| + ( r+1
− k) − 1
Questions:
1. What is the smallest field size needed for an (n, k, r) MR code?
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
• By the bound on the field size of an MDS code
nr
r+1
nr
≤ |F| + ( r+1
− k) − 1
Questions:
1. What is the smallest field size needed for an (n, k, r) MR code?
- For optimal LRC order n is sufficient
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
• By the bound on the field size of an MDS code
nr
r+1
nr
≤ |F| + ( r+1
− k) − 1
Questions:
1. What is the smallest field size needed for an (n, k, r) MR code?
- For optimal LRC order n is sufficient
- Contains the MDS conjecture as a subproblem
Maximally Recoverable Codes
14 / 17
Lower bounds on the field size
F(n, k, r) is upper bounded by a superpolynomial function of n
Theorem [Gopalan, Huang, Jenkins, Yekhanin 14]
F(n, k, r) ≥ k + 1
Proof:
nr
• Puncturing the code by one coordinate from each recovery set results an ( r+1
, k)
MDS code
• By the bound on the field size of an MDS code
nr
r+1
nr
≤ |F| + ( r+1
− k) − 1
Questions:
1. What is the smallest field size needed for an (n, k, r) MR code?
- For optimal LRC order n is sufficient
- Contains the MDS conjecture as a subproblem
2. Are MR codes inherently different from optimal LRC codes?
Maximally Recoverable Codes
14 / 17
MR codes under arbitrary topologies
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
1. E.g. R1 = {1, ..., r + 1}, R2 = {r + 2, ..., 2(r + 1)}, ..., R
Maximally Recoverable Codes
nr
r+1
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
1. E.g. R1 = {1, ..., r + 1}, R2 = {r + 2, ..., 2(r + 1)}, ..., R
nr
r+1
2. Can be arbitrary code constraints
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
• CR = a code with a set of coordinate constraints R
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
• CR = a code with a set of coordinate constraints R
• Definition: Given a code CR , its Information sets
I = {S ⊆ [n] : CR is decodable from S, |S| = k}
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
• CR = a code with a set of coordinate constraints R
• Definition: Given a code CR , its Information sets
I = {S ⊆ [n] : CR is decodable from S, |S| = k}
• Goal: Construct a code CR with local constraints R that is “highly” decodable
(large set I)
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
• R = {Ri ⊆ [n] : i = 1, ..., m} constraints on code’s coordinates,
• CR = a code with a set of coordinate constraints R
• Definition: Given a code CR , its Information sets
I = {S ⊆ [n] : CR is decodable from S, |S| = k}
• Goal: Construct a code CR with local constraints R that is “highly” decodable
(large set I)
0
• Definition: CR with information sets I is MR if for any CR
with I 0
I 0 ⊆ I ( i.e. I is a maximum)
Maximally Recoverable Codes
15 / 17
MR codes under arbitrary topologies
0
• Definition: CR with I is MR if for any CR
with I 0
I0 ⊆ I
Maximally Recoverable Codes
16 / 17
MR codes under arbitrary topologies
0
• Definition: CR with I is MR if for any CR
with I 0
I0 ⊆ I
• Example:
Maximally Recoverable Codes
16 / 17
MR codes under arbitrary topologies
0
• Definition: CR with I is MR if for any CR
with I 0
I0 ⊆ I
• Example: R1 = {1, ..., r + 1}
Maximally Recoverable Codes
16 / 17
MR codes under arbitrary topologies
0
• Definition: CR with I is MR if for any CR
with I 0
I0 ⊆ I
• Example: R1 = {1, ..., r + 1} , ..., Rn/(r+1) = {n − r, ..., n}
Maximally Recoverable Codes
16 / 17
MR codes under arbitrary topologies
0
• Definition: CR with I is MR if for any CR
with I 0
I0 ⊆ I
• Example: R1 = {1, ..., r + 1} , ..., Rn/(r+1) = {n − r, ..., n}
CR is MR if I = {S ⊂ [n] : Ri * S, |S| = k}
Maximally Recoverable Codes
16 / 17
MR codes under arbitrary topologies
0
• Definition: CR with I is MR if for any CR
with I 0
I0 ⊆ I
• Example: R1 = {1, ..., r + 1} , ..., Rn/(r+1) = {n − r, ..., n}
CR is MR if I = {S ⊂ [n] : Ri * S, |S| = k}
Lemma
For any set R there exists an MR code CR over a large enough field
Maximally Recoverable Codes
16 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Questions:
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Questions:
1. Given an MR code with set of constraints R characterize its information sets I
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Questions:
1. Given an MR code with set of constraints R characterize its information sets I
2. MR code constructions for other topologies (e.g. grid-like topologies)
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Questions:
1. Given an MR code with set of constraints R characterize its information sets I
2. MR code constructions for other topologies (e.g. grid-like topologies)
3. Are the known constructions extendable to other topologies?
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Questions:
1. Given an MR code with set of constraints R characterize its information sets I
2. MR code constructions for other topologies (e.g. grid-like topologies)
3. Are the known constructions extendable to other topologies?
4. Field size?
Maximally Recoverable Codes
17 / 17
MR codes under arbitrary topologies - cont’d
“Maximally Recoverable Codes for Grid-like Topologies” [Gopalan, Hu, Saraf, Wang,
Yekhanin 16]
1. Good problem introduction
2. The first non-trivial lower bound on |F| (for grid-like topologies)
3. More results...
Questions:
1. Given an MR code with set of constraints R characterize its information sets I
2. MR code constructions for other topologies (e.g. grid-like topologies)
3. Are the known constructions extendable to other topologies?
4. Field size?
5. Is it possible to construct MR codes using the RS-like construction ?
Maximally Recoverable Codes
17 / 17
LRC codes on graphs
In this part we used materials provided to us by Fatemeh Arbabjolfaei (University of California, San Diego),
Søren Riis (Queen Mary, University of London ) and Arya Mazumdar (University of Massachusetts at
Amherst), with their kind permission.
LRC codes
1 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
x2
x3
LRC codes
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
LRC codes
3
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
3
• Each node can recover its content from its incoming neighbors
LRC codes
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
3
• Each node can recover its content from its incoming neighbors
• Recovery sets:
LRC codes
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
3
• Each node can recover its content from its incoming neighbors
• Recovery sets: A1 = {2},
LRC codes
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
3
• Each node can recover its content from its incoming neighbors
• Recovery sets: A1 = {2}, A2 = {1, 3}, A3 = {1}
LRC codes
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
3
• Each node can recover its content from its incoming neighbors
• Recovery sets: A1 = {2}, A2 = {1, 3}, A3 = {1}
• How much data can be stored?
LRC codes
2 / 15
LRC codes on graphs [Mazumdar 2014, Shanmugam and Dimakis 2014]
x1
• Storage recovery graph G
1
x2
x3
2
3
• Each node can recover its content from its incoming neighbors
• Recovery sets: A1 = {2}, A2 = {1, 3}, A3 = {1}
• How much data can be stored?
• Which coding scheme achieves the limit?
LRC codes
2 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
2. n recovery functions fi , s.t. for any (x1 , ..., xn ) ∈ C
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
2. n recovery functions fi , s.t. for any (x1 , ..., xn ) ∈ C
fi (xj : j ∈ N(i)) = xi
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
2. n recovery functions fi , s.t. for any (x1 , ..., xn ) ∈ C
fi (xj : j ∈ N(i)) = xi
• The storage capacity of G over Fq
Capq (G) =
max
C⊆Fnq is a
storage code for G
LRC codes
logq |C|
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
2. n recovery functions fi , s.t. for any (x1 , ..., xn ) ∈ C
fi (xj : j ∈ N(i)) = xi
• The storage capacity of G over Fq
Capq (G) =
max
C⊆Fnq is a
storage code for G
LRC codes
logq |C| ≤ n
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
2. n recovery functions fi , s.t. for any (x1 , ..., xn ) ∈ C
fi (xj : j ∈ N(i)) = xi
• The storage capacity of G over Fq
Capq (G) =
max
C⊆Fnq is a
storage code for G
logq |C| ≤ n
• The storage capacity of G is
Cap(G) = sup Capq (G)
q
LRC codes
3 / 15
Storage Capacity
• The network is modeled by a (directed) graph G = (V, E), |V| = n
• Each node (vertex) stores a symbol from Fq
• Storage code:
1. A set of vectors C ⊆ Fnq
2. n recovery functions fi , s.t. for any (x1 , ..., xn ) ∈ C
fi (xj : j ∈ N(i)) = xi
• The storage capacity of G over Fq
Capq (G) =
max
C⊆Fnq is a
storage code for G
logq |C| ≤ n
• The storage capacity of G is
Cap(G) = sup Capq (G) = lim Capq (G) (Fekete’s lemma)
q
q→∞
LRC codes
3 / 15
LRC codes on graphs - Example
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N1
N3
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N1
N3
• Each node stores a single bit
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N1
N3
• Each node stores a single bit
• Two bits of information b1 , b2 can be stored (Cap2 (G) = 2)
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N1
N3
• Each node stores a single bit
• Two bits of information b1 , b2 can be stored (Cap2 (G) = 2)
• Store:
• N 1 = b1
• N 2 = b2
• N 3 = b1 + b2
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N1
N3
• Each node stores a single bit
• Two bits of information b1 , b2 can be stored (Cap2 (G) = 2)
• Store:
• N 1 = b1
• N 2 = b2
• N 3 = b1 + b2
• Cap(G) = supq Capq (G)
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N1
N3
• Each node stores a single bit
• Two bits of information b1 , b2 can be stored (Cap2 (G) = 2)
• Store:
• N 1 = b1
• N 2 = b2
• N 3 = b1 + b2
• Cap(G) = supq Capq (G) = 2
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
• N1 = N2 ∧ N5 ,
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
• N1 = N2 ∧ N5 , N2 = N1 ∨ N3 , N3 = N2 ∧ N4 , N4 = N3 ∧ N5 , N5 = N1 ∨ N4
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
• N1 = N2 ∧ N5 , N2 = N1 ∨ N3 , N3 = N2 ∧ N4 , N4 = N3 ∧ N5 , N5 = N1 ∨ N4
• Cap2 (G) ≥ log2 (5) = 2.32...
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
• N1 = N2 ∧ N5 , N2 = N1 ∨ N3 , N3 = N2 ∧ N4 , N4 = N3 ∧ N5 , N5 = N1 ∨ N4
• Cap2 (G) ≥ log2 (5) = 2.32...
• In fact Cap2 (G) = log2 (5) = 2.32...
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
• N1 = N2 ∧ N5 , N2 = N1 ∨ N3 , N3 = N2 ∧ N4 , N4 = N3 ∧ N5 , N5 = N1 ∨ N4
• Cap2 (G) ≥ log2 (5) = 2.32...
• In fact Cap2 (G) = log2 (5) = 2.32...
• However Cap(G) = Cap4 (G)
LRC codes
4 / 15
LRC codes on graphs - Example
N2
N3
N1
N4
N5
• C = {(0, 0, 0, 0, 0), (0, 1, 1, 0, 0), (0, 0, 0, 1, 1), (1, 1, 0, 1, 1), (1, 1, 1, 0, 1)}
• N1 = N2 ∧ N5 , N2 = N1 ∨ N3 , N3 = N2 ∧ N4 , N4 = N3 ∧ N5 , N5 = N1 ∨ N4
• Cap2 (G) ≥ log2 (5) = 2.32...
• In fact Cap2 (G) = log2 (5) = 2.32...
• However Cap(G) = Cap4 (G) = 2.5
[Blasiak, Kleinberg, Lubetzky 13, Christofides, Markstrom 11]
LRC codes
4 / 15
LRC codes on graphs - Example
LRC codes on graphs - Example
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
LRC codes on graphs - Example
1, 5
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
LRC codes on graphs - Example
1, 2
1, 5
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
LRC codes on graphs - Example
1, 2
2, 3
1, 5
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
LRC codes on graphs - Example
1, 2
2, 3
1, 5
3, 4
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
LRC codes on graphs - Example
1, 2
2, 3
1, 5
3, 4
4, 5
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
LRC codes
4 / 15
LRC codes on graphs - Example
1, 2
2, 3
1, 5
3, 4
4, 5
• Cap4 (G) = 2.5 → 5 bits can be stored in the system {1, 2, 3, 4, 5}
• Cap(G) ≤ 2.5?
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
H(N1 , N2 , N3 , N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
H(N1 , N2 , N3 , N4 , N5 ) = H(N1 , N2 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
H(N1 , N2 , N3 , N4 , N5 ) = H(N1 , N2 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
= H(N1 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
H(N1 , N2 , N3 , N4 , N5 ) = H(N1 , N2 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
= H(N1 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
≤ H(N1 , N4 , N5 ) + H(N3 , N4 , N5 ) − H(N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
H(N1 , N2 , N3 , N4 , N5 ) = H(N1 , N2 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
= H(N1 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
≤ H(N1 , N4 , N5 ) + H(N3 , N4 , N5 ) − H(N4 , N5 )
= H(N1 , N4 ) + H(N3 , N5 ) − H(N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
H(N1 , N2 , N3 , N4 , N5 ) = H(N1 , N2 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
= H(N1 , N3 , N4 , N5 ) + H(N4 , N5 ) − H(N4 , N5 )
≤ H(N1 , N4 , N5 ) + H(N3 , N4 , N5 ) − H(N4 , N5 )
= H(N1 , N4 ) + H(N3 , N5 ) − H(N4 , N5 )
≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
H(N1 , N2 , N3 , N4 , N5 ) = H(N2 , N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
H(N1 , N2 , N3 , N4 , N5 ) = H(N2 , N4 , N5 )
= H(N4 , N5 ) + H(N2 |N4 , N5 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
H(N1 , N2 , N3 , N4 , N5 ) = H(N2 , N4 , N5 )
= H(N4 , N5 ) + H(N2 |N4 , N5 )
≤ H(N4 , N5 ) + H(N2 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N4 , N5 ) + H(N2 )
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N4 , N5 ) + H(N2 )
• ⇒ 2H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N2 ) + H(N3 ) + H(N4 ) + H(N5 ) ≤ 5
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N4 , N5 ) + H(N2 )
• ⇒ 2H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N2 ) + H(N3 ) + H(N4 ) + H(N5 ) ≤ 5
• ⇒ Cap(G) ≤ 2.5
LRC codes
4 / 15
LRC codes on graphs - Example
• H(X, Y, Z) + H(Z) ≤ H(X, Z) + H(Y, Z)
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N3 ) + H(N4 ) + H(N5 ) − H(N4 , N5 )
• H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N4 , N5 ) + H(N2 )
• ⇒ 2H(N1 , N2 , N3 , N4 , N5 ) ≤ H(N1 ) + H(N2 ) + H(N3 ) + H(N4 ) + H(N5 ) ≤ 5
• ⇒ Cap(G) ≤ 2.5
• Question: Given an arbitrary graph G how to calculate Cap(G)?
LRC codes
4 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
1. Exists i s.t. vi 6= ui and
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
1. Exists i s.t. vi 6= ui and
2. v|N(i) = u|N(i)
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
1. Exists i s.t. vi 6= ui and
2. v|N(i) = u|N(i)
• v, u ∈ Fnq can both be contained in a storage code ⇔ v ≁ u in Conf(G)
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
1. Exists i s.t. vi 6= ui and
2. v|N(i) = u|N(i)
• v, u ∈ Fnq can both be contained in a storage code ⇔ v ≁ u in Conf(G)
• C ⊆ Fnq is a storage code ⇔ C is an independent set of Conf(G)
LRC codes
5 / 15
The Confusion Graph [Bar-Yossef, Birk, Jayram, and Kol 2006]
Confusion Graph
• Given a graph G on n vertices and q ∈ N
• The confusion graph Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
1. Exists i s.t. vi 6= ui and
2. v|N(i) = u|N(i)
• v, u ∈ Fnq can both be contained in a storage code ⇔ v ≁ u in Conf(G)
• C ⊆ Fnq is a storage code ⇔ C is an independent set of Conf(G)
• Corollary:
Capq (G) = logq α(Conf(G)),
where α(·) is the independence number
LRC codes
5 / 15
Computing Cap(G)
• Capq (G) = logq α(Conf(G))
LRC codes
6 / 15
Computing Cap(G)
• Capq (G) = logq α(Conf(G))
• Computing the independence number is hard
LRC codes
6 / 15
Computing Cap(G)
• Capq (G) = logq α(Conf(G))
• Computing the independence number is hard
• Need to compute it for arbitrarily large confusion graphs (q → ∞)
LRC codes
6 / 15
Computing Cap(G)
• Capq (G) = logq α(Conf(G))
• Computing the independence number is hard
• Need to compute it for arbitrarily large confusion graphs (q → ∞)
• Bounds on Cap(G)?
LRC codes
6 / 15
Index Coding [Birk, Kol 1998]
? x2
x1 x2 x3
x1
?
x1 ? x3
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
x1 x2 x3
x1
?
x1 ? x3
• Side information A1 = {2}, A2 = {1, 3}, A3 = {1}
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Side information A1 = {2}, A2 = {1, 3}, A3 = {1}
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Side information A1 = {2}, A2 = {1, 3}, A3 = {1}
• Send x1 + x2 and x3
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Side information A1 = {2}, A2 = {1, 3}, A3 = {1}
• Send x1 + x2 and x3
• What is the fundamental limit on the number of transmissions?
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Side information A1 = {2}, A2 = {1, 3}, A3 = {1}
• Send x1 + x2 and x3
• What is the fundamental limit on the number of transmissions?
• Which coding scheme achieves the limit?
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Side information A1 = {2}, A2 = {1, 3}, A3 = {1}
• Send x1 + x2 and x3
• What is the fundamental limit on the number of transmissions?
• Which coding scheme achieves the limit?
• Many generalizations...
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• G = (V, E), |V| = n receivers
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• G = (V, E), |V| = n receivers
• Receiver i requests message xi ∈ Fq
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• G = (V, E), |V| = n receivers
• Receiver i requests message xi ∈ Fq
• An incoming edge represents the side information
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• G = (V, E), |V| = n receivers
• Receiver i requests message xi ∈ Fq
• An incoming edge represents the side information
• An index code of length L is
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• G = (V, E), |V| = n receivers
• Receiver i requests message xi ∈ Fq
• An incoming edge represents the side information
• An index code of length L is
1. An encoding function E : Fnq → FLq
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• G = (V, E), |V| = n receivers
• Receiver i requests message xi ∈ Fq
• An incoming edge represents the side information
• An index code of length L is
1. An encoding function E : Fnq → FLq
2. n decoding functions Di s.t. for any x ∈ Fnq
Di (E(x), x|N(i) ) = xi .
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Indexq (G) = the shortest length of an index code over Fq
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Indexq (G) = the shortest length of an index code over Fq
• Index(G) = infq Indexq (G)
LRC codes
7 / 15
Index Coding [Birk, Kol 1998]
? x2
• Side information graph G
1
x1 x2 x3
2
x1
3
?
x1 ? x3
• Indexq (G) = the shortest length of an index code over Fq
• Index(G) = infq Indexq (G) = limq→∞ Indexq (G)
LRC codes
7 / 15
Index Coding - Continued
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
Exists i s.t.vi 6= ui
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
Exists i s.t.vi 6= ui and v|N(i) = u|N(i)
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
Exists i s.t.vi 6= ui and v|N(i) = u|N(i)
• Fact: E(·) is an index code iff it is a proper coloring of Conf(G)
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
Exists i s.t.vi 6= ui and v|N(i) = u|N(i)
• Fact: E(·) is an index code iff it is a proper coloring of Conf(G)
Corollary
Given a graph G
LRC codes
8 / 15
Index Coding - Continued
• Index code: an encoding function E : Fnq → FLq
• The confusion of G, Conf(G) has
• Vertex set: Fnq
• Edge set: v = (v1 , ..., vn ) ∼ u = (u1 , ..., un ) if
Exists i s.t.vi 6= ui and v|N(i) = u|N(i)
• Fact: E(·) is an index code iff it is a proper coloring of Conf(G)
Corollary
Given a graph G
Indexq (G) = ⌈logq χ(Conf(G))⌉,
where χ(·) is the chromatic number
LRC codes
8 / 15
Duality between Storage Capacity and Index Coding
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Observations:
• Storage Capacity and Index coding are dual problems
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Observations:
• Storage Capacity and Index coding are dual problems
• upper bound on Index(G) ⇒ lower bound on Cap(G)
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Observations:
• Storage Capacity and Index coding are dual problems
• upper bound on Index(G) ⇒ lower bound on Cap(G)
• lower bound on Index(G) ⇒ upper bound on Cap(G)
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• χf = fractional chromatic number
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• χf = fractional chromatic number
• Hq = Confq (G) is a Cayley Graph of (Fnq , +)
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• χf = fractional chromatic number
• Hq = Confq (G) is a Cayley Graph of (Fnq , +) → is vertex transitive
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• χf = fractional chromatic number
• Hq = Confq (G) is a Cayley Graph of (Fnq , +) → is vertex transitive
⇒ α(Hq ) · χf (Hq ) = |V(Hq )| = qn
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• χf = fractional chromatic number
• Hq = Confq (G) is a Cayley Graph of (Fnq , +) → is vertex transitive
⇒ α(Hq ) · χf (Hq ) = |V(Hq )| = qn
⇒ lim logq α(Hq ) + lim logq χf (Hq ) = n
q→∞
q→∞
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• χf = fractional chromatic number
• Hq = Confq (G) is a Cayley Graph of (Fnq , +) → is vertex transitive
⇒ α(Hq ) · χf (Hq ) = |V(Hq )| = qn
⇒ lim logq α(Hq ) + lim logq χf (Hq ) = n
q→∞
q→∞
⇒ Cap(G) + lim logq χf (Hq ) = n
q→∞
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• Cap(G) + limq→∞ logq χf (Hq ) = n
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• Cap(G) + limq→∞ logq χf (Hq ) = n
• For any graph, and in particular, for Hq
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• Cap(G) + limq→∞ logq χf (Hq ) = n
• For any graph, and in particular, for Hq
χf (Hq ) ≤ χ(Hq ) ≤ χf (Hq ) · ln |V(Hq )|
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• Cap(G) + limq→∞ logq χf (Hq ) = n
• For any graph, and in particular, for Hq
χf (Hq ) ≤ χ(Hq ) ≤ χf (Hq ) · ln |V(Hq )|
⇒ lim logq χf (Hq ) = lim logq χ(Hq )
q→∞
q→∞
LRC codes
9 / 15
Duality between Storage Capacity and Index Coding
Theorem [Mazumdar 14, Shanmugam and Dimakis 14]
Let G = (V, E), |V| = n, then
Cap(G) + Index(G) = n
Proof:
• Cap(G) + limq→∞ logq χf (Hq ) = n
• For any graph, and in particular, for Hq
χf (Hq ) ≤ χ(Hq ) ≤ χf (Hq ) · ln |V(Hq )|
⇒ lim logq χf (Hq ) = lim logq χ(Hq ) = Index(G)
q→∞
q→∞
LRC codes
9 / 15
Bounds on Cap(G)
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• First proof: Index(G) ≥ α(G)
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• First proof: Index(G) ≥ α(G)
• Second Proof:
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• First proof: Index(G) ≥ α(G)
• Second Proof:
1. S ⊆ V(G), a maximal independent set
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• First proof: Index(G) ≥ α(G)
• Second Proof:
1. S ⊆ V(G), a maximal independent set
2. The information in S can be recovered by the nodes V(G)\S
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
• Theorem [Baber, Chistofides,Dang, Riis, Vaughan]:
There exists a graph G on 10 vertices s.t.
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
• Theorem [Baber, Chistofides,Dang, Riis, Vaughan]:
There exists a graph G on 10 vertices s.t.
• Shannon type inequalities give: Cap(G) ≤ 114/17 = 6.705...
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
• Theorem [Baber, Chistofides,Dang, Riis, Vaughan]:
There exists a graph G on 10 vertices s.t.
• Shannon type inequalities give: Cap(G) ≤ 114/17 = 6.705...
• Non-Shannon type inequalities give: Cap(G) ≤ 20/3 = 6.666...
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
• Theorem [Baber, Chistofides,Dang, Riis, Vaughan]:
There exists a graph G on 10 vertices s.t.
• Shannon type inequalities give: Cap(G) ≤ 114/17 = 6.705...
• Non-Shannon type inequalities give: Cap(G) ≤ 20/3 = 6.666...
Lower bounds:
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
• Theorem [Baber, Chistofides,Dang, Riis, Vaughan]:
There exists a graph G on 10 vertices s.t.
• Shannon type inequalities give: Cap(G) ≤ 114/17 = 6.705...
• Non-Shannon type inequalities give: Cap(G) ≤ 20/3 = 6.666...
Lower bounds:
• Index(G) ≤ min{χf (G), Minrank(G)}
LRC codes
10 / 15
Bounds on Cap(G)
• Any bound on Index(G) ⇒ a bound on Cap(G)
Upper Bounds:
• Cap(G) ≤ n − α(G) = VertexCover(G)
• Shannon type inequalities (Recall the proof for the pentagon)
• Non-Shannon type inequalities
• Theorem [Baber, Chistofides,Dang, Riis, Vaughan]:
There exists a graph G on 10 vertices s.t.
• Shannon type inequalities give: Cap(G) ≤ 114/17 = 6.705...
• Non-Shannon type inequalities give: Cap(G) ≤ 20/3 = 6.666...
Lower bounds:
• Index(G) ≤ min{χf (G), Minrank(G)}
• Cap(G) ≥ n − min{χf (G), Minrank(G)}
LRC codes
10 / 15
Guessing game on graphs [Riis2005]
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
LRC codes
(
0 if blue
1 if red
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
1. observes only the hats of his mates (complete graph)
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
1. observes only the hats of his mates (complete graph)
2. makes a guess on the color of his hat
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
1. observes only the hats of his mates (complete graph)
2. makes a guess on the color of his hat
• What is the probability that all players guessed correctly?
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
1. observes only the hats of his mates (complete graph)
2. makes a guess on the color of his hat
• What is the probability that all players guessed correctly?
• Random guess → Psuccess =
1 n
2
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
1. observes only the hats of his mates (complete graph)
2. makes a guess on the color of his hat
• What is the probability that all players guessed correctly?
• Random guess → Psuccess =
• Strategy: assume
Pn
j=1
xj = 0
1 n
2
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
• n players
• Red/Blue hat is assigned independently to each player. xj =
(
0 if blue
1 if red
• Each player:
1. observes only the hats of his mates (complete graph)
2. makes a guess on the color of his hat
• What is the probability that all players guessed correctly?
• Random guess → Psuccess =
• Strategy: assume
Pn
j=1
1 n
2
xj = 0 → Psuccess =
LRC codes
1
2
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Guessing game on graphs [Riis2005]
x1
x2
x3
• Arbitrary graphs?
LRC codes
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Guessing game on graphs [Riis2005]
x1
x2
x3
• Arbitrary graphs?
• Neighbors A1 = {2}, A2 = {1, 3}, A3 = {1}
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
x1
x2
x3
• Arbitrary graphs?
• Neighbors A1 = {2}, A2 = {1, 3}, A3 = {1}
• Fundamental limit on the amount of improvement over random guesses?
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
x1
x2
x3
• Arbitrary graphs?
• Neighbors A1 = {2}, A2 = {1, 3}, A3 = {1}
• Fundamental limit on the amount of improvement over random guesses?
• What is the optimal strategy?
LRC codes
11 / 15
Guessing game on graphs [Riis2005]
x1
x2
x3
• Arbitrary graphs?
• Neighbors A1 = {2}, A2 = {1, 3}, A3 = {1}
• Fundamental limit on the amount of improvement over random guesses?
• What is the optimal strategy?
• Connection to index coding [Yi–Sun–Jafar–Gesbert 2015]
LRC codes
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Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
LRC codes
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Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
LRC codes
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Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
• Let X = (x1 , ..., xn ), xi ∈ [q] be a hat assignment
LRC codes
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Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
• Let X = (x1 , ..., xn ), xi ∈ [q] be a hat assignment
• A strategy f = (f1 , ..., fn ) composed of n functions fi : (xj : j ∈ N(i)) → [q]
LRC codes
12 / 15
Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
• Let X = (x1 , ..., xn ), xi ∈ [q] be a hat assignment
• A strategy f = (f1 , ..., fn ) composed of n functions fi : (xj : j ∈ N(i)) → [q]
• The strategy is successful iff
LRC codes
12 / 15
Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
• Let X = (x1 , ..., xn ), xi ∈ [q] be a hat assignment
• A strategy f = (f1 , ..., fn ) composed of n functions fi : (xj : j ∈ N(i)) → [q]
• The strategy is successful iff
fi (xj : j ∈ N(i)) = xi for all i
LRC codes
12 / 15
Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
• Let X = (x1 , ..., xn ), xi ∈ [q] be a hat assignment
• A strategy f = (f1 , ..., fn ) composed of n functions fi : (xj : j ∈ N(i)) → [q]
• The strategy is successful iff
fi (xj : j ∈ N(i)) = xi for all i
• Equivalently: The hat assignment X is a fixed point for the mapping
X = (x1 , ..., xn )
LRC codes
12 / 15
Guessing Number [Riis2005]
• Given a (directed) graph G = (V, E), |V| = n
• Each player has a hat of q possible colors
• Let X = (x1 , ..., xn ), xi ∈ [q] be a hat assignment
• A strategy f = (f1 , ..., fn ) composed of n functions fi : (xj : j ∈ N(i)) → [q]
• The strategy is successful iff
fi (xj : j ∈ N(i)) = xi for all i
• Equivalently: The hat assignment X is a fixed point for the mapping
X = (x1 , ..., xn ) 7→ f (X) = (f1 (X), ..., fn (X))
fi (X) is a function of the hat colors of the neighbors of i
LRC codes
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Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
Prand
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
Prand
• For a fixed strategy f
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
Prand
• For a fixed strategy f
logq
P(f is successful)
= logq P(f is successful) + n
Prand
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
Prand
• For a fixed strategy f
logq
P(f is successful)
= logq P(f is successful) + n
Prand
| fixed points of f |
+n
= logq
qn
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
Prand
• For a fixed strategy f
logq
P(f is successful)
= logq P(f is successful) + n
Prand
| fixed points of f |
+n
= logq
qn
= logq | fixed points of f |
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
= max logq |fixed points of f |
f is a
Prand
strategy
• For a fixed strategy f
logq
P(f is successful)
= logq P(f is successful) + n
Prand
| fixed points of f |
+n
= logq
qn
= logq | fixed points of f |
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
= max logq |fixed points of f |
f is a
Prand
strategy
LRC codes
13 / 15
Guessing Number [Riis2005]
q-Guessing Number - Definition
The q-guessing number of G
Guessq (G) = max logq
f is a
strategy
P(f is successful)
= max logq |fixed points of f |
f is a
Prand
strategy
Guessing Number - Definition
The guessing number of G
Guess(G) = sup Guessq (G)
q
LRC codes
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Guessing games and LRC codes on graphs
LRC codes
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Guessing games and LRC codes on graphs
• Given a graph G
LRC codes
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
max
f is a strategy
LRC codes
logq |fixed points of f |
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
LRC codes
logq |C|
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
logq |C|
• The set of fixed points of a strategy forms a storage code C for G
LRC codes
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
logq |C|
• The set of fixed points of a strategy forms a storage code C for G
Guessq (G) ≤ Capq (G)
LRC codes
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
logq |C|
• The set of fixed points of a strategy forms a storage code C for G
Guessq (G) ≤ Capq (G)
• The codewords of a storage code form the set of fixed points of some strategy for
the hat game
LRC codes
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
logq |C|
• The set of fixed points of a strategy forms a storage code C for G
Guessq (G) ≤ Capq (G)
• The codewords of a storage code form the set of fixed points of some strategy for
the hat game
Capq (G) ≤ Guessq (G)
LRC codes
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
logq |C|
• The set of fixed points of a strategy forms a storage code C for G
Guessq (G) ≤ Capq (G)
• The codewords of a storage code form the set of fixed points of some strategy for
the hat game
Capq (G) ≤ Guessq (G)
• Guessq (G) = Capq (G)
LRC codes
14 / 15
Guessing games and LRC codes on graphs
• Given a graph G
Guessq (G) =
Capq (G) =
max
f is a strategy
max
logq |fixed points of f |
C⊆Fnq is a
storage code for G
logq |C|
• The set of fixed points of a strategy forms a storage code C for G
Guessq (G) ≤ Capq (G)
• The codewords of a storage code form the set of fixed points of some strategy for
the hat game
Capq (G) ≤ Guessq (G)
• Guessq (G) = Capq (G)
• Guess(G) = Cap(G)
LRC codes
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Open questions
LRC codes
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Open questions
• Derive improved bounds on Cap(G)
LRC codes
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Open questions
• Derive improved bounds on Cap(G)
• How expansion properties of the graph (expander graphs) affect the storage
capacity?
LRC codes
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Open questions
• Derive improved bounds on Cap(G)
• How expansion properties of the graph (expander graphs) affect the storage
capacity?
• Approximate the storage capacity using spectral properties of the graph
LRC codes
15 / 15
Open questions
• Derive improved bounds on Cap(G)
• How expansion properties of the graph (expander graphs) affect the storage
capacity?
• Approximate the storage capacity using spectral properties of the graph
• Given a graph G, characterize the tradeoff between the distance and the rate of
the code
LRC codes
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Open questions
• Derive improved bounds on Cap(G)
• How expansion properties of the graph (expander graphs) affect the storage
capacity?
• Approximate the storage capacity using spectral properties of the graph
• Given a graph G, characterize the tradeoff between the distance and the rate of
the code
• Networks with time varying topologies: Develop algorithms that efficiently shift
between storage codes for different topologies
LRC codes
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