Fundamentals of Coding for Network Coding
Marcus Greferath and Angeles Vazquez-Castro
Aalto University School of Science, Finland
Autonomous University of Barcelona, Spain
[email protected] and [email protected]
June 27, 2016
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Fundamentals of Coding for Network Coding
Part 1 – Introduction/overview
Part 2 – Fundamentals
Part 3 – From theory to practice
#"
Fundamentals of Coding for Network Coding
Part 1 – Introduction/overview
Part 2 – Fundamentals
Part 3 – From theory to practice
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Structure of the presentation
Network coding
Packet level
Physical level
Coding for network coding
Network error correction
Subspace coding
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Structure of the presentation
Network coding
Packet level
Physical level
Coding for network coding
Network error correction
Subspace coding
&"
Network graph model
o! Directed graph.
o! Source rate.
o! Edge e has capacity ce.
o! Edge receives flow fe ! ce.
o! The net flow into any non-source
non-sink vertex is zero (flow
conservation).
o! A cut between source s and a sink t
is a set of graph edges whose
removal disconnects s from t.
'"
Elias, Feinstein, and Shannon (1956)
Theorem. Max-flow Min-cut.
In an information flow network, the maximum amount of
information flow passing from a source s to a sink t is equal
to the capacity of the minimum cut-set.
o! Generalizes Menger’s theorem, 1927: maximum number
of edge-disjoint paths from s to t in a directed graph is
equal to the minimum s-t cut.
("
Elias, Feinstein, and Shannon (1956)
)"
Elias, Feinstein, and Shannon (1956)
o! Ford and Fulkerson proved it independently the same
year.
o! Ford-Fulkerson algorithm computes maximum s-t flow in
polynomial time.
o! Underlying assumptions:
–! Kirchhoff law.
–! Energy conservation law.
–! Information flow is a commodity.
*"
Commodity flows
o! An information flow just like:
o! Fluids in pipes.
o! Cars in a road system.
o! Electrical currents.
o! Etc.
!+"
Internet
o! Origins (60s)
–! Paul Baran (RAND Corp.): survivable networks of nuclear
war: "message-blocks".
–! Leonard Kleinrock (MIT) developed a mathematical theory
for packet-switching.
o! Internet has successfully evolved based on information
flow as a commodity:
–! Nodes store-and-forward information packets.
!!"
Ahlswede, Cai, Li, and Yeung (1998/2000)
Theorem. Multicast Max-flow Min-cut theorem.
The maximum amount of information flow passing from a
source s to every ti multicast destination is equal to the
minimum value among all cut-sets.
o! A multicast flow pattern can be found iff nodes do not
only store-and-forward, but do “network coding”.
o! Hence, information flow is NOT a commodity.
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Butterfly network: simple experimental test
Performance metrics
o!
o!
Average throughput: Average number of packets per second at
every multicast receiver.
Average delay: Average delay for a receiver to obtain packets
generated at the same time from the source.
Experiment
o!
o!
o!
o!
o!
ce : 1 packet/sec.
": 2 packets/sec.
Delay: 0 ms.
IP packet size.
Averaging over 1000 sec.
!$"
Butterfly network: routing
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Butterfly network: network coding
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Butterfly network: simulation results
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Network coding advantage
o! Routing capacity #(R).
o! Coding capacity $(R).
o! Coding advantage is $(R)/#(R).
o! Routing is also a coding strategy, $(R) % #(R).
o! For directed butterfly, $(R)/#(R)=1.333.
o! For directed networks, coding advantage is unbounded.
o! For undirected networks, coding advantage is bounded
$(R)! 2&#(R).
!("
Li et Al. (2003) and Ho et Al. (2003/06)
o! Linear coding is enough to achieve multicast capacity.
o! Linear coding means for some mathematical
representation of packets
o! Random linear coding is enough to achieve multicast
capacity.
o! Random linear coding means: random selection of !ij.
!)"
Koetter, Medard (2002/03)
o! Algebraic matricial model.
o! Networks with non ergodic failures.
o! Provide solvability conditions in purely algebraic terms.
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Koetter, Medard (2002/03)
o! Solving linear relations between X and Z we get
Z=MX
where
o! X is the vector of input processes,
o! Z is the vector of output processes,
o! M is the transfer matrix and can be factorized.
M = ATBT.
#+"
Koetter, Medard (2002/03)%
Z=MX
Z=MX
M = ATBT
Z=MX
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Koetter, Medard (2002/03)
Lemma Let '[X1,X2, . . . , Xn] be the ring of polynomials
over an infinite field in variables X1,X2, . . . , Xn. For any
non-zero element f '[X1,X2, . . . , Xn] there exists an
infinite set of n-tuples (x1,x2, . . . , xn) 'n such that
f(x1,x2, . . . , xn) (0.
o! This lemma is relevant when considering the algebraic closure over
extension fields.
o! Assignments to the coefficients )e,e with det(M) (0, provides infinite
number of solutions.
##"
General benefits and cost
o!
o!
o!
o!
o!
Increases throughput, reduces delay.
Better use of network resources (efficiency).
Reduced computational complexity compared to routing.
Robustness to/detection of link failures.
Enables in-network engineering of packet flows.
o! Additional overhead due to coding coefficients.
o! Requires novel design paradigms and possibly gradual
implementation/deployment.
#$"
Examples
o! TCP/IP native leads to network congestion
o! NC enables faster TCP and lower congestion.
o! P2P file sharing/content distribution
o! Avalanche (Microsoft): BitTorrent-like with NC.
o! Big file to randomly coded small pieces .
o! Participants share coded pieces.
o! Higher throughput, easy scheduling, lower delay.
o! Instant messaging
o! Alternative to flooding (informative packets).
o! (Passive) network tomography
o! Topology inference.
o! Distributed storage.
#%"
Structure of the presentation
Network coding
Packet level
Physical level
Coding for network coding
Error sources
Operator linear channel
Subspace coding
#&"
Physical layer network coding
o! The concept of network coding can be applied at any
communication layer, including physical layer.
o! Bit level.
and
o! Signal level.
o! At signal level electromagnetic waves superimpose:
network coding performed by Nature.
#'"
Physical layer network coding
o! “Physical” due to isomorphism between strings of bits
and a finite field.
o! Additions of signals can be mapped to GF(2n) additions
of digital bit streams: interference becomes part of the
arithmetic operation in network coding.
o! Wireless traditional: interference is bad.
o! (Physical layer) network coding: interference is good.
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Wu, Chou, Kung, 2004
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Relaying vs. computation
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Amplify and Forward:
Transmit a scaled version of received signals.
Compress and Forward:
Compress and forward without decoding.
Decode and Forward:
Decode and re-encode incoming signals.
Compute and Forward:
Decode and transmit “optimal” functions/computations
of lattice signals.
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Lattice-based signal constellations
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Example: Katti et All. 2007 (COPE)
o! Well advertised proposal of “opportunistic” coding
protocol known as COPE.
o! A shim between MAC and IP layers, protocol sets the
nodes to a promiscuous mode.
o! Agnostic to protocols above/below.
$#"
Benefits/cost
o! Increases throughput, reduces delay.
o! Better use of wireless resources (efficiency).
o! Bigger gains than theoretical (protocol-dependant).
o! Requires novel design of signal constellations (rich
underlying structure) and protocols.
$$"
Beyond classic information
o! Network coding in quantum networks. Early results:
o! Hayashi, Iwama, Nishimura, Raymond, and Yamashita, 2007, showed that
network coding for quantum communication does not give any benefit on the
butterfly network.
o! Leung, Oppenheim, and Winter, 2006, generalized this result to more types
of networks.
o! Network coding in molecular networks. The molecules
diffuse in the environment (broadcasting).
o! A. Aijaz, H. Aghvami, and M. R. Nakhai, “On error performance of network
coding in diffusion-based molecular nanonetworks,” Nanotechnology, IEEE
transactions on, vol. 13, no. 5, pp. 871–874, 2014.
$%"
Structure of the presentation
Network coding
Packet level
Physical level
Coding for network coding
Network error correction
Subspace coding
$&"
Types of errors in network-coded networks
o! Random errors
o! Tackled classically on a link-by-link basis.
o! Either errors are corrected or the packet is discarded.
o! Erasure errors
o! Due to underlying protocols (e.g. congestion).
o! Malicious nodes
o! Packets may be altered by malicious nodes.
o! Exogenous packets may be injected to interfere communication.
o! Errors in headers
o! Any error in the header may cause crucial information to be lost.
$'"
Cai and Yeung (2006, part I and part II)
o! Classical error correction codes: redundancy in the time
domain, link-by-link.
o! Network error correction: redundancy in the space
domain as well, across links.
o! The work generalizes classic coding theory results:
singleton bound, Hamming bound and Gilbert-Vashamov
bound.
o! Linear network coding with errors:
$("
Network error correction
Different approaches
o! Noncoherent single-source multicast
o! Jaggi, Langberg, Katti, Ho, Katabi, Médard 07.
o! Koetter & Kschischang, 08, 10.
o! Montanari & Urbanke, 13.
o! Coherent and noncoherent multi-source multicast
o!Vyetrenko, Ho,Effros, Kliewer & Erez 09.
$)"
Dana, Palanki, Hassibi, Effros, 2006
o! Cut-capacity in terms of wireless broadcast property.
o! When erasure pattern is known at receiver nodes, the
capacity region coincides with the cutset bound and is
achievable with network coding.
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Structure of the presentation
Network coding
Packet level
Physical level
Coding for network coding
Network error correction
Subspace coding
%+"
Kötter and Kschischang (2008)
R. Kötter "
F. Kschischang "
Danilo Silva"
1.! R. Kötter and F. R. Kschischang, “Coding for errors and erasures in random network coding,”
IEEE Trans. Inf. Theory, 2008.
2.! D. Silva, F. R. Kschischang, and R. Kötter, “A rank-metric approach to error control in
random network coding,” IEEE Trans. Inf. Theory, 2008.
3.! D. Silva, F. R. Kschischang, and R. Kötter, “Communication over finite-field matrix channels”,
Information Theory, IEEE Trans. Inf. Theory, 2010.
%!"
Kötter, Silva, Kschischang, 2008-2011
o! Two separate problems (“non-coherent”):
o! Network coding problem.
o! Error correction problem.
o! Network matricial channel (linear operator channel)
o! Adversarial error model.
From F. Kschischang’s talk at COST Action IC
1104 Training School on Network coding. &
%#"
Kötter, Silva, Kschischang, 2008-2011
o! Observe:
o! Linear network codes achieve multicast capacity (large q).
o! Linear network codes can be chosen randomly.
o! Sender transmits X, the receiver collects Y=AX.
o! Left multiplication by A performs row operations on X.
o! The rows of AX lie in the subspace spanned by X.
o! The transmitter encodes information via the selection of
a vector space from some appropriate codebook of
subspaces.
From F. Kschischang’s talk at COST Action IC
1104 Training School on Network coding. &
%$"
Classic coding vs. subspace coding
o! Classical algebraic coding theory
o! Transmitter: emits a vector, an element of Fqn.
o! Receiver: receives a vector, possibly corrupted by noise.
o! Code design goal: to construct a large collection of wellseparated vectors according to some metric (e.g., Hamming).
o! Algebraic subspace coding
o! Transmitter: emits a vector space, an element of Pq(n).
o! Receiver: receives a vector space, possibly corrupted by noise.
o! Code design goal: to construct a large collection of wellseparated vector spaces according to some metric.
From F. Kschischang’s talk at COST Action IC
1104 Training School on Network coding. &
%%"
Rank metric codes
o! Let A and B be two n x m matrices over Fq. The distance
between A and B is defined as d(A,B)= rank(A-B).
o! Delsarte codes: a set of n x m matrices in Fqnxm.
o! Gabidulin codes: a set of n x m matrices in Fnqm.
o! For a minimum rank distance of a rank-metric code d,
the Singleton bound is
%&"
Lifted Gabidulin codes
o! Codes that achieve the bound are called maximum rankdistance (MRD) codes. Linear MRD codes are known to
exist for all choices of parameters q, n, m.
o! A simple construction of (constant dimension) subspace
codes is the so-called lift of Gabidulin codes (CG).
%'"
More constructions of subspace codes
o! A large number of subspace codes constructions have been
developed within the framework of COST Action IC 1104
Random Network Coding and Designs over GF(q).
o! Novel construction approaches, e.g.:
o!
o!
o!
o!
With certain automorphism group. Etzion, Vardy, 2011.
Spread codes. Manganiello, Gorla, Rosenthal, 2008.
Orbit codes. Magianello, Trautmann, Rosenthal, 2011.
Based on Schubert Calculus and Plücker coordinates.
Trautmann, Silberstein, Rosenthal, 2013.
o! Multilevel construction. Etzion, Silberstein, 2009.
o! Constructions based on q-analog designs (Part 2 of this tutorial).
Full list at www.network-coding.eu/
%("
Fundamentals of Coding for Network Coding
End of Part 1 – Introduction/overview
%)"