A Complete Characterization of Irreducible Cyclic Orbit Codes
A Complete Characterization of Irreducible
Cyclic Orbit Codes
Anna-Lena Trautmann
Institute of Mathematics
University of Zurich
The Seventh International Workshop on Coding and
Cryptography
Paris, April 11-15 2011
joint work with Joachim Rosenthal
A Complete Characterization of Irreducible Cyclic Orbit Codes
Outline
1
Random Linear Network Coding
2
Orbit Codes and Subgroups of GLn
3
ICOCs in Extension Field Representation
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Linear Network Coding
channel
sources
sinks
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Linear Network Coding
sources
inner nodes
sinks
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
When sending information through a network we can optimize
the throughput by doing linear combinations on the
intermediate nodes.
Example (The Butterfly Network):
S1
R1
a
a
a
b
a
a
b
R2
S2
Received:
R1 : (a, a)
,
R2 : (a, b)
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
When sending information through a network we can optimize
the throughput by doing linear combinations on the
intermediate nodes.
Example (The Butterfly Network):
S1
R1
a
a
a+b
b
a+b
a+b
b
R2
S2
Received:
R1 : (a, a + b) ,
R2 : (b, a + b)
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Random (linear) network coding
Use: dynamically varying connections, large networks,
unknown network topology, network security, etc.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Random (linear) network coding
Use: dynamically varying connections, large networks,
unknown network topology, network security, etc.
Inner nodes forward a random linear combination of the
incoming information.
Choose linear subspaces of Fnq as codewords since these
stay invariant under linear operations on the basis vectors.
It it useful to restrict to constant dimension codes (CDC),
i.e. subsets of the Grassmannian Gq (k, n).
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Random (linear) network coding
Use: dynamically varying connections, large networks,
unknown network topology, network security, etc.
Inner nodes forward a random linear combination of the
incoming information.
Choose linear subspaces of Fnq as codewords since these
stay invariant under linear operations on the basis vectors.
It it useful to restrict to constant dimension codes (CDC),
i.e. subsets of the Grassmannian Gq (k, n).
Represent the codewords as row spaces of k × n matrices
over Fq .
Unique representation is the reduced row echelon form.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Definition
A metric on the Grassmannian Gq (k, n) is the subspace distance:
dS (U, V ) := dim(U + V ) − dim(U ∩ V )
=2(k − dim(U ∩ V ))
A Complete Characterization of Irreducible Cyclic Orbit Codes
Random Linear Network Coding
Definition
A metric on the Grassmannian Gq (k, n) is the subspace distance:
dS (U, V ) := dim(U + V ) − dim(U ∩ V )
=2(k − dim(U ∩ V ))
If the distance between any two elements of a CDC is greater
than or equal to 2d we say that the code has minimum distance
dmin = 2d.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Orbit Codes
Let U ∈ M atk×n be a matrix representation of U ∈ Gq (k, n).
Then we have the following group operation from the right on
the Grassmannian:
Gq (k, n) × GLn −→
Gq (k, n)
(U, A)
7−→ UA := row space(U A)
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Orbit Codes
Let U ∈ M atk×n be a matrix representation of U ∈ Gq (k, n).
Then we have the following group operation from the right on
the Grassmannian:
Gq (k, n) × GLn −→
Gq (k, n)
(U, A)
7−→ UA := row space(U A)
Definition
Let U ∈ Gq (k, n) be fixed and G a subgroup of GLn . Then
C = {UA | A ∈ G}
is called an orbit code .
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Theorem
Gq (k, n) ∼
= GLn /Stab(U),
=⇒ It is possible that different subgroups generate the same
orbit code.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Theorem
Gq (k, n) ∼
= GLn /Stab(U),
=⇒ It is possible that different subgroups generate the same
orbit code.
Definition
If an orbit code can be defined by an irreducible cyclic subgroup
G ≤ GLn , it is called an irreducible cyclic orbit code.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Theorem
Gq (k, n) ∼
= GLn /Stab(U),
=⇒ It is possible that different subgroups generate the same
orbit code.
Definition
If an orbit code can be defined by an irreducible cyclic subgroup
G ≤ GLn , it is called an irreducible cyclic orbit code.
Theorem
Let C = {UA | A ∈ G} be an orbit code. Then
dmin (C) = min{dS (UA, UA′ ) | A, A′ ∈ G/Stab(U)}
= min{dS (U, UA) | A ∈ G/Stab(U)}.
(“Linearity”)
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Definition
A ∈ GLn (resp. G ≤ GLn ) is called irreducible if there is no
non-trivial A- (resp. G-) invariant subspace of Fnq .
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Definition
A ∈ GLn (resp. G ≤ GLn ) is called irreducible if there is no
non-trivial A- (resp. G-) invariant subspace of Fnq .
Theorem
A cyclic subgroup of GLn is irreducible if and only if its
generator matrix is irreducible.
An invertible matrix is irreducible if and only if its
characteristic polynomial is irreducible.
Irreducible invertible matrices with the same characteristic
polynomial are similar.
Irreducible cyclic subgroups of GLn are conjugate if and
only if the characteristic polynomials of their generators
have the same order.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Theorem
Let G be an irreducible matrix, G = hGi and H = hS −1 GSi for
an S ∈ GLn . Moreover, let U ∈ Gq (k, n) and V := US. Then
the orbit codes
C := {UA|A ∈ G} and C ′ := {VB|B ∈ H}
have the same cardinality and minimum distance.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Orbit Codes and Subgroups of GLn
Theorem
Let G be an irreducible matrix, G = hGi and H = hS −1 GSi for
an S ∈ GLn . Moreover, let U ∈ Gq (k, n) and V := US. Then
the orbit codes
C := {UA|A ∈ G} and C ′ := {VB|B ∈ H}
have the same cardinality and minimum distance.
=⇒ It is sufficient to characterize the orbits of cyclic groups
generated by companion matrices of irreducible polynomials of
degree n.
The results are then carried over to any irreducible cyclic orbit
code via the choice of starting point of the orbit.
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Extension Field Representation
Let α be a root of an irreducible polynomial p(x) of degree n
over Fq . Then
Fnq ∼
= Fq n ∼
= Fq [α].
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Extension Field Representation
Let α be a root of an irreducible polynomial p(x) of degree n
over Fq . Then
Fnq ∼
= Fq n ∼
= Fq [α].
If p(x) is primitive, then
Fq [α]\{0} = hαi = {αi |i = 0, ..., q n − 2}.
=⇒ hαi acts transitively on Fqn \{0}.
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Theorem
Let p(x) be an irreducible polynomial over Fq of degree n and P
its companion matrix. Furthermore let α ∈ Fqn be a root of p(x)
and φ be the canonical homomorphism
φ : Fnq −→ Fqn
n
X
vi αi−1 .
(v1 , . . . , vn ) 7−→
i=1
Then the following diagram commutes (for v ∈ Fnq ):
P
v −→ vP
φ↓
↓φ
v ′ −→ v ′ α
α
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Example
Over the binary field let p(x) := x4 + x + 1 be primitive, α a
root of p(x) and P its companion matrix:


0 1 0 0
 0 0 1 0 

P =
 0 0 0 1 
1 1 0 0
Vector space representation:
0 1 0 0
0 0 1 0
U = rs
=⇒ UP = rs
.
0 0 0 1
1 1 0 0
In field representation:
φ((0100)) = α
φ((0001)) = α3
=⇒
α·α
= α2 = φ((0010))
3
4
α · α = α = α + 1 = φ((1100))
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Theorem
Over Fq let p(x) be a primitive polynomial and α a root of it.
Assume U = {0, u1 , . . . , uqk −1 } ∈ Gq (k, n),
φ(ui ) = αbi
∀i = 1, . . . , q k − 1
and d < k be minimal such that any element of the set
{bm − bl
mod q n − 1|l, m ∈ Zqk −1 , l 6= m}
has multiplicity less than q d − 1, i.e. a quotient of two elements
in the field representation appears at most q d − 1 times in the
set of all pairwise quotients. Then the orbit of the group
generated by the companion matrix P of p(x) on U is an orbit
code of cardinality q n − 1 and minimum distance 2k − 2d.
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Example
Over the binary field let p(x) := x4 + x + 1 be primitive, α a
root of p(x) and P its companion matrix. Let
U = {0, u1 , u2 , u3 } be the starting point of the orbit with
u1 = (1000) = φ−1 (1) = φ−1 (α0 )
u2 = (0011) = φ−1 (α2 + α3 ) = φ−1 (α6 )
u3 = (1011) = φ−1 (1 + α2 + α3 ) = φ−1 (α13 )
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Example
Over the binary field let p(x) := x4 + x + 1 be primitive, α a
root of p(x) and P its companion matrix. Let
U = {0, u1 , u2 , u3 } be the starting point of the orbit with
u1 = (1000) = φ−1 (1) = φ−1 (α0 )
u2 = (0011) = φ−1 (α2 + α3 ) = φ−1 (α6 )
u3 = (1011) = φ−1 (1 + α2 + α3 ) = φ−1 (α13 )
The difference set of the exponents
mod 24 − 1 = 15 is
{6, 9, 13, 2, 7, 8}
where each element has multiplicity 1 = 21 − 1. Thus, the
minimum distance of the orbit code U hP i is
2k − 2d = 2 · 2 − 2 · 1 = 2.
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Theorem
If k|n, c :=
q n −1
q k −1
and α a primitive element of Fqn , then the
vector space generated by 1, αc , ..., α(k−1)c is equal to
{αic |i = 0, ..., q k − 2} ∪ {0} = Fqk .
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Theorem
If k|n, c :=
q n −1
q k −1
and α a primitive element of Fqn , then the
vector space generated by 1, αc , ..., α(k−1)c is equal to
{αic |i = 0, ..., q k − 2} ∪ {0} = Fqk .
Spread codes are CDCs with maximal distance 2k and
n −1
(hence optimal).
cardinality qqk −1
Theorem
The set
S = αi · Fqk | i = 0, . . . , c − 1
defines a spread code.
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Example
Over the binary field let p(x) := x6 + x + 1 be primitive, α a
root of p(x) and P its companion matrix. For the 3-dimensional
spread compute c = 63
7 = 9 and construct a basis for the
starting point of the orbit:
u1 = φ−1 (α0 ) = φ−1 (1) = (100000)
u2 = φ−1 (αc ) = φ−1 (α9 ) = φ−1 (α4 + α3 ) = (000110)
u3 = φ−1 (α2c ) = φ−1 (α18 ) = φ−1 (α3 + α2 + α + 1) = (111100)
A Complete Characterization of Irreducible Cyclic Orbit Codes
ICOCs in Extension Field Representation
Example
Over the binary field let p(x) := x6 + x + 1 be primitive, α a
root of p(x) and P its companion matrix. For the 3-dimensional
spread compute c = 63
7 = 9 and construct a basis for the
starting point of the orbit:
u1 = φ−1 (α0 ) = φ−1 (1) = (100000)
u2 = φ−1 (αc ) = φ−1 (α9 ) = φ−1 (α4 + α3 ) = (000110)
u3 = φ−1 (α2c ) = φ−1 (α18 ) = φ−1 (α3 + α2 + α + 1) = (111100)
The starting point

1 0
U = rs  0 0
1 1
is



1 0 0 0 0 0
0 0 0 0
0 1 1 0  = rs  0 1 1 0 1 0 
0 0 0 1 1 0
1 1 0 0
and the orbit of the group generated by P on U is a spread code.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Résumé and Outlook
1
Analogous result for non-primitive irreducible polynomials.
=⇒ For given irreducible cyclic group and vector space we
showed how to compute cardinality and minimum distance
via field representation.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Résumé and Outlook
1
Analogous result for non-primitive irreducible polynomials.
=⇒ For given irreducible cyclic group and vector space we
showed how to compute cardinality and minimum distance
via field representation.
2
The theory can be used to construct orbit codes for a given
cardinality and minimum distance (e.g. spread codes for
any k and n).
A Complete Characterization of Irreducible Cyclic Orbit Codes
Résumé and Outlook
1
Analogous result for non-primitive irreducible polynomials.
=⇒ For given irreducible cyclic group and vector space we
showed how to compute cardinality and minimum distance
via field representation.
2
The theory can be used to construct orbit codes for a given
cardinality and minimum distance (e.g. spread codes for
any k and n).
3
Use this theory to characterize all cyclic orbit codes.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Résumé and Outlook
1
Analogous result for non-primitive irreducible polynomials.
=⇒ For given irreducible cyclic group and vector space we
showed how to compute cardinality and minimum distance
via field representation.
2
The theory can be used to construct orbit codes for a given
cardinality and minimum distance (e.g. spread codes for
any k and n).
3
Use this theory to characterize all cyclic orbit codes.
4
Use the group structure for decoding algorithms.
A Complete Characterization of Irreducible Cyclic Orbit Codes
Thank you for your attention.