Isometry and Automorphisms of Constant Dimension Codes

Home , Isometry

Anna-Lena Trautmann

Institute of Mathematics University of Zurich

“Crypto and Coding” Z¨urich, March 12th 2012

1 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction

1 Introduction

2 Isometry of Random Network Codes

3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes

2 / 25 isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes

Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation

constant dimension codes are used for random network coding

3 / 25 automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes

Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation

constant dimension codes are used for random network coding isometry classes are equivalence classes

3 / 25 automorphism groups are canonical representative of orbit codes

Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation

constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding

3 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction Motivation

constant dimension codes are used for random network coding isometry classes are equivalence classes automorphism groups of linear codes are useful for decoding automorphism groups are canonical representative of orbit codes

3 / 25 Deﬁnition Subspace metric:

dS(U, V) = dim(U + V) − dim(U ∩ V)

Injection metric:

dI (U, V) = max(dim U, dim V) − dim(U ∩ V)

Isometry and Automorphisms of Constant Dimension Codes Introduction Random Network Codes

Deﬁnition n n The projective geometry P(Fq ) is the set of all subspaces of Fq . n A random network code is a subset of P(Fq ).

4 / 25 Isometry and Automorphisms of Constant Dimension Codes Introduction Random Network Codes

Deﬁnition n n The projective geometry P(Fq ) is the set of all subspaces of Fq . n A random network code is a subset of P(Fq ).

Deﬁnition Subspace metric:

dS(U, V) = dim(U + V) − dim(U ∩ V)

Injection metric:

dI (U, V) = max(dim U, dim V) − dim(U ∩ V)

4 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

1 Introduction

2 Isometry of Random Network Codes

3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes

5 / 25 Any isometry ι is injective:

U 6= V ⇐⇒ d(U, V) 6= 0 ⇐⇒ d(ι(U), ι(V)) 6= 0 ⇐⇒ ι(U) 6= ι(V)

and hence, if the domain is equal to the codomain, bijective. The inverse map ι−1 is an isometry as well.

Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Deﬁnition n n A distance-preserving map ι : P(Fq ) → P(Fq ) i.e. fulﬁlling

n d(U, V) = d(ι(U), ι(V)) ∀ U, V ∈ P(Fq ).

n is called an isometry on P(Fq ).

6 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Deﬁnition n n A distance-preserving map ι : P(Fq ) → P(Fq ) i.e. fulﬁlling

n d(U, V) = d(ι(U), ι(V)) ∀ U, V ∈ P(Fq ).

n is called an isometry on P(Fq ).

Any isometry ι is injective:

U 6= V ⇐⇒ d(U, V) 6= 0 ⇐⇒ d(ι(U), ι(V)) 6= 0 ⇐⇒ ι(U) 6= ι(V)

and hence, if the domain is equal to the codomain, bijective. The inverse map ι−1 is an isometry as well.

6 / 25 Lemma

n Let ι be as before and U ∈ P(Fq ) arbitrary. Then

ι({0}) = {0} =⇒ dim(U) = d({0}, U) = d({0}, ι(U)) = dim(ι(U))

and on the other hand

n n ι({0}) = Fq =⇒ dim(U) = d({0}, U) = d(Fq , ι(U)) = n−dim(ι(U)).

The isometries with ι({0}) = {0} are exactly the isometries that keep the dimension of a codeword.

Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Lemma n n n If ι : P(Fq ) → P(Fq ) is an isometry, then ι({0}) ∈ {0}, Fq .

7 / 25 The isometries with ι({0}) = {0} are exactly the isometries that keep the dimension of a codeword.

Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Lemma n n n If ι : P(Fq ) → P(Fq ) is an isometry, then ι({0}) ∈ {0}, Fq .

Lemma

n Let ι be as before and U ∈ P(Fq ) arbitrary. Then

ι({0}) = {0} =⇒ dim(U) = d({0}, U) = d({0}, ι(U)) = dim(ι(U))

and on the other hand

n n ι({0}) = Fq =⇒ dim(U) = d({0}, U) = d(Fq , ι(U)) = n−dim(ι(U)).

7 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Lemma n n n If ι : P(Fq ) → P(Fq ) is an isometry, then ι({0}) ∈ {0}, Fq .

Lemma

n Let ι be as before and U ∈ P(Fq ) arbitrary. Then

ι({0}) = {0} =⇒ dim(U) = d({0}, U) = d({0}, ι(U)) = dim(ι(U))

and on the other hand

n n ι({0}) = Fq =⇒ dim(U) = d({0}, U) = d(Fq , ι(U)) = n−dim(ι(U)).

The isometries with ι({0}) = {0} are exactly the isometries that keep the dimension of a codeword.

7 / 25 Theorem n n For n > 2 a map ι : P(Fq ) → P(Fq ) is an order-preserving n bijection (with respect to the subset relation) of P(Fq ) if and only if it is an isometry with ι({0}) = {0}.

Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Theorem (Fundamental Theorem of Projective Geometry) n n Every order-preserving bijection f : P(Fq ) → P(Fq ), where n > 2, is induced by a semilinear transformation (A, α) ∈

PΓLn = (GLn/Zn) o Aut(Fq)

∗ where Zn = {µIn | µ ∈ Fq} is the set of scalar transformations.

8 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Theorem (Fundamental Theorem of Projective Geometry) n n Every order-preserving bijection f : P(Fq ) → P(Fq ), where n > 2, is induced by a semilinear transformation (A, α) ∈

PΓLn = (GLn/Zn) o Aut(Fq)

∗ where Zn = {µIn | µ ∈ Fq} is the set of scalar transformations.

Theorem n n For n > 2 a map ι : P(Fq ) → P(Fq ) is an order-preserving n bijection (with respect to the subset relation) of P(Fq ) if and only if it is an isometry with ι({0}) = {0}.

8 / 25 Remark: 1 All isometric codes are equivalent from a coding point of view (i.e. same rate and error correction capability). 2 There are more equivalence maps than order-preserving isometries.

Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Deﬁnition n 1 Two codes C1, C2 ⊆ P(Fq ) are linearly isometric if there exists A ∈ PGLn (or GLn) such that C1 = C2A.

2 We call C1 and C2 semilinearly isometric if there exists (A, α) ∈ PΓLn (or ΓLn) such that C1 = C2(A, α).

9 / 25 2 There are more equivalence maps than order-preserving isometries.

Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Deﬁnition n 1 Two codes C1, C2 ⊆ P(Fq ) are linearly isometric if there exists A ∈ PGLn (or GLn) such that C1 = C2A.

2 We call C1 and C2 semilinearly isometric if there exists (A, α) ∈ PΓLn (or ΓLn) such that C1 = C2(A, α).

Remark: 1 All isometric codes are equivalent from a coding point of view (i.e. same rate and error correction capability).

9 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry of Random Network Codes

Deﬁnition n 1 Two codes C1, C2 ⊆ P(Fq ) are linearly isometric if there exists A ∈ PGLn (or GLn) such that C1 = C2A.

2 We call C1 and C2 semilinearly isometric if there exists (A, α) ∈ PΓLn (or ΓLn) such that C1 = C2(A, α).

Remark: 1 All isometric codes are equivalent from a coding point of view (i.e. same rate and error correction capability). 2 There are more equivalence maps than order-preserving isometries.

9 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions

1 Introduction

2 Isometry of Random Network Codes

3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes

10 / 25 Deﬁnition

The Grassmannian Gq(k, n) is the set of all k-dimensional n subspaces of Fq .A constant dimension code is a subset of Gq(k, n).

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions

Deﬁnition n For a given code C ⊆ P(Fq ),

Aut(C) = {A ∈ GLn|CA = C}

is called the (linear) automorphism group of the code.

11 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions

Deﬁnition n For a given code C ⊆ P(Fq ),

Aut(C) = {A ∈ GLn|CA = C}

is called the (linear) automorphism group of the code.

Deﬁnition

The Grassmannian Gq(k, n) is the set of all k-dimensional n subspaces of Fq .A constant dimension code is a subset of Gq(k, n).

11 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

1 Introduction

2 Isometry of Random Network Codes

3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes

12 / 25 l Proof: Since there is only one spread of lines in Fqk , different n Desarguesian spreads of Fq can only arise from the different k isomorphisms between Fqk and Fq . As the isomorphisms are linear maps, there exists a linear map between the different spreads arising from them.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

Theorem All Desarguesian spread codes are linearly isometric.

13 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

Theorem All Desarguesian spread codes are linearly isometric.

l Proof: Since there is only one spread of lines in Fqk , different n Desarguesian spreads of Fq can only arise from the different k isomorphisms between Fqk and Fq . As the isomorphisms are linear maps, there exists a linear map between the different spreads arising from them.

13 / 25 Proof: Let l := n/k. We want to ﬁnd all Fq-linear bijections of l−1 k P (Fqk ). We know that PGLl(q ) is the groups of all l−1 Fqk -linear bijections of P (Fqk ) and that Aut(Fqk ) is the set of all automorphisms of Fqk that stabilize Fq. Thus, k PGLl(q ) × Aut(Fqk ) is the set of all Fq-linear bijections of l−1 P (Fqk ). It follows that in the aﬃne space the linear automorphism group of such a spread is isomorphic to k GLl(q ) × Aut(Fqk ).

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

Theorem The linear automorphism group of a Desarguesian spread code k C ⊆ Gq(k, n) is isomorphic to GL n (q ) × Aut( k ). k Fq

14 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

Theorem The linear automorphism group of a Desarguesian spread code k C ⊆ Gq(k, n) is isomorphic to GL n (q ) × Aut( k ). k Fq

Proof: Let l := n/k. We want to ﬁnd all Fq-linear bijections of l−1 k P (Fqk ). We know that PGLl(q ) is the groups of all l−1 Fqk -linear bijections of P (Fqk ) and that Aut(Fqk ) is the set of all automorphisms of Fqk that stabilize Fq. Thus, k PGLl(q ) × Aut(Fqk ) is the set of all Fq-linear bijections of l−1 P (Fqk ). It follows that in the aﬃne space the linear automorphism group of such a spread is isomorphic to k GLl(q ) × Aut(Fqk ).

14 / 25 Another point of view: the generator matrices of the code words are of the type U = B1 B2 ...Bl

where the blocks Bi are an element of Fq[P ]. To stay inside this structure (i.e. to apply an automorphism) we can permute the blocks, do block-wise multiplications or do block-wise additions with elements from Fq[P ]. This coincides with the structure of the automorphism groups from before.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

Corollary The automorphism group of a Desarguesian spread code in Gq(k, n) is generated by all elements in GLn where the k × k-blocks are elements of Fq[P ] and block diagonal matrices where the blocks represent an automorphism of Fqk .

15 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes

Another point of view: the generator matrices of the code words are of the type U = B1 B2 ...Bl

15 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Spread codes Example

Consider G3(2, 4) and the irreducible polynomial p(x) = x2 + x + 2, i.e.

0 1 P = 1 2

C = rs I 0 ∪ {rs IP i | i = 0,..., 7} ∪ rs 0 I Its automorphism group has 11520 elements:

I I IP Q Aut(C) = , , , I P I Q

1 0 where Q = ∈ GL2. Here Q represents the only 2 2 3 non-trivial automorphism of F32 , i.e. x 7→ x .

16 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes

1 Introduction

2 Isometry of Random Network Codes

3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes

17 / 25 Two lifted Fqk -linearly isometric codes in the rank-metric space are not automatically linearly isometric in the Grassmannian!

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes

Lemma (Berger)

1 m The set of Fqk -linear isometries on Fqk equipped with the lin m ∗ rank-metric is R (Fqk ) := GLm(q) × Fqk . 2 m The set of Fqk -semilinear isometries on Fqk equipped with the rank-metric is semi m ∗ R (Fqk ) := GLm(q) × Fqk o Aut(Fqk ).

18 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes

Lemma (Berger)

Two lifted Fqk -linearly isometric codes in the rank-metric space are not automatically linearly isometric in the Grassmannian!

18 / 25 Proof: It holds that I { I B | B ∈ C } k = { I BR | B ∈ C }. k R R k R

Since R ∈ AutR(CR), this set is equal to the original one.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes

We can use the knowledge of the automorphism group of a rank-metric code for ﬁnding the automorphism group of the respective lifted rank-metric code. Theorem

k×(n−k) Let CR ⊆ F be a rank-metric code and C its lifted code. Then I k | R ∈ Aut (C ) ⊆ Aut(C). R R R

19 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes

We can use the knowledge of the automorphism group of a rank-metric code for ﬁnding the automorphism group of the respective lifted rank-metric code. Theorem

k×(n−k) Let CR ⊆ F be a rank-metric code and C its lifted code. Then I k | R ∈ Aut (C ) ⊆ Aut(C). R R R

Proof: It holds that I { I B | B ∈ C } k = { I BR | B ∈ C }. k R R k R

Since R ∈ AutR(CR), this set is equal to the original one.

19 / 25 Proof: I rs I B k = rs I B k 1 A k 2 I ⇐⇒ ∃C ,C ∈ GLk : C C B k = C C B 1 2 1 1 1 A 2 2 2

⇐⇒ C1 = C2 ∧ B1A = B2 I i.e. if k ∈ Aut(C), then A ∈ Aut (C ). ⊇ is clear. A R R =⇒ If we know the automorphism group of a lifted rank-metric code, we also know the automorphism group of the rank-metric code itself.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes Theorem

k×(n−k) Let CR ⊆ F be a rank-metric code and C its lifted code. I I k ∈ Aut(C) = k | R ∈ Aut (C ) . A R R R

20 / 25 =⇒ If we know the automorphism group of a lifted rank-metric code, we also know the automorphism group of the rank-metric code itself.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes Theorem

k×(n−k) Let CR ⊆ F be a rank-metric code and C its lifted code. I I k ∈ Aut(C) = k | R ∈ Aut (C ) . A R R R

Proof: I rs I B k = rs I B k 1 A k 2 I ⇐⇒ ∃C ,C ∈ GLk : C C B k = C C B 1 2 1 1 1 A 2 2 2

⇐⇒ C1 = C2 ∧ B1A = B2 I i.e. if k ∈ Aut(C), then A ∈ Aut (C ). ⊇ is clear. A R R

20 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes Theorem

k×(n−k) Let CR ⊆ F be a rank-metric code and C its lifted code. I I k ∈ Aut(C) = k | R ∈ Aut (C ) . A R R R

Proof: I rs I B k = rs I B k 1 A k 2 I ⇐⇒ ∃C ,C ∈ GLk : C C B k = C C B 1 2 1 1 1 A 2 2 2

⇐⇒ C1 = C2 ∧ B1A = B2 I i.e. if k ∈ Aut(C), then A ∈ Aut (C ). ⊇ is clear. A R R =⇒ If we know the automorphism group of a lifted rank-metric code, we also know the automorphism group of the

rank-metric code itself. 20 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Lifted rank-metric codes

Example 1 0 1 1 0 1 0 0 C = , , , R 0 1 0 1 0 1 0 1 1 b Aut (C ) = | b ∈ R R 0 1 F2  1 0 0 0   1 0 0 0  *  0 1 0 0   0 1 0 0  lifted code: Aut(C) =   ,   ,  0 1 1 0   0 0 1 1  0 0 0 1 0 0 0 1  1 0 0 0   1 1 0 0  +  0 0 1 0   0 1 0 0    ,    0 1 0 0   0 1 1 0  0 1 1 1 0 0 0 1

21 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Orbit codes

1 Introduction

2 Isometry of Random Network Codes

3 Isometry and Automorphisms of Known Code Constructions Spread codes Lifted rank-metric codes Orbit codes

22 / 25 Proof: If C = UG, then CG = UGG = UG. Let G be a generating group of C and G ≤ H ≤ Aut(C). Hence, C = UG and CH = C. This implies that UH = UGH = CH = C, since G is a subgroup of H.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Orbit codes

Theorem Every generating group of an orbit code is a subgroup of the automorphism group. Every subgroup of the automorphism group containing a generating group is a generating group. Hence, the automorphism group is a generating group of the orbit code.

23 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Orbit codes

Proof: If C = UG, then CG = UGG = UG. Let G be a generating group of C and G ≤ H ≤ Aut(C). Hence, C = UG and CH = C. This implies that UH = UGH = CH = C, since G is a subgroup of H.

23 / 25 Proof:

A ∈ Aut(C) ⇐⇒ CA = C ⇐⇒ ∀B0 ∈ G ∃B∗ ∈ G : UB0A = UB∗ ⇐⇒ ∀B0 ∈ G ∃B∗ ∈ G : UB0AB∗−1 = U

The statement follows with B00 := B∗−1 ∈ G.

Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Orbit codes

Theorem

A ∈ GLn is in the automorphism group of C = UG if and only if 0 00 for every B ∈ GLn there exists a B ∈ GLn such that

0 00 B AB ∈ StabGLn (U).

24 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Orbit codes

Theorem

A ∈ GLn is in the automorphism group of C = UG if and only if 0 00 for every B ∈ GLn there exists a B ∈ GLn such that

0 00 B AB ∈ StabGLn (U).

Proof:

A ∈ Aut(C) ⇐⇒ CA = C ⇐⇒ ∀B0 ∈ G ∃B∗ ∈ G : UB0A = UB∗ ⇐⇒ ∀B0 ∈ G ∃B∗ ∈ G : UB0AB∗−1 = U

The statement follows with B00 := B∗−1 ∈ G.

24 / 25 Isometry and Automorphisms of Constant Dimension Codes Isometry and Automorphisms of Known Code Constructions Orbit codes

Merci vielmal.

25 / 25