On the Kernel of Z2s -Linear Codes

Carlos Vela Cabellojoint work with Prof. C. Fernández-Córdoba and Prof. M. Villanueva

Department of Information and Communications EngineeringUniversitat Autònoma de Barcelona, Spain.

cristina.fernandez,carlos.vela,merce.villanueva@uab.cat

5th International Castle Meeting on Coding Theory and ApplicationsVihula Manor, EstoniaAugust 28-31, 2017.

Outline

1 Introduction

2 Construction of Z2s -Linear Hadamard Codes

3 Partial classification. Kernel.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 2 / 21

Binary Hadamard codes

A binary code C of length n is a non-empty subset of Zn2.

A binary code C of length n is a Hadamard code if,

C has 2n codewords andC has minimum distance n/2.

The minimum distance d , of a code C , ismindH(u, v) : u, v ∈ C , u 6= v. The Hamming distance dH betweenu, v ∈ Zn

2, dH(u, v) is the number of coordinates in which u and v differ.Linear binary Hadamard codes are also know as First Order Reed-Müllercodes

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 3 / 21

Binary Hadamard codes

A binary code C of length n is a non-empty subset of Zn2.

A binary code C of length n is a Hadamard code if,

C has 2n codewords andC has minimum distance n/2.

The minimum distance d , of a code C , ismindH(u, v) : u, v ∈ C , u 6= v. The Hamming distance dH betweenu, v ∈ Zn

2, dH(u, v) is the number of coordinates in which u and v differ.Linear binary Hadamard codes are also know as First Order Reed-Müllercodes

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 3 / 21

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 4 / 21

Z2s -Additive code

Let Z2s be the ring of integers modulo 2s with s ≥ 2. The set ofvectors of length n over Z2s is denoted by Zn

2s . A code over Z2s oflength n is a non-empty subset C of Zn

2s . If C has group structure,then. . .

Z2s -additive code.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 5 / 21

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 6 / 21

Z2s -Additive codes

Since C is a subgroup of Zn2s , it is isomorphic to an abelian

structure Zt12s × Zt2

2s−1 × · · · × Zts−14 × Zts

2 , and we say that C is oftype (n; t1, . . . , ts).The standard form of its generator matrix is

t1 generators of order 2s ,t2 generators of order 2s−1,

...ts generators of order 2.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 7 / 21

Gray map

The classical Gray map is φ : Z4 → Z22

φ(0) = (0, 0)φ(1) = (0, 1)φ(2) = (1, 1)φ(3) = (1, 0)

ISOMETRY

Let u ∈ Z2s . The generalized Gray map image of u is (Carlet, 1998),

φ(u) = (us , . . . , us) + (u1, . . . , us−1)Y ,

where:[u1, u2, . . . , us ]2 is the binary expansion of uY is a matrix of size (s − 1)× 2s−1 which columns are theelements of Zs−1

2 .but. . .

ISOMETRYC.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 8 / 21

Gray map

The classical Gray map is φ : Z4 → Z22

φ(0) = (0, 0)φ(1) = (0, 1)φ(2) = (1, 1)φ(3) = (1, 0)

ISOMETRY

Let u ∈ Z2s . The generalized Gray map image of u is (Carlet, 1998),

φ(u) = (us , . . . , us) + (u1, . . . , us−1)Y ,

where:[u1, u2, . . . , us ]2 is the binary expansion of uY is a matrix of size (s − 1)× 2s−1 which columns are theelements of Zs−1

2 .but. . .

ISOMETRYC.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 8 / 21

Gray map

The classical Gray map is φ : Z4 → Z22

φ(0) = (0, 0)φ(1) = (0, 1)φ(2) = (1, 1)φ(3) = (1, 0)

ISOMETRY

Let u ∈ Z2s . The generalized Gray map image of u is (Carlet, 1998),

φ(u) = (us , . . . , us) + (u1, . . . , us−1)Y ,

where:[u1, u2, . . . , us ]2 is the binary expansion of uY is a matrix of size (s − 1)× 2s−1 which columns are theelements of Zs−1

2 .but. . .

ISOMETRYC.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 8 / 21

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 9 / 21

We construct Z2s -Linear Hadamard codes!

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 10 / 21

We construct Z2s -Linear Hadamard codes!

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 10 / 21

We construct Z2s -Linear Hadamard codes!

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 10 / 21

We construct Z2s -Linear Hadamard codes!

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 10 / 21

Construction of Z2s -Linear Hadamard codes

How are their generator matrices?

Let Ti = j · 2s−i : j ∈ 0, 1, . . . , 2i − 1, for all i ∈ 1, . . . , s. Note thatTs = 0, . . . , 2s − 1.

Let t1, t2,. . . ,ts be nonnegative integers with t1 ≥ 1. Consider the matrixAt1,...,ts whose columns are of the form zT with

z ∈ 1 × T t1−1s × T t2

s−1 × · · · × T ts1 .

We can also construct these matrices, recursively, in the following way.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 11 / 21

Construction of Z2s -Linear Hadamard codes

How are their generator matrices?

Let Ti = j · 2s−i : j ∈ 0, 1, . . . , 2i − 1, for all i ∈ 1, . . . , s. Note thatTs = 0, . . . , 2s − 1.

Let t1, t2,. . . ,ts be nonnegative integers with t1 ≥ 1. Consider the matrixAt1,...,ts whose columns are of the form zT with

z ∈ 1 × T t1−1s × T t2

s−1 × · · · × T ts1 .

We can also construct these matrices, recursively, in the following way.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 11 / 21

Construction of Z2s -Linear Hadamard codes

How are their generator matrices?

Let Ti = j · 2s−i : j ∈ 0, 1, . . . , 2i − 1, for all i ∈ 1, . . . , s. Note thatTs = 0, . . . , 2s − 1.

Let t1, t2,. . . ,ts be nonnegative integers with t1 ≥ 1. Consider the matrixAt1,...,ts whose columns are of the form zT with

z ∈ 1 × T t1−1s × T t2

s−1 × · · · × T ts1 .

We can also construct these matrices, recursively, in the following way.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 11 / 21

Construction of Z2s -Linear Hadamard codes

How are their generator matrices?

Let Ti = j · 2s−i : j ∈ 0, 1, . . . , 2i − 1, for all i ∈ 1, . . . , s. Note thatTs = 0, . . . , 2s − 1.

Let t1, t2,. . . ,ts be nonnegative integers with t1 ≥ 1. Consider the matrixAt1,...,ts whose columns are of the form zT with

z ∈ 1 × T t1−1s × T t2

s−1 × · · · × T ts1 .

We can also construct these matrices, recursively, in the following way.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 11 / 21

Construction of Z2s -Linear Hadamard codes

A little algorithm:Start with the matrix A1,0,...,0 = (1).Given a matrix At1,...,ts we construct:

At′1,...,t′s =

(At1,...,ts At1,...,ts . . . At1,...,ts

0 · 2i−1 1 · 2i−1 . . . (2s−i+1 − 1) · 2i−1

)where i ∈ 1, . . . , s, t ′j = tj for j 6= i and t ′i = ti + 1.

TheoremLet t1, . . . , ts be nonnegative integers with t1 ≥ 1. The Z2s -linear codeΦ(Ht1,...,ts ) = Ht1,...,ts of type (n; t1, t2, . . . , ts) is a binary Hadamard codeof length 2t , with t = (

∑si=1(s − i + 1) · ti )− 1.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 12 / 21

Construction of Z2s -Linear Hadamard codes

A little algorithm:Start with the matrix A1,0,...,0 = (1).Given a matrix At1,...,ts we construct:

At′1,...,t′s =

(At1,...,ts At1,...,ts . . . At1,...,ts

0 · 2i−1 1 · 2i−1 . . . (2s−i+1 − 1) · 2i−1

)where i ∈ 1, . . . , s, t ′j = tj for j 6= i and t ′i = ti + 1.

TheoremLet t1, . . . , ts be nonnegative integers with t1 ≥ 1. The Z2s -linear codeΦ(Ht1,...,ts ) = Ht1,...,ts of type (n; t1, t2, . . . , ts) is a binary Hadamard codeof length 2t , with t = (

∑si=1(s − i + 1) · ti )− 1.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 12 / 21

Construction of Z2s -Linear Hadamard codes

A little algorithm:Start with the matrix A1,0,...,0 = (1).Given a matrix At1,...,ts we construct:

At′1,...,t′s =

(At1,...,ts At1,...,ts . . . At1,...,ts

0 · 2i−1 1 · 2i−1 . . . (2s−i+1 − 1) · 2i−1

)where i ∈ 1, . . . , s, t ′j = tj for j 6= i and t ′i = ti + 1.

TheoremLet t1, . . . , ts be nonnegative integers with t1 ≥ 1. The Z2s -linear codeΦ(Ht1,...,ts ) = Ht1,...,ts of type (n; t1, t2, . . . , ts) is a binary Hadamard codeof length 2t , with t = (

∑si=1(s − i + 1) · ti )− 1.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 12 / 21

Construction of Z2s -Linear Hadamard codes

A little algorithm:Start with the matrix A1,0,...,0 = (1).Given a matrix At1,...,ts we construct:

At′1,...,t′s =

(At1,...,ts At1,...,ts . . . At1,...,ts

0 · 2i−1 1 · 2i−1 . . . (2s−i+1 − 1) · 2i−1

)where i ∈ 1, . . . , s, t ′j = tj for j 6= i and t ′i = ti + 1.

TheoremLet t1, . . . , ts be nonnegative integers with t1 ≥ 1. The Z2s -linear codeΦ(Ht1,...,ts ) = Ht1,...,ts of type (n; t1, t2, . . . , ts) is a binary Hadamard codeof length 2t , with t = (

∑si=1(s − i + 1) · ti )− 1.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 12 / 21

Example of Construction (s = 3)

A1,0,0 = (1)

A1,0,1 =

(1 10 4

), A1,1,0 =

(11 1102 46

), A2,0,0 =

(11 11 11 1101 23 45 67

),

A1,1,1 =

11 11 11 1102 46 02 4600 00 44 44

, A2,0,1 =

11 11 11 11 11 11 11 1101 23 45 67 01 23 45 6700 00 00 00 44 44 44 44

,

A2,1,0 =

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 1101 23 45 67 01 23 45 67 01 23 45 67 01 23 45 6700 00 00 00 22 22 22 22 44 44 44 44 66 66 66 66

.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 13 / 21

Example of Construction (s = 3)

A1,0,0 = (1)

A1,0,1 =

(1 10 4

), A1,1,0 =

(11 1102 46

), A2,0,0 =

(11 11 11 1101 23 45 67

),

A1,1,1 =

11 11 11 1102 46 02 4600 00 44 44

, A2,0,1 =

11 11 11 11 11 11 11 1101 23 45 67 01 23 45 6700 00 00 00 44 44 44 44

,

A2,1,0 =

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 1101 23 45 67 01 23 45 67 01 23 45 67 01 23 45 6700 00 00 00 22 22 22 22 44 44 44 44 66 66 66 66

.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 13 / 21

Example of Construction (s = 3)

A1,0,0 = (1)

A1,0,1 =

(1 10 4

), A1,1,0 =

(11 1102 46

), A2,0,0 =

(11 11 11 1101 23 45 67

),

A1,1,1 =

11 11 11 1102 46 02 4600 00 44 44

, A2,0,1 =

11 11 11 11 11 11 11 1101 23 45 67 01 23 45 6700 00 00 00 44 44 44 44

,

A2,1,0 =

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 1101 23 45 67 01 23 45 67 01 23 45 67 01 23 45 6700 00 00 00 22 22 22 22 44 44 44 44 66 66 66 66

.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 13 / 21

Example of Construction (s = 3)

A1,0,0 = (1)

A1,0,1 =

(1 10 4

), A1,1,0 =

(11 1102 46

), A2,0,0 =

(11 11 11 1101 23 45 67

),

A1,1,1 =

11 11 11 1102 46 02 4600 00 44 44

, A2,0,1 =

11 11 11 11 11 11 11 1101 23 45 67 01 23 45 6700 00 00 00 44 44 44 44

,

A2,1,0 =

11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 1101 23 45 67 01 23 45 67 01 23 45 67 01 23 45 6700 00 00 00 22 22 22 22 44 44 44 44 66 66 66 66

.

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 13 / 21

Rank & Kernel

To classify the constructed codes we will use two structural properties ofbinary codes which are the rank and the dimension of the kernel.

The rank of a binary code C is the dimension of the linear span, 〈C 〉, of C .

The kernel of a binary code C is define as K (C ) = x ∈ Zn2 : x + C = C.

If the all-zero vector belongs to C , then K (C ) is a linear subcode of C .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 14 / 21

Rank & Kernel

To classify the constructed codes we will use two structural properties ofbinary codes which are the rank and the dimension of the kernel.

The rank of a binary code C is the dimension of the linear span, 〈C 〉, of C .

The kernel of a binary code C is define as K (C ) = x ∈ Zn2 : x + C = C.

If the all-zero vector belongs to C , then K (C ) is a linear subcode of C .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 14 / 21

Rank & Kernel

To classify the constructed codes we will use two structural properties ofbinary codes which are the rank and the dimension of the kernel.

The rank of a binary code C is the dimension of the linear span, 〈C 〉, of C .

The kernel of a binary code C is define as K (C ) = x ∈ Zn2 : x + C = C.

If the all-zero vector belongs to C , then K (C ) is a linear subcode of C .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 14 / 21

The case s = 2

A construction of Z4-Linear Hadamard codes is given in:Krotov, D. S., Z4-linear Hadamard and extended perfect codes, WCC2001, International Workshop on Coding

and Cryptography, ser. Electron. Notes Discrete Math., 6 (2001), 107–112.

An a classification by using the rank and the dimension of the kernel isgiven in:

Phelps, K. T., J. Rifà, and M. Villanueva, On the additive (Z4-linear and non-Z4-linear) Hadamard codes:

rank and kernel, IEEE Trans. Inf. Theory, 52 (2006), no. 1, 316–319.

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Kernel of Z2s -Linear Hadamard codes

We define σ ∈ 1, . . . , s as the integer such that ord(w2) = 2s+1−σ.

Theorem

Let H = Ht1,...,ts be a Z2s -additive Hadamard code of type (n; t1, . . . , ts)such that Φ(H) is nonlinear. Let Hb be the subcode of H which containsall codewords of order two. Let P = 2pσ−2

p=0 if σ ≥ 2, and P = ∅ if σ = 1.Then, ⟨

Φ(Hb),Φ(P),Φ(s−2∑i=0

2i )

⟩= K (Φ(H)),

and ker(Φ(H)) = σ +∑s

i=1 ti .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 16 / 21

Example of Kernel

Let H3,0 be a Z4-additive code. In this case σ = 1

A3,0 =

11 11 11 11 11 11 11 1101 23 01 23 01 23 01 2300 00 11 11 22 22 33 33

.

H3,0b =

(2222222222222222),(0202020202020202),(0000222200002222)

P = (1111111111111111)

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 17 / 21

Example of Kernel

Let H3,0 be a Z4-additive code. In this case σ = 1

A3,0 =

11 11 11 11 11 11 11 1101 23 01 23 01 23 01 2300 00 11 11 22 22 33 33

.

H3,0b =

(2222222222222222),(0202020202020202),(0000222200002222)

P = (1111111111111111)

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 17 / 21

Example of Kernel

Let H3,0 be a Z4-additive code. In this case σ = 1

A3,0 =

11 11 11 11 11 11 11 1101 23 01 23 01 23 01 2300 00 11 11 22 22 33 33

.

H3,0b =

(2222222222222222),(0202020202020202),(0000222200002222)

P = (1111111111111111)

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 17 / 21

t = 8 t = 9type (r , k) type (r , k)

Z4

(27; 1, 7) (9,9) (28; 1, 8) (10,10)(27; 2, 5) (9,9) (28; 2, 6) (10,10)(27; 3, 3) (10,7) (28; 3, 4) (11,8)(27; 4, 1) (12,6) (28; 4, 2) (13,7)

(28; 5, 0) (16,6)

Z8

(26; 1, 0, 6) (9,9) (27; 1, 0, 7) (10,10)(26; 1, 1, 4) (9,9) (27; 1, 1, 5) (10,10)(26; 1, 2, 2) (10,7) (27; 1, 2, 3) (11,8)(26; 1, 3, 0) (12,6) (27; 1, 3, 1) (13,7)(26; 2, 0, 3) (11,6) (27; 2, 0, 4) (12,7)(26; 2, 1, 1) (13,5) (27; 2, 1, 2) (14,6)(26; 3, 0, 0) (17,4) (27; 2, 2, 0) (17,5)

(27; 3, 0, 1) (18,5)

Type, rank and kernel of all Z2s -linear Hadamard codes of length 2t .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 18 / 21

t = 8 t = 9type (r , k) type (r , k)

Z4

(27; 1, 7) (9,9) (28; 1, 8) (10,10)(27; 2, 5) (9,9) (28; 2, 6) (10,10)(27; 3, 3) (10,7) (28; 3, 4) (11,8)(27; 4, 1) (12,6) (28; 4, 2) (13,7)

(28; 5, 0) (16,6)

Z8

(26; 1, 0, 6) (9,9) (27; 1, 0, 7) (10,10)(26; 1, 1, 4) (9,9) (27; 1, 1, 5) (10,10)(26; 1, 2, 2) (10,7) (27; 1, 2, 3) (11,8)(26; 1, 3, 0) (12,6) (27; 1, 3, 1) (13,7)(26; 2, 0, 3) (11,6) (27; 2, 0, 4) (12,7)(26; 2, 1, 1) (13,5) (27; 2, 1, 2) (14,6)(26; 3, 0, 0) (17,4) (27; 2, 2, 0) (17,5)

(27; 3, 0, 1) (18,5)

Type, rank and kernel of all Z2s -linear Hadamard codes of length 2t .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 19 / 21

t 3 4 5 6 7 8 9 10 11Z4 1 1 2 2 3 3 4 4 5Z8 1 1 2 3 4 6 7 9 11Z16 1 1 1 2 4 5 8 10 14

Number of nonequivalent Z2s -linear Hadamard codes of length 2t .

C.Fernández-Córdoba, C.Vela, M.Villanueva Z2s -linear Hadamard codes 20 / 21

Thank you for your attention

Tänan teid tähelepanu eest

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