Construction Methodsfor Self-Dual Codes

Stefka Bouyuklieva Iliya Bouyukliev

Veliko Tarnovo University Bulgarian Academy of Sciences

BULGARIA

10 July 2014 – p. 1/26

Main Problems

1. Construction of self-dual codes with given lengthand minimum distance.

2. Classification of all self-dual codes with givenlength and minimum distance.

3. Classification of all self-dual codes with givenlength and minimum distance, invariant under agiven permutation group.

4. Classification of all self-dual codes for a givenlength.

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Definitions• Fq - finite field with q elements;• F

nq - n-dimensional vector space overFq;

• Weightof a vectorx ∈ Fnq: wt(x) = |i|xi 6= 0|;

• Linear codeof lengthn and dimensionk -k-dimensional subspace ofFn

q;

• Minimum weightof a linear codeC:

d(C) = minwt(x)|x ∈C, x 6= 0

• C - a linear[n,k,d]q code.

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C - a linear [n,k,d] code• Generator matrixG of C - a k×n matrix whose

rows form a basis ofC as a linear subspace ofFnq;

• Check matrixH of C - a (n− k)×n matrix suchthatx ∈C ⇐⇒ HxT = 0;

Example: The repetition codeC1 = 0000,1111

G(C1)= (1111), H(C1) =

1 1 0 01 0 1 01 0 0 1

=G(C2).

C2 = 0000,1100,1010,1001, . . . ,0011,1111.C2 is a linear[4,3,2] code,C2 =C⊥

1 .

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C - a binary linear [n,k,d] code• C - self-orthogonal code ifC ⊆C⊥

• C - self-dual code ifC =C⊥

• Any self-dual code has dimensionk = n/2• All codewords in a binary self-orthogonal code

have even weights• Doubly-even code - if 4| wt(v) ∀v ∈C• Singly-even self-dual code - if∃v ∈C :

wt(v)≡ 2 (mod 4)

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Self-dual codesExample: The extended Hamming code -[8,4,4]

G =

1 0 0 0 0 1 1 10 1 0 0 1 0 1 10 0 1 0 1 1 0 10 0 0 1 1 1 1 0

.

H8 consists of: 14 codewords of weight 4,the all-ones vector (weight 8), the zero vector

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Extremal self-dual codesIf C is a binary self-dual[n,n/2,d] code then

d ≤ 4[n/24]+4

except whenn ≡ 22 (mod 24) when

d ≤ 4[n/24]+6

Whenn is a multiple of 24, any code meeting thebound must be doubly-even.

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Extremal self-dual codes• n=8, d=4 - extended Hamming code• n=16, d=4 - two codes• n=24, d=8 - extended Golay code

• n=32, d=8 - five codes• n=40, d=8 - many codes• n=48,d=12 extended quadratic-residue code

• n=56, n=64, d=12 - many codes• n=72, d=16 - ?

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Optimal self-dual codesA self-dual code is called optimal if it has the largestminimum weight among all self-dual codes of thatlength.

• Any extremal self-dual code is optimal.• For some lengths, no extremal self-dual codes

exist!• There are no extremal self-dual codes of lengths

2, 4, 6, 10, 26, 28, 30, 34, 50, 52, 54, 58, ...

Conjecture: The optimal self-dual codes of lengths24m+ r for r = 2, 4, 6, and 10 are not extremal.

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Equivalent codes; Aut(C)• Two binary codesC andC′ are equivalent if there

is a permutationπ ∈ Sn: C′ = π(C)

• Automorphism of C is a permutation of thecoordinates that preserves C

• All automorphisms of C form a groupAut(C)

• Extended Golay code:Aut(g24) = M24 -5-transitive and|M24|= 210.33.5.7.11.23

• Extended quadratic-residue [48,24,12] code:Aut(q48) = PSL(2,47) - 2-transitive and|PSL(2,47)|= 25.3.23.47

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History• 1975, Vera Pless –n ≤ 20• 1980-90, Conway, Pless, Sloane –n ≤ 30• 2006, Bilous, Van Rees, –n = 32,34• 2008, Melchor, Gaborit –n = 36 (Optimal)• 2011, Harada, Munemasa –n = 36• 2011, Harada, Munemasa;

C. Aguilar-Melchor, Ph. Gaborit, Jon-Lark Kim,L. Sok, P. Sole –n = 38 (Optimal)

• 2011, Betsumiya, Harada, Munemasa –n = 40(Doubly even )

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Gluing theory

G1 0G2

.. .0 Gt

X

C1,C2, . . . ,Ct - self-orthogonal codesIf u ∈ X thenu = (u1u2 . . .ut), ui ∈C⊥

i

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Gluing theory - examples[18,9,4] self-dual codes:

111100001111

111100001111

111100001111

010101 000011 010110010110 010101 000011000011 000011 000011

1111000000001111000000001111000000001111

111010001110100011101

01010101010000000 101010101101111111 0

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The number of binary SD codes

n ♯I ♯II dmax,I ♯max,I dmax,II ♯max,II

24 46 9 6 1 8 1

26 103 6 1

28 261 6 3

30 731 6 13

32 3 210 85 8 3 8 5

34 24 147 6 938

36 519 492 8 41

38 38 682 183*BB 8 2 744

40 8 250 058 081*BDM 94 343 8 10 200 655*BBH 8 16 470

*BBH - Bouyuklieva, Bouyukliev, Harada*BDM - Bouyukliev, Dzhumalieva-Stoeva, Monev

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Main constructionIf C is a binary[n = 2k > 2,k,d] SD code, thenC is equivalent to a code with a generator matrix

G =

x1 . . .xk−1 00. . .0 1 0x1 x1

Ik−1 A... ...

xk−1 xk−1

where the matrix(Ik−1|A) generates a self-dual codeof lengthn−2.

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Main constructionHow to construct only inequivalent[n,k] SD codesstarting from all inequivalent[n−2,k−1] SD codes?

1. How to construct only inequivalent child-codes ofone[n−2,k−1] code?

2. If we take two inequivalent[n−2,k−1] codes,how to extend them such that the constructedchild-codes to be inequivalent?

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Equivalence of the child-codesLet C1 be a self-dual code of lengthn−2 with a

generator matrixG1 and automorphism groupAut(C1). ThenAut(C1) acts on the setFk−1

2 :

G1P = ApG1, P ∈ Aut(C1), AP ∈ GL(k−1,2)

a 7→ APaT , a ∈ Fk−12

Theorem:The matrices(G1 aT aT ) and(G1 bT bT )generate equivalent codes⇐⇒ a andb belong tothe same orbit under the action ofAut(C1) onFk−1

2 .

Child(C1) - the set of all inequivalen child-codes ofC1

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The main idea• G - finite group• G acts on a setΩ and defines an equivalence

relation:g(a)∼= a; g ∈ G

• ρ : Ω 7→ Ω - canonical map

b ∼= a ⇒ ρ(b)≡ ρ(a)≡ ra ∈ Ω• ra - canonical representative of the equivalence

class

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Ω = SD codes of length≤ n′

σC ∈ Sn - a canonical permutation ofC, ifσC(C) = ρ(C)

Example:fC : Sn → Z

π(C) =C′ 7→ G′C 7→ v′C ∈ F

2kn2 7→ fC(π)

G′C - matrix whose rows are the lexicographically

ordered codewordsv′C - a vector constructed by gluing the rows ofG′

C

If fC(σ) = max fC(π)|π ∈ Sn thenρ(C) = σ(C)

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Parent test• C1 - a self-dual[n−2,n/2−1] code;• B ∈Child(C1);• ρ(B) - the canonical representative ofB;• σ - a canonical permutation ofB;• 1≤ c1,c2 ≤ n, σ(c1) = n−1, σ(c2) = n;

If there is an automorphismτ of B such thatτ(c1),τ(c2)= n−1,n then the codeB passesthe parent test.

Obviously, the canonical representativeB of oneequivalence class passes the parent test.

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Parent testLemma 1: If B1 andB2 are two equivalent self-dual[n = 2k,k] codes which pass the parent test,their parent-codes are also equivalent.

Lemma 2: Let C1 andC2 be two equivalent SDcodes of dimensionr. Then for any child-codeB1of C1 which passes the parent test, there isa child-codeB2 of C2, equivalent toB1,such thatB2 also passes the parent test.

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The algorithmProcedure Augmentation(A: self-dual code;k: dimension);

If the dimension ofA is equal tok then

Uk := Uk ∪A; PRINT (A, |Aut(A)|); ;

If the dimension ofA is less thank then

find the setChild(A) of all inequivalent children ofA;

(using already knownAut(A))

For all codesB from the setChild(A) do the following:

if B passes the parent test then Augmentation(B,k);

Procedure Main;

INPUT: Ur – all NBSDC[2r,r]; OUTPUT:Uk – all NBSDC[2k,k];

Uk := (the empty set);

for all codesA ∈Ur do the following:

find the automorphism group ofA; Augmentation(A,k);

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Advantages of the algorithm• Construction and test for equivalence in one.• Possibilities for use of invariants in the search of

canonical representative and canonicalpermutation.

• Easy for parallelization.• Recursive construction (we can start from the

trivial code).

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The number of binary SD codesThe number of all binary SD codes of even lengthn

is

N(n) =n/2−1

∏i=1

(2i +1) =r(n)

∑i=1

n!| Aut(Ci) |

,

whereC1,C2, . . . ,Cr(n) are the inequivalent binary SDcodes of lengthn

⇒ r(n)≥∏n/2−1

i=1 (2i +1)

n!

∑C∈U

n!|Aut(C)|

|x∈C|wt(x) = d|=

(

nd

)n/2−2

∏i=1

(2i+1).

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d = 2There is one-to-one correspondence between the setof all inequivalent self-dual[n,n/2] codes and the setof all inequivalent self-dual[n+2,n/2+1,2] codes

C 7→ (00|C)∪ (11|C)

r(n,d) - the number of the inequivalent binary[n,n/2,d] self-dual codes

⇒ r(n+2,2) = r(n)

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d = 4If C is a binary[n = 2k > 2,k,4] SD code, thenCis equivalent to a code with a generator matrix

G =

11 00· · ·0 00· · ·0 1 101 00· · ·0 v 0 100 Ik−2 A aT aT

where the matrix(Ik−2|A) generates a self-dual codeof lengthn−4.

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