On the Classification of Extremal Doubly Even Self-Dual Codes with 2-Transitive Automorphism Group
aa r X i v : . [ m a t h . C O ] F e b On the Classification of Extremal DoublyEven Self-Dual Codes with 2-TransitiveAutomorphism Groups
Naoki Chigira ∗ , Masaaki Harada † and Masaaki Kitazume ‡ April 4, 2018
Abstract
In this note, we complete the classification of extremal doubly evenself-dual codes with 2-transitive automorphism groups.
Keywords extremal doubly even self-dual code, automorphism group, 2-transitive group
Mathematics Subject Classification
As described in [5], self-dual codes are an important class of linear codes forboth theoretical and practical reasons. It is a fundamental problem to classifyself-dual codes of modest lengths and determine the largest minimum weightamong self-dual codes of that length (see [2, 5]). It was shown in [4] that theminimum weight d of a doubly even self-dual code of length n is boundedby d ≤ ⌊ n ⌋ + 4. A doubly even self-dual code meeting the bound is called ∗ Department of Mathematics, Kumamoto University, Kumamoto 860–8555, Japan.email: [email protected] † Department of Mathematical Sciences, Yamagata University, Yamagata 990–8560,Japan. email: [email protected] ‡ Department of Mathematics and Informatics, Chiba University, Chiba 263–8522,Japan. email: [email protected] xtremal . A common strategy for the problem whether there is an extremaldoubly even self-dual code for a given length is to classify extremal doublyeven self-dual codes with a given nontrivial automorphism group (see [2, 5]).Recently, Malevich and Willems [3] have shown that if C is an extremaldoubly even self-dual code with a 2-transitive automorphism group then C isequivalent to one of the extended quadratic residue codes of lengths 8 , , , , T ⋊ SL(2 , ), where T is an elementary abelian group of order 1024.The aim of this note is to complete the classification of extremal doublyeven self-dual codes with 2-transitive automorphism groups. This is com-pleted by excluding the open case in the above characterization [3], usingTheorem A in [1]. Theorem 1.
Let C be an extremal doubly even self-dual code with a -transitive automorphism group. Then C is equivalent to one of the the ex-tended quadratic residue codes of lengths , , , , , or the second-order Reed–Muller code of length . For an n -element set Ω, the power set P (Ω) – the family of all subsets of Ω –is regarded as an n -dimensional binary vector space with the inner product( X, Y ) ≡ | X ∩ Y | (mod 2) for X, Y ∈ P (Ω). The weight of X is defined tobe the integer | X | . A subspace C of P (Ω) is called a code of length n . Notethat all codes in this note are binary. The dual code C ⊥ of C is the set ofall X ∈ P (Ω) satisfying ( X, Y ) = 0 for all Y ∈ C . A code C is said to be self-orthogonal if C ⊂ C ⊥ , and self-dual if C = C ⊥ . A doubly even code is acode whose codewords have weight a multiple of 4.Let G be a permutation group on an n -element set Ω. We define the code C ( G, Ω) by C ( G, Ω) = h Fix( σ ) | σ ∈ I ( G ) i ⊥ , where I ( G ) denotes the set of involutions of G and Fix( σ ) is the set of fixedpoints of σ on Ω. Theorem 2 (Chigira, Harada and Kitazume [1]) . Let C be a binary self-orthogonal code of length n invariant under the group G . Then C ⊂ C ( G, Ω) .
2y using Theorem 2, some self-dual codes invariant under sporadic almostsimple groups were constructed in [1]. In this note, we apply Theorem 2 toa family of 2-transitive groups containing the group (2 ) ⋊ SL(2 , ).Let r, s be positive integers. We consider the following group GG = T ⋊ H ( T = (2 r ) s , H = SL(2 s, r )) , where the group T is regarded as the natural module GF (2 r ) s of H . Here T acts regularly on T itself and H acts on T as the stabilizer of the unit of T , which is regarded as the zero vector of GF (2 r ) s . Then G naturally acts2-transitively on T . Lemma 3.
There is no self-dual code of length rs invariant under G = T ⋊ H .Proof. By the fundamental theory of Jordan canonical forms in basic linearalgebra, the dimension of the subspace of GF (2 r ) s spanned by the vectorsfixed by an involution in H = SL(2 s, r ) is equal to or greater than s . Thenit is easily seen that there exist two involutions σ, τ in H such that eachof them fixes some s -dimensional subspace of GF (2 r ) s , and the zero vectoris the only vector fixed by both of them (i.e. T = Fix( σ ) ⊕ Fix( τ )). Ascodewords in C ( G, Ω) ⊥ , the inner product (Fix( σ ) , Fix( τ )) is equal to 1,since | Fix( σ ) ∩ Fix( τ ) | = 1. This yields that C ( G, T ) ⊥ is not self-orthogonal.Suppose that B is a self-dual code invariant under G . By Theorem 2, B ⊂ C ( G, T ). Since B ⊥ ⊃ C ( G, T ) ⊥ and B = B ⊥ , C ( G, T ) ⊥ is self-orthogonal.This is a contradiction.The case ( r, s ) = (5 ,
1) in the above lemma completes the proof of Theo-rem 1.
Remark . In the above proof, the cardinality of the fixed subspace of di-mension s is 2 rs , which is smaller than the value 4 ⌊ rs ⌋ + 4, except forthe cases ( r, s ) = (1 , , (2 , rs invariant under the group G = T ⋊ SL(2 s, r ) if rs > s − , r ) is 2 rs . If s > ⌊ (2 s − r ⌋ + 4, except for the small cases ( r, s ) = (1 , , (1 , , (2 , r, s ) = (1 ,
2) or (1 , C ( G, T ), for G = T ⋊ SL(2 s − , r )where T = (2 r ) s − , is equivalent to the extended Hamming code of length 8,3r the second-order Reed–Muller code of length 32 (see [1, Example 2.10]),respectively. For the remaining case ( r, s ) = (2 ,
2) (i.e. G = T ⋊ SL(3 , ), T =2 ), the smallest cardinality of the fixed subspace of an involution is 16 ( > C ( G, T ) ⊥ is self-orthogonal with minimum weight 16.) Acknowledgment.
This work is supported by JSPS KAKENHI GrantNumbers 23340021, 24340002, 24540024.
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Finite Fields Appl. (2005), 451–490.[3] A. Malevich and W. Willems, On the classification of the extremal self-dual codes over small fields with 2-transitive automorphism groups, Des.Codes Cryptogr. , (to appear), DOI 10.1007/s10623-012-9655-9.[4] C.L. Mallows and N.J.A. Sloane, An upper bound for self-dual codes,
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