On involutions in extremal self-dual codes and the dual distance of semi self-dual codes

A classical result of Conway and Pless is that a natural projection of the fixed code of an automorphism of odd prime order of a self-dual binary linear code is self-dual 13. In this paper we prove that the same holds for involutions under some (quite strong) conditions on the codes.In order to prove it, we introduce a new family of binary codes: the semi self-dual codes. A binary self-orthogonal code is called semi self-dual if it contains the all-ones vector and is of codimension 2 in its dual code. We prove upper bounds on the dual distance of semi self-dual codes.As an application we get the following: let C be an extremal self-dual binary linear code of length 24m and ź ź Aut ( C ) be a fixed point free automorphism of order 2. If m is odd or if m = 2 k with ( 5 k - 1 k - 1 ) odd then C is a free F 2 { ź } -module. This result has quite strong consequences on the structure of the automorphism group of such codes.

[1]  Shengyuan Zhang.  On the nonexsitence of extremal self-dual codes , 1999 .

[2]  L. Chouinard.  Projectivity and relative projectivity over group rings , 1976 .

[3]  P. Gaborit.  A bound for certain s-extremal lattices and codes , 2007 .

[4]  Gabriele Nebe.  An extremal [72, 36, 16] binary code has no automorphism group containing Z2 x Z4, Q8, or Z10 , 2012, Finite Fields Their Appl..

[5]  E. J. Farrell.  Cycle decompositions of complete graphs , 1982, Discret. Math..

[6]  Vassil Yorgov,et al.  The Automorphism Group of a Self Dual Binary [72,36,16] Code Does Not Contain Z4 , 2013, 1310.2570.

[7]  Martino Borello.  The automorphism group of a self-dual [72,36,16] code is not an elementary abelian group of order 8 , 2013, ArXiv.

[8]  Stefka Bouyuklieva,et al.  On the automorphism group of a binary self-dual $$[120, 60, 24]$$[120,60,24] code , 2013, Applicable Algebra in Engineering, Communication and Computing.

[9]  Stefka Bouyuklieva.  On the Automorphisms of Order 2 with Fixed Points for the Extremal Self-Dual Codes of Length 24"m , 2002, Des. Codes Cryptogr..

[10]  Gabriele Nebe,et al.  Automorphisms of doubly even self‐dual binary codes , 2008, ArXiv.

[11]  Martino Borello,et al.  Automorphisms of Order $2p$ in Binary Self-Dual Extremal Codes of Length a Multiple of 24 , 2013, IEEE Transactions on Information Theory.

[12]  Philippe Gaborit,et al.  On the Classification of Extremal $[36,18,8]$ Binary Self-Dual Codes , 2008, IEEE Transactions on Information Theory.

[13]  Christine Bachoc,et al.  Designs and self-dual codes with long shadows , 2004, J. Comb. Theory, Ser. A.

[14]  Eric M. Rains.  Bounds for Self-Dual Codes Over Z4 , 2000 .

[15]  Martino Borello,et al.  The automorphism group of a self-dual [72, 36, 16] code does not contain S3, A4 or D8 , 2013, Adv. Math. Commun..

[16]  N. J. A. Sloane,et al.  An Upper Bound for Self-Dual Codes , 1973, Inf. Control..

[17]  N. J. A. Sloane.  Is there a (72, 36) d = 16 self-dual code? (Corresp.) , 1973, IEEE Trans. Inf. Theory.

[18]  Martino Borello.  The Automorphism Group of a Self-Dual $[72, 36, 16]$ Binary Code Does Not Contain Elements of Order 6 , 2012, IEEE Transactions on Information Theory.

[19]  Stefka Bouyuklieva,et al.  On the automorphisms of order 15 for a binary self-dual $$[96, 48, 20]$$[96,48,20] code , 2016, Des. Codes Cryptogr..

[20]  Martino Borello.  The automorphism group of a self-dual [72, 36, 16][72, 36, 16] code is not an elementary abelian group of order 8 , 2014, Finite Fields Their Appl..

[21]  Stefka Bouyuklieva,et al.  On the automorphism group of a binary self-dual [120,60,24]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[120, 60, 2 , 2012, Applicable Algebra in Engineering, Communication and Computing.

[22]  Martino Borello.  Automorphism groups of self-dual binary linear codes with a particular regard to the extremal case of length 72 , 2014 .

[23]  John H. Conway,et al.  On primes dividing the group order of a doubly-even (72, 36, 16) code and the group order of a quaternary (24, 12, 10) code , 1982, Discret. Math..

[24]  Eric M. Rains,et al.  Shadow Bounds for Self-Dual Codes , 1998, IEEE Trans. Inf. Theory.

[25]  Christine Bachoc.  On Harmonic Weight Enumerators of Binary Codes , 1999, Des. Codes Cryptogr..