Automorphisms of Extremal Self-Dual Codes

Let <i>C</i> be a binary extremal self-dual code of length <i>n</i> ¿ 48. We prove that for each <i>¿ ¿ Aut(C</i>) of prime order <i>p</i> ¿ 5 the number of fixed points in the permutation action on the coordinate positions is bounded by the number of <i>p</i>-cycles. It turns out that large primes <i>p</i>, i.e., <i>n</i>-<i>p</i> small, seem to occur in <i>|Aut(C</i>)| very rarely. Examples are the extended quadratic residue codes. We further prove that doubly even extended quadratic residue codes of length <i>n</i> = <i>p</i> + 1 are extremal only in the cases <i>n</i> =8, 24, 32, 48, 80, and 104.

[1]  Masaaki Harada,et al.  New extremal self-dual codes of length 68 , 1999, IEEE Trans. Inf. Theory.

[2]  E. M. Rains,et al.  Self-Dual Codes , 2002, math/0208001.

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

[4]  Radka Russeva,et al.  On binary self-dual codes of lengths 60, 62, 64 and 66 having an automorphism of order 9 , 2007, Des. Codes Cryptogr..

[5]  Radinka Yorgova,et al.  Binary self-dual codes with automorphisms of order 23 , 2008, Des. Codes Cryptogr..

[6]  Vera Pless,et al.  A classification of self-orthogonal codes over GF(2) , 1972, Discret. Math..

[7]  N. J. A. Sloane,et al.  On the Classification and Enumeration of Self-Dual Codes , 1975, J. Comb. Theory, Ser. A.

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

[9]  Vassil Y. Yorgov.  A method for constructing inequivalent self-dual codes with applications to length 56 , 1987, IEEE Trans. Inf. Theory.

[10]  Wolfgang Willems,et al.  A note on self-dual group codes , 2002, IEEE Trans. Inf. Theory.

[11]  Iliya Bouyukliev.  About the code equivalence , 2007 .

[12]  R. Brualdi,et al.  Handbook Of Coding Theory , 2011 .

[13]  Wolfgang Willems,et al.  Self-dual extended cyclic codes , 2005, Applicable Algebra in Engineering, Communication and Computing.

[14]  Masaaki Harada,et al.  Some Extremal Self-Dual Codes with an Automorphism of Order 7 , 2003, Applicable Algebra in Engineering, Communication and Computing.

[15]  Masaaki Harada,et al.  New extremal self-dual codes of length 62 and related extremal self-dual codes , 2002, IEEE Trans. Inf. Theory.

[16]  Andrew M. Gleason,et al.  WEIGHT POLYNOMIALS OF SELF-DUAL CODES AND THE MacWILLIAMS IDENTITIES , 1970 .

[17]  W. Cary Huffman.  Automorphisms of codes with applications to extremal doubly even codes of length 48 , 1982, IEEE Trans. Inf. Theory.

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

[19]  N. J. A. Sloane,et al.  The Binary Self-Dual Codes of Length up to 32: A Revised Enumeration , 1992, J. Comb. Theory, Ser. A.

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

[21]  S. Dougherty,et al.  New Extremal Self-Dual Codes of Length , 1999 .

[22]  Stefka Bouyuklieva,et al.  On the structure of binary self-dual codes having an automorphism of order a square of an odd prime , 2005, IEEE Transactions on Information Theory.

[23]  Zhang Shengyuan.  On the Nonexistence of Extremal Self-dual Codes , 1999, Discret. Appl. Math..

[24]  Jonathan L. Alperin,et al.  Groups and Representations , 1995 .

[25]  Stefka Bouyuklieva,et al.  Classification of the binary self-dual [42, 21, 8] codes having an automorphism of order 3 , 2007, Finite Fields Their Appl..