The Automorphism Group of an Extremal [72,36,16] Code does not contain elements of order 6

The existence of an extremal code of length 72 is a long-standing open problem. Let C be a putative extremal code of length 72 and suppose that C has an automorphism g of order 6. We show that C, as an F_2-module, is the direct sum of two modules, one easily determinable and the other one which has a very restrictive structure. We use this fact to do an exhaustive search and we do not find any code. This proves that the automorphism group of an extremal code of length 72 does not contain elements of order 6.Comment: 15 pages, 0 figures. A revised version of the paper is published on IEEE Transactions on Information Theor

Some MDS Codes over Gf(64) Connected with the Binary Doubly-Even [72,36,16] Code

* The author is supported by a Return Fellowship from the Alexander von Humboldt Foundation.MDS [8,4,5] codes over a field with 64 elements are constructed. All such codes which are self-dual under a Hermitian type inner product are classified. The connection between these codes and a putative binary self- dual [72,36,16] code is considered

On the Structure of the Linear Codes with a Given Automorphism

The purpose of this paper is to present the structure of the linear codes over a finite field with q elements that have a permutation automorphism of order m. These codes can be considered as generalized quasi-cyclic codes. Quasi-cyclic codes and almost quasi-cyclic codes are discussed in detail, presenting necessary and sufficient conditions for which linear codes with such an automorphism are self-orthogonal, self-dual, or linear complementary dual

Self-dual codes with an automorphism of order 17

In this paper we study optimal binary self-dual codes with minimum distance 12 having an automorphism of order 17. We prove that all such codes have parameters [68 + f; 34 + f=2; 12]; f = 0; 2; 4 and automorphism of type 17- (4; f), f = 0; 2; 4 and provide a full classication of these codes. This classication gives: new values b = 17, 153, 170, 187, 221, 255 for = 0 in the weight enumerator W68,2 of [68, 34, 12] codes; new values b = 102, 136, 170, 204, 238, 272, 306, 340, 374, 408, 442, 476, 510, 544, 578, and 612 for g = 0 in W70,1 of [70, 35, 12] codes; numerous singly- and doubly-even [72, 36, 12] codes with new parameters in their weight enumerators

The extremal codes of length 42 with automorphism of order 7

AbstractAll [42, 21, 8] binary self-dual codes with automorphisms of order 7 are enumerated. Up to equivalence there are 16 such codes. These codes are defined with their generator matrices

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page