Self-orthogonal codes over a non-unital ring and combinatorial matrices
Résumé
There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E = a, b | 2a = 2b = 0, a 2 = a, b 2 = b, ab = a, ba = b. We study a special construction of self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), and Doubly Regular Tournaments (DRT). We construct quasi self-dual codes over E, and Type IV codes, that is, quasi self-dual codes whose all codewords have even Hamming weight. All these codes can be represented as formally self-dual additive codes over F 4. The classical invariant theory bound for the weight enumerators of this class of codes improves the known bound on the minimum distance of Type IV codes over E.
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