Self-orthogonal codes over a non-unital ring and combinatorial matrices

초록

There is a local ring E of order 4, without identity for the multiplication, defined by generators and relations as E = < a, b vertical bar 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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