Self-orthogonal codes over a non-unital ring and combinatorial matrices
Abstract
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,\, a2=a,\, b2=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 4. The classical invariant theory bound for the weight enumerators of this class of codesimproves the known bound on the minimum distance of Type IV codes over E.
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