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

06/14/2021

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by   Minjia Shi, et al.

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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 _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.