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 $E=⟨a,b\mid 2a=2b=0,a^2=a,b^2=b,ab=a,ba=b⟩.$ 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 codesimproves the known bound on the minimum distance of Type IV codes over $E.$

18 pages