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

There is a local ring E of order 4,  without identity for the multiplication, defined by generators and relations as \(E=\langle a,b \mid 2a=2b=0,\, a^2=a,\, b^2=b,\,ab=a,\, ba=b\rangle .\) 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 codewords all have even Hamming weight. All these codes can be represented as formally self-dual additive codes over \(\mathbb {F}_4.\) 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.