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 $E=langle a,b mid 2a=2b=0,, a^2=a,, b^2=b,,ab=a,, ba=brangle.$ 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 $E.$