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 special construction self-orthogonal codes over E, based on combinatorial matrices related to two-class association schemes, Strongly Regular Graphs (SRG), Doubly Tournaments (DRT). construct quasi self-dual Type IV codes, that is, whose all codewords have even Hamming weight. All these can be represented formally additive $$\mathbb {F}_4.$$ The classical invariant theory bound weight enumerators this class improves known minimum distance E.