Quasi Type IV codes over a non-unital ring

There is a local ring I of order 4, without identity for the multiplication , defined by generators and relations as I = a, b | 2a = 2b = 0, a 2 = b, ab = 0. We study the algebraic structure of linear codes over this non-unital local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes (QSD) over I, and Type IV I-codes, that is, QSD codes all codewords of which have even Hamming weight. Further, we define quasi Type IV codes over I as those QSD codes with an even torsion code. We give a mass formula for QSD codes, and another quasi Type IV codes, and classify both of them in short lengths.