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 vertical bar 2a = 2b = 0, a(2) = b, ab = 0 >. We give a natural map between linear codes over I and additive codes over F-4, that allows for efficient computations. We study the algebraic structure of linear codes over this non-unital local ring, their generator and parity-check matrices. A canonical form for these matrices is given in the case of so-called nice codes. By analogy with Z(4)-codes, we define residue and torsion codes attached to a linear I-code. 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. This is the natural analogue of Type IV codes over the field F-4. 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 for quasi Type IV codes, and classify both types of codes, up to coordinate permutation equivalence, in short lengths.