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.

Automated summary

The article investigates the algebraic structure of linear codes over non-unital rings, introduces concepts such as quasi self-dual codes and different types of codes, and provides formulas and classifications for these codes in short lengths.