Quasi type IV codes over a non-unital ring

Abstract

There is a local ring I of order 4,  without identity for the multiplication, defined by generators and relations as

$$\begin{aligned} I=\langle a,b \mid 2a=2b=0,\, a^{2}=b,\, \,ab=0 \rangle . \end{aligned}$$

We give a natural map between linear codes over I and additive codes over ${\mathbb{F}}_{4},$ 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 ${\mathbb{Z}}_{4}$-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 ${\mathbb{F}}_{4}.$ 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.