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27-01-2021 | Original Paper | Issue 3/2021

Applicable Algebra in Engineering, Communication and Computing 3/2021

Quasi type IV codes over a non-unital ring

Journal:
Applicable Algebra in Engineering, Communication and Computing > Issue 3/2021
Authors:
Adel Alahmadi, Alaa Altassan, Widyan Basaffar, Alexis Bonnecaze, Hatoon Shoaib, Patrick Solé
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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.

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