On the Classification of Codes over Non-Unital Ring of Order 4

TL;DR

This paper classifies optimal and well-structured codes of length up to 7 over a specific non-unital, non-commutative ring of order 4, expanding coding theory beyond traditional finite fields and rings.

Contribution

It provides the first classification of codes over a particular non-unital, non-commutative ring of order 4, including weight enumerators for lengths up to 7.

Findings

Classification of optimal codes over the ring E for lengths ≤ 7.

Explicit weight and complete weight enumerators for these codes.

Extension of coding theory to non-unital, non-commutative rings.

Abstract

In the last 60 years coding theory has been studied a lot over finite fields $\mathbb{F}_q$ or commutative rings $\mathcal{R}$ with unity. Although in $1993$, a study on the classification of the rings (not necessarily commutative or ring with unity) of order $p^2$ had been presented, the construction of codes over non-commutative rings or non-commutative non-unital rings surfaced merely two years ago. In this letter, we extend the diverse research on exploring the codes over the non-commutative and non-unital ring $E= \langle 2a=2b=0, a^2=a, b^2=b, ab=a, ba=b \rangle$ by presenting the classification of optimal and nice codes of length $n\leq7$ over $E$, along-with respective weight enumerators and complete weight enumerators.