Classification of LCD and self-dual codes over a finite non-unital local ring

TL;DR

This paper classifies LCD and self-dual codes over a specific non-unital, noncommutative ring, providing conditions for MDS and AMDS codes and classifying them for small lengths over rings E_2 and E_3.

Contribution

It introduces new classifications and conditions for LCD and self-dual codes over a noncommutative ring, extending the understanding of MDS and AMDS codes in this context.

Findings

Classified MDS and AMDS LCD codes over E_2 and E_3 for lengths up to 6.

Classified MDS and AMDS left self-dual codes over E_2 and E_3 for lengths up to 12.

Derived necessary and sufficient conditions for a code to be MDS or AMDS over the ring.

Abstract

This work explores LCD and self-dual codes over a noncommutative non-unital ring $ E_p= \langle r,s ~|~ pr =ps=0,~ r^2=r,~ s^2=s,~ rs=r,~ sr=s \rangle$ of order $p^2$ where $p$ is a prime. Initially, we study the monomial equivalence of two free $E_p$-linear codes. In addition, a necessary and sufficient condition is derived for a free $E_p$-linear code to be MDS and almost MDS (AMDS). Then, we use these results to classify MDS and AMDS LCD codes over $E_2$ and $E_3$ under monomial equivalence for lengths up to $6$. Subsequently, we study left self-dual codes over the ring $E_p$ and classify MDS and AMDS left self-dual codes over $E_2$ and $E_3$ for lengths up to $12$. Finally, we study self-dual codes over the ring $E_p$ and classify MDS and AMDS self-dual codes over $E_2$ and $E_3$ for smaller lengths.