Negacyclic codes of odd length over the ring $\mathbb{F}_p[u,v]/\langle u^2,v^2,uv-vu\rangle$

TL;DR

This paper investigates the structure of negacyclic codes of odd length over a specific non-commutative ring, providing explicit descriptions of their generators, rank, and minimum distance.

Contribution

It introduces a detailed analysis of negacyclic codes over a non-commutative ring, including their unique generators, rank, and minimum distance, which was not previously established.

Findings

Explicit form of the unique generating set

Determination of the rank of the codes

Calculation of the minimum distance

Abstract

We discuss the structure of negacyclic codes of odd length over the ring $\mathbb{F}_p[u, v]/ \langle u^2, v^2, uv-vu \rangle$. We find the unique generating set, the rank and the minimum distance for these negacyclic codes.