Cyclic codes over the ring $\mathbb{F}_p[u,v,w]/\langle u^2, v^2, w^2, uv-vu, vw-wv, uw-wu \rangle$

TL;DR

This paper explores the structure and properties of cyclic codes over a specific Frobenius ring, providing generator characterization, analyzing code parameters, and constructing good p-ary codes via Gray images.

Contribution

It introduces a unique generator set for cyclic codes over the ring and characterizes free cyclic codes, advancing understanding of codes over Frobenius rings.

Findings

Identified a unique set of generators for these cyclic codes

Characterized free cyclic codes over the ring

Constructed good p-ary codes via Gray images

Abstract

In this paper, we investigate cyclic codes over the ring $ \mathbb{F}_p[u,v,w]\langle u^2,$ $v^2, w^2$, $uv-vu, vw-wv, uw-wu \rangle$, where $p$ is a prime number. Which is a part of family of Frobenius rings. We find a unique set of generators for these codes and characterize the free cyclic codes. We also study the rank and the Hamming distance of these codes. We also constructs some good $p-ary$ codes as the Gray images of these cyclic codes.