Cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$

TL;DR

This paper investigates cyclic codes over a specific non-commutative ring, providing generator sets, analyzing their rank and Hamming distance, and identifying optimal codes via Gray images.

Contribution

It introduces a unique set of generators for cyclic codes over the ring and characterizes their Gray images, including nearly all optimal ternary codes of length 12.

Findings

Derived a unique generator set for these cyclic codes.

Analyzed the rank and Hamming distance of the codes.

Identified all but one optimal ternary code of length 12 as Gray images.

Abstract

Let $p$ be a prime number. In this paper, we study cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$. We find a unique set of generators for these codes. We also study the rank and the Hamming distance of these codes. We obtain all except one ternary optimal code of length 12 as the Gray image of the cyclic codes over the ring $ \Z_p[u, v]/\langle u^2, v^2, uv-vu\rangle$. We also characterize the $p$-ary image of these cyclic codes under the Gray map.