Repeated-Root Constacyclic Codes of Length $3p^s$ over the Finite Non-Chain Ring $\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$ and their Duals

TL;DR

This paper investigates the algebraic structure of repeated-root constacyclic codes of length 3p^s over a specific non-chain ring, analyzing cases based on whether the unit alpha is a cube in the ring, and also determines code duals.

Contribution

It provides a detailed classification of alpha-constacyclic codes over a non-chain ring for different units alpha, including their duals and codeword counts, which was previously unexplored.

Findings

Classified the structure of constacyclic codes when alpha is a cube in the ring.

Analyzed the structure when alpha is not a cube, considering multiple subcases.

Determined the number of codewords and dual codes for these classes.

Abstract

This study aims to determine the algebraic structures of $\alpha$-constacyclic codes of length $3p^s$ over the finite commutative non-chain ring $\mathcal{R}=\frac{\mathbb{F}_{p^m}[u, v]}{\langle u^2, v^2, uv-vu\rangle}$, for a prime $p \neq 3.$ For the unit $\alpha$, we consider two different instances: when $\alpha$ is a cube in $\mathcal{R}$ and when it is not. Analyzing the first scenario is relatively easy. When $\alpha$ is not a unit in $\mathcal{R}$, we consider several subcases and determine the algebraic structures of constacyclic codes in those cases. Further, we also provide the number of codewords and the duals of $\alpha$-constacyclic codes.