A class of constacyclic codes over $\mathbb{F}_{p^m}[u]/\left<u^2\right>$

TL;DR

This paper classifies all constacyclic codes of length 4p^s over a specific finite chain ring, including their duals, sizes, and some isodual codes, expanding understanding of code structures over such rings.

Contribution

It provides a complete characterization of constacyclic codes of length 4p^s over the ring R, including duals, sizes, and examples of isodual codes, which was previously unexplored.

Findings

All constacyclic codes of length 4p^s over R are determined.

Dual codes of these constacyclic codes are explicitly characterized.

Some isodual constacyclic codes of length 4p^s over R are identified.

Abstract

Let $p$ be an odd prime, and let $m$ be a positive integer satisfying $p^m \equiv 3~(\text{mod }4).$ Let $\mathbb{F}_{p^m}$ be the finite field with $p^m$ elements, and let $R=\mathbb{F}_{p^m}[u]/\left<u^2\right>$ be the finite commutative chain ring with unity. In this paper, we determine all constacyclic codes of length $4p^s$ over $R$ and their dual codes, where $s$ is a positive integer. We also determine their sizes and list some isodual constacyclic codes of length $4p^s$ over $R.$