On $(\alpha+u\beta)$-constacyclic codes of length $p^sn$ over $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$

TL;DR

This paper characterizes all $(eta+ueta)$-constacyclic codes over a specific finite ring of length $p^sn$, including their duals, with detailed classifications for certain parameters over $ ext{F}_3+u ext{F}_3$.

Contribution

It provides a complete classification of $(eta+ueta)$-constacyclic codes over $ ext{F}_{p^m}+u ext{F}_{p^m}$ of length $p^sn$, including explicit descriptions of dual codes.

Findings

Explicit representations of all such codes and their duals.

Special enumeration of $(2+u)$-constacyclic codes of length $6 ext{·}5^t$ over $ ext{F}_3+u ext{F}_3$.

Complete classification for codes with parameters satisfying gcd conditions.

Abstract

Let $\mathbb{F}_{p^m}$ be a finite field of cardinality $p^m$ and $R=\mathbb{F}_{p^m}[u]/\langle u^2\rangle=\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$ $(u^2=0)$, where $p$ is an odd prime and $m$ is a positive integer. For any $\alpha,\beta\in \mathbb{F}_{p^m}^{\times}$, the aim of this paper is to represent all distinct $(\alpha+u\beta)$-constacyclic codes over $R$ of length $p^sn$ and their dual codes, where $s$ is a nonnegative integer and $n$ is a positive integer satisfying ${\rm gcd}(p,n)=1$. Especially, all distinct $(2+u)$-constacyclic codes of length $6\cdot 5^t$ over $\mathbb{F}_{3}+u\mathbb{F}_3$ and their dual codes are listed, where $t$ is a positive integer.