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

Abstract

Let ${\mathbb{F}}_{p^m}$ be a finite field of cardinality $p^m$ and $R={\mathbb{F}}_{p^m}[u]/⟨u^2⟩={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}$ $(u^2=0)$, 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 $p^{s}n$ and their dual codes, where $s$ is a nonnegative integer and $n$ is a positive integer satisfying $\gcd (p,n)=1$. 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.