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

Let $\mathbb{F}{p^m}$ be a finite field of cardinality $p^m$ and $R=\mathbb{F}{p^{m}[u]/\langle} u^{2\rangle=\mathbb{F}{pm}+u\mathbb{F}{pm}$} $(u^2=0)$, 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.