All $α+uβ$-constacyclic codes of length $np^{s}$ over $\mathbb{F}_{p^{m}}+u\mathbb{F}_{p^{m}}$

Let $\mathbb{F}{p^{m}}$ be a finite field with cardinality $p^{m}$ and $R=\mathbb{F}{p^{{m}}+u\mathbb{F}_{p{m}}$} with $u^{2}=0$. We aim to determine all $\alpha+u\beta$-constacyclic codes of length $np^{s}$ over $R$, where $\alpha,\beta\in\mathbb{F}{p^{m}}{*}$, $n, s\in\mathbb{N}{+}$ and $\gcd(n,p)=1$. Let $\alpha{0}\in\mathbb{F}{p^{m}}{*}$ and $\alpha{0}^{{p{s}}=\alpha$.} The residue ring $R[x]/\langle x^{{np{s}}-\alpha-u\beta\rangle$} is a chain ring with the maximal ideal $\langle x^{n}-\alpha{0}\rangle$ in the case that $x^{{n}-\alpha_{0}$} is irreducible in $\mathbb{F}{p^{m}}[x]$. If $x^{n}-\alpha{0}$ is reducible in $\mathbb{F}_{p^{m}}[x]$, we give the explicit expressions of the ideals of $R[x]/\langle x^{{np{s}}-\alpha-u\beta\rangle$.} Besides, the number of codewords and the dual code of every $\alpha+u\beta$-constacyclic code are provided.