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

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