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 $α+uβ$-constacyclic codes of length $np^{s}$ over $R$, where $α,β\in\mathbb{F}_{p^{m}}^{*}$, $n, s\in\mathbb{N}_{+}$ and $\gcd(n,p)=1$. Let $α_{0}\in\mathbb{F}_{p^{m}}^{*}$ and $α_{0}^{p^{s}}=α$. The residue ring $R[x]/\langle x^{np^{s}}-α-uβ\rangle$ is a chain ring with the maximal ideal $\langle x^{n}-α_{0}\rangle$ in the case that $x^{n}-α_{0}$ is irreducible in $\mathbb{F}_{p^{m}}[x]$. If $x^{n}-α_{0}$ is reducible in $\mathbb{F}_{p^{m}}[x]$, we give the explicit expressions of the ideals of $R[x]/\langle x^{np^{s}}-α-uβ\rangle$. Besides, the number of codewords and the dual code of every $α+uβ$-constacyclic code are provided.

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