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

Abstract

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 $n{p}^s$ over $R$, where $\alpha ,\beta \in {\mathbb{F}}_{p^m}^{\ast }$, $n,s\in {\mathbb{N}}_{+}$ and $\gcd (n,p)=1$. Let ${\alpha }_0\in {\mathbb{F}}_{p^m}^{\ast }$ and ${\alpha }_0^{p^s}=\alpha$. The residue ring $R[x]/⟨x^{n{p}^s}-\alpha -u\beta ⟩$ is a chain ring with the maximal ideal $⟨x^n-{\alpha }_0⟩$ 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 $R[x]/⟨x^{n{p}^s}-\alpha -u\beta ⟩$. Besides, the number of codewords and the dual code of every $\alpha +u\beta$-constacyclic code are provided.

arXiv admin note: text overlap with arXiv:1512.01406 by other authors