Constacyclic codes of length $(p^r,p^s)$ over mixed alphabets

Abstract

For any prime p and positive integer m, let R be the finite commutative ring ${\mathbb {F}}_{p^m}+u{\mathbb {F}}_{p^m}+v{\mathbb {F}}_{p^m}+uv{\mathbb {F}}_{p^m}$, where$u^2=0,v^2=0$ and $uv=vu$. Let $\lambda =\lambda _1+u\lambda _2+v\lambda _3+uv\lambda _4$ be a unit of R, where $\lambda _1, \lambda _2,\lambda _3, \lambda _4 \in \mathbb F_{p^m}$ and $\lambda _1\ne 0$. We know that $\lambda$-constacyclic codes of length $p^s$ over R are exactly ideals of the ring $\frac{R[x]}{ \langle x^{p^s} -\lambda \rangle }$. For all possible values of $\lambda$, we study $\lambda$-constacyclic codes of length $p^s$ over R. We also extend structures of codes from single alphabet to mixed alphabet, and determine separable constacyclic codes of length $(p^r, p^s)$ over ${\mathbb {F}}_{p^m}R$.