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 \({\mathbb {F}}_{p^m}R\).

对于任何素数p和正整数m，令R为有限交换环\（{\ mathbb {F}} _ {p ^ m} + u {\ mathbb {F}} _ {p ^ m} + v {\ mathbb {F}} _ {p ^ m} + uv {\ mathbb {F}} _ {p ^ m} \），其中\（u ^ 2 = 0，v ^ 2 = 0 \）和\（uv = vu \）。令\（\ lambda = \ lambda _1 + u \ lambda _2 + v \ lambda _3 + uv \ lambda _4 \）为R的单位，其中\（\ lambda _1，\ lambda _2，\ lambda _3，\ lambda _4 \ in \ mathbb F_ {p ^ m} \）和\（\ lambda _1 \ ne 0 \）。我们知道在R上长度为\（p ^ s \）的\（\ lambda \）-恒定周期代码恰好是环的理想状态\（\ frac {R [x]} {\ langle x ^ {p ^ s}-\ lambda \ rangle} \）。对于所有可能的值\（\拉姆达\） ，我们研究\（\拉姆达\）长度的-constacyclic码\（第2 -S \）过- [R 。我们还将代码结构从单个字母扩展到混合字母，并确定长度为\（（p ^ r，p ^ s）\）的\（{\ mathbb {F}} _ {p ^ m} R \）。