Constacyclic codes of length $n{p}^s$ over $\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^t⟩}$: Torsions and Cardinalities

Abstract

The purpose of this article is to study constacyclic codes of length $n{p}^s$ over $R^t:=\frac{{\mathbb{F}}_{p^m}[u]}{⟨u^t⟩},$ where $t$ is a natural number and $\gcd (n,p)=1$. We give generators of all the ideals of $R_{\delta }^{t,n}:=\frac{R^t[x]}{⟨x^{n{p}^s}-\delta ⟩},$ where $\delta ={\delta }_0+u{\delta }_1+\cdots +u^{t-1}{\delta }_{t-1}$ is a unit in $R^t$. For $n=1,2,3$ and $t=3$, we provide all types of ideals (constacyclic codes) and also give the torsional degrees as well as cardinalities of these codes.