On a class of $(\delta +\alpha u^2)$-constacyclic codes over ${\mathbb{F}}_q[u]/⟨u^4⟩$

Abstract

Let ${\mathbb{F}}_q$ be a finite field of cardinality $q$, $R={\mathbb{F}}_q[u]/⟨u^4⟩={\mathbb{F}}_q+u{\mathbb{F}}_q+u^2{\mathbb{F}}_q+u^3{\mathbb{F}}_q$ $(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer satisfying $\gcd (q,n)=1$. For any $\delta ,\alpha \in {\mathbb{F}}_q^{\times }$, an explicit representation for all distinct $(\delta +\alpha u^2)$-constacyclic codes over $R$ of length $n$ is given, and the dual code for each of these codes is determined. For the case of $q=2^m$ and $\delta =1$, all self-dual $(1+\alpha u^2)$-constacyclic codes over $R$ of odd length $n$ are provided.