On a class of $(\delta+\alpha u^2)$-constacyclic codes over $\mathbb{F}_{q}[u]/\langle u^4\rangle$

TL;DR

This paper classifies and explicitly describes a class of constacyclic codes over a finite chain ring, including their duals and self-dual codes for specific parameters, advancing coding theory over rings.

Contribution

It provides explicit representations of all $( ext{}oldsymbol{ extdelta}+oldsymbol{ extalpha} u^2 ext{)}$-constacyclic codes over the ring $R$, including duals and self-dual codes for certain cases.

Findings

Explicit classification of $( extdelta+ extalpha u^2)$-constacyclic codes over $R$

Determination of dual codes for each class

Construction of all self-dual codes when $q=2^m$ and $ extdelta=1$

Abstract

Let $\mathbb{F}_{q}$ be a finite field of cardinality $q$, $R=\mathbb{F}_{q}[u]/\langle u^4\rangle=\mathbb{F}_{q}+u\mathbb{F}_{q}+u^2\mathbb{F}_{q}+u^3\mathbb{F}_{q}$ $(u^4=0)$ which is a finite chain ring, and $n$ be a positive integer satisfying ${\rm gcd}(q,n)=1$. 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.