A note on the duals of skew constacyclic codes

TL;DR

This paper investigates the algebraic and geometric properties of duals of skew constacyclic codes over finite fields, providing new insights into their structure, duality, and specific classes like MDS codes.

Contribution

It offers a detailed analysis of the dual codes of skew constacyclic codes, including properties, consequences, and results on special subclasses such as 1-generator skew quasi-twisted and MDS skew constacyclic codes.

Findings

Characterization of duals of skew constacyclic codes

Properties of 1-generator skew quasi-twisted codes

Results on MDS skew constacyclic codes

Abstract

Let $\mathbb{F}_q$ be a finite field with $q$ elements and denote by $\theta : \mathbb{F}_q\to\mathbb{F}_q$ an automorphism of $\mathbb{F}_q$. In this paper, we deal with skew constacyclic codes, that is, linear codes of $\mathbb{F}_q^n$ which are invariant under the action of a semi-linear map $\Phi_{\alpha,\theta}:\mathbb{F}_q^n\to\mathbb{F}_q^n$, defined by $\Phi_{\alpha,\theta}(a_0,...,a_{n-2}, a_{n-1}):=(\alpha \theta(a_{n-1}),\theta(a_0),...,\theta(a_{n-2}))$ for some $\alpha\in\mathbb{F}_q\setminus\{0\}$ and $n\geq 2$. In particular, we study some algebraic and geometric properties of their dual codes and we give some consequences and research results on $1$-generator skew quasi-twisted codes and on MDS skew constacyclic codes.