A note on the duals of 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 $n\ge 2$. 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.

31 pages, 3 tables; this is a revised version that includes improvements to the presentation of the main results, a new subsection and an appendix which is an extension of Section 2 of the previous version