Dual codes of product semi-linear 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 linear codes of ${\mathbb{F}}_q^n$ invariant under a semi-linear map $T:{\mathbb{F}}_q^n\to {\mathbb{F}}_q^n$ for some $n\ge 2$. In particular, we study three kind of their dual codes, some relations between them and we focus on codes which are products of module skew codes in the non-commutative polynomial ring ${\mathbb{F}}_q[X,\theta ]$ as a subcase of linear codes invariant by a semi-linear map $T$. In this setting we give also an algorithm for encoding, decoding and detecting errors and we show a method to construct codes invariant under a fixed $T$.

arXiv commentv1: 37 pages; v2: 31 pages, the presentation of the main topics is improved by some minor changes and additional results, and some mistakes and typos have been corrected