Dual codes of product semi-linear codes

TL;DR

This paper investigates dual codes of product semi-linear codes over finite fields, exploring their properties, relations, and construction methods, including algorithms for encoding, decoding, and error detection.

Contribution

It introduces new insights into duals of semi-linear invariant codes, especially those formed as products of module skew codes, and provides practical algorithms for their implementation.

Findings

Analyzed relations between different dual codes.

Developed algorithms for encoding, decoding, and error detection.

Proposed methods to construct codes invariant under specific semi-linear maps.

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 $n\geq 2$. 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 $T$.