On a class of left metacyclic codes

TL;DR

This paper develops a system theory for left metacyclic codes over finite fields, characterizing their structure as concatenated codes and analyzing dual and self-orthogonal codes.

Contribution

It introduces a novel framework for analyzing left metacyclic codes using cyclic and skew cyclic codes, providing explicit code descriptions and duality properties.

Findings

Codes are direct sums of concatenated cyclic and skew cyclic codes.

Explicit expressions for outer codes in the concatenation are derived.

Dual and self-orthogonal code characterizations are provided.

Abstract

Let $G_{(m,3,r)}=\langle x,y\mid x^m=1, y^3=1,yx=x^ry\rangle$ be a metacyclic group of order $3m$, where ${\rm gcd}(m,r)=1$, $1<r<m$ and $r^3\equiv 1$ (mod $m$). Then left ideals of the group algebra $\mathbb{F}_q[G_{(m,3,r)}]$ are called left metacyclic codes over $\mathbb{F}_q$ of length $3m$, and abbreviated as left $G_{(m,3,r)}$-codes. A system theory for left $G_{(m,3,r)}$-codes is developed for the case of ${\rm gcd}(m,q)=1$ and $r\equiv q^\epsilon$ for some positive integer $\epsilon$, only using finite field theory and basic theory of cyclic codes and skew cyclic codes. The fact that any left $G_{(m,3,r)}$-code is a direct sum of concatenated codes with inner codes ${\cal A}_i$ and outer codes $C_i$ is proved, where ${\cal A}_i$ is a minimal cyclic code over $\mathbb{F}_q$ of length $m$ and $C_i$ is a skew cyclic code of length $3$ over an extension field of $\mathbb{F}_q$. Then an explicit expression for each outer code in any concatenated code is provided. Moreover, the dual code of each left $G_{(m,3,r)}$-code is given and self-orthogonal left $G_{(m,3,r)}$-codes are determined.