On a class of left metacyclic codes

Abstract

Let $G_{(m,3,r)}=⟨x,y\mid x^m=1,y^3=1,yx=x^{r}y⟩$ be a metacyclic group of order $3m$, where $\gcd (m,r)=1$, $1<r<m$ 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 $\gcd (m,q)=1$ 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 ${\mathcal{A}}_i$ and outer codes $C_i$ is proved, where ${\mathcal{A}}_i$ 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.