On a class of left metacyclic codes

Let $G{(m,3,r)}=\langle x,y\mid x^m=1, y^{3=1,yx=xry\rangle$} 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 $Ci$ is proved, where ${\cal A}i$ is a minimal cyclic code over $\mathbb{F}q$ of length $m$ and $Ci$ 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.