On a Class of Left Metacyclic Codes.

Let G(m,3,r)=\langle x,y\mid x^{m}=1, y^{3}=1,yx=x^{r}y\rangle be a metacyclic group of order 3m , where {\mathrm{ gcd}}(m,r)=1 , 1<r<m and r^3\equiv 1 (mod m ). Then, left ideals of the group algebra \mathbb Fq[G(m,3,r)] are called left metacyclic codes over \mathbb Fq 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 \mathrm gcd(m,q)=1 and r\equiv q^\epsilon (mod m ) 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 Ci is proved, where \mathcal Ai is a minimal cyclic code over \mathbb Fq of length m and G_{(m,3,r)} -code is given and self-orthogonal left G_{(m,3,r)}$ -codes are determined. [ABSTRACT FROM PUBLISHER]