On skew cyclic codes over $F_q+v{F}_q+v^2{F}_q$

Abstract

In the present paper, we study skew cyclic codes over the ring $F_q+v{F}_q+v^2{F}_q$, where $v^3=v,q=p^m$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $F_q+v{F}_q+v^2{F}_q$ using decomposition method. By defining a Gray map from $F_q+v{F}_q+v^2{F}_q$ to $F_q^3$, it has been proved that the Gray image of a skew cyclic code of length $n$ over $F_q+v{F}_q+v^2{F}_q$ is a skew $3$-quasi cyclic code of length $3n$ over $F_q$. Further, it is shown that the skew cyclic codes over $F_q+v{F}_q+v^2{F}_q$ are principally generated. Finally, the idempotent generators of skew cyclic codes over $F_q+v{F}_q+v^2{F}_q$ are also obtained.