On skew cyclic codes over $F_q+vF_q+v^2F_q$

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