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

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