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

TL;DR

This paper explores the structure and properties of skew cyclic codes over a specific ring extension, demonstrating their principal generation, Gray map images, and idempotent generators, thus advancing coding theory over non-field rings.

Contribution

It introduces a decomposition approach for skew cyclic codes over the ring $F_q+vF_q+v^2F_q$, and establishes their Gray images as skew 3-quasi cyclic codes, along with principal and idempotent generators.

Findings

Gray image of skew cyclic codes are skew 3-quasi cyclic codes.

Skew cyclic codes over the ring are principally generated.

Idempotent generators of these codes are explicitly obtained.

Abstract

In the present paper, we study skew cyclic codes over the ring $F_{q}+vF_{q}+v^2F_{q}$, where $v^3=v,~q=p^m$ and $p$ is an odd prime. We investigate the structural properties of skew cyclic codes over $F_{q}+vF_{q}+v^2F_{q}$ 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.