Skew cyclic and skew $({\alpha }_1+u{\alpha }_2+v{\alpha }_3+uv{\alpha }_4)$-constacyclic codes over $F_q+u{F}_q+v{F}_q+uv{F}_q$

Abstract

In this note, we study skew cyclic and skew constacyclic codes over the ring $\mathcal{R}=F_q+u{F}_q+v{F}_q+uv{F}_q$ where $q=p^m,$ $p$ is an odd prime, $u^2=u,v^2=v,uv=vu$. We show that Gray images of a skew cyclic and skew $\alpha$-constacyclic code of length $n$ are skew quasi-cyclic code of length $4n$ over $F_q$ of index 4. Also, it is shown that skew $\alpha$-constacyclic codes are either equivalent to $\alpha$-constacyclic codes or $\alpha$-quasi-twisted codes over $\mathcal{R}$. Further, structural properties, specially, generating polynomials and idempotent generators for skew cyclic and skew constacyclic codes are determined by decomposition method.

Communicated to International Journal of Information and Coding Theory