$(\theta , \delta _\theta )$-Cyclic codes over $\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle$

Abstract

Let $\mathbb {F}_q$ be the finite field of order $q=p^m$, where p is a prime, m is a positive integer, and $\mathcal {R}=\mathbb {F}_q[u,v]/\langle u^2-u, v^2-v, uv-vu \rangle$. Thus $\mathcal {R}[x;\theta ,\delta _\theta ]$ is a noncommutative ring, known as skew polynomial ring, where $\theta$ is an automorphism of $\mathcal {R}$ and $\delta _\theta$ is a $\theta$-derivation of $\mathcal {R}$. The main concern of this work is to characterize $(\theta , \delta _\theta )$-cyclic codes over the ring $\mathcal {R}$. Towards this, first we establish existence of the right division algorithm in $\mathcal {R}[x;\theta ,\delta _\theta ]$. Then we find generating polynomials and idempotent generators for $(\theta , \delta _\theta )$-cyclic codes over the ring $\mathcal {R}$. Moreover, it is shown that $(\theta , \delta _\theta )$-cyclic codes are principally generated. Finally, by using the decomposition method, we have provided several examples of $(\theta , \delta _\theta )$-cyclic codes of different lengths over $\mathcal {R}$ out of them many are optimal as per the available database (Grassl, Code Tables: bounds on the parameters of various types of codes. http://www.codetables.de/).