$(\Theta ,{\Delta }_{\Theta },\mathbf{a})$-cyclic codes over ${\mathbb{F}}_q^l$ and their applications in the construction of quantum codes

Abstract

In this article, for a finite field ${\mathbb{F}}_q$ and a natural number $l,$ let $\mathcal{R}$ denote the product ring ${\mathbb{F}}_q^l.$ Firstly, for an automorphism $\Theta$ of $\mathcal{R},$ a $\Theta$-derivation ${\Delta }_{\Theta }$ of $\mathcal{R}$ and for a unit $\mathbf{a}$ in $\mathcal{R},$ we study $(\Theta ,{\Delta }_{\Theta },\mathbf{a})$-cyclic codes over $\mathcal{R}.$ In this direction, we give an algebraic characterization of a $(\Theta ,{\Delta }_{\Theta },\mathbf{a})$-cyclic code over $\mathcal{R}$, determine its generator polynomial, and find its decomposition over ${\mathbb{F}}_q.$ Secondly, we give a necessary and sufficient condition for a $(\Theta ,0,\mathbf{a})$-cyclic code to be Euclidean dual-containing code over $\mathcal{R}.$ Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over ${\mathbb{F}}_q$ as Gray images of $(\Theta ,{\Delta }_{\Theta },\mathbf{a})$-cyclic codes over $\mathcal{R}.$ Moreover, we determine orthogonality preserving Gray maps and construct Euclidean dual-containing codes with good parameters. Lastly, as an application, we construct MDS and almost MDS quantum codes by employing the Euclidean dual-containing and annihilator dual-containing CSS constructions.