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

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