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

TL;DR

This paper introduces a new class of cyclic codes over product rings using automorphisms and derivations, characterizes their algebraic structure, and applies them to construct optimal quantum codes.

Contribution

It defines and analyzes $( heta, riangle_ heta, a)$-cyclic codes over product rings, providing their algebraic characterization, duality conditions, and applications in quantum code construction.

Findings

Characterization of $( heta, riangle_ heta, a)$-cyclic codes over $ ext{F}_q^l$

Construction of MDS and optimal linear codes via Gray maps

Development of quantum codes with good parameters from classical 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.