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

Akanksha, Anuj Kumar Bhagat, Ritumoni Sarma

TL;DR

The paper advances the theory of codes over the product ring $\mathcal{R}=\mathbb{F}_q^l$ by introducing $(\Theta, \Delta_{\Theta}, \mathbf{a})$-cyclic codes, deriving their algebraic characterization and generator structures, and showing a direct decomposition into skew-constacyclic components over $\mathbb{F}_q$. It develops Gray maps from $\mathcal{R}^n$ to $\mathbb{F}_q^{nl}$ that preserve distance and, under suitable matrix conditions, dual containment, enabling the construction of high-performance linear codes. The authors then apply Euclidean and annihilator CSS constructions to produce MDS and almost-MDS quantum codes with explicit parameters, supported by numerous examples computed in MAGMA and SageMath. The work significantly broadens the toolkit for generating quantum codes with good parameters by leveraging ring-skew structures and Gray-mapped images.

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 $Θ$ of $\mathcal{R},$ a $Θ$-derivation $Δ_Θ$ of $\mathcal{R}$ and for a unit $\mathbf{a}$ in $\mathcal{R},$ we study $(Θ, Δ_Θ, \mathbf{a})$-cyclic codes over $\mathcal{R}.$ In this direction, we give an algebraic characterization of a $(Θ, Δ_Θ, \mathbf{a})$-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 $(Θ, 0, \mathbf{a})$-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 $(Θ, Δ_Θ, \mathbf{a})$-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.

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

TL;DR

The paper advances the theory of codes over the product ring by introducing -cyclic codes, deriving their algebraic characterization and generator structures, and showing a direct decomposition into skew-constacyclic components over . It develops Gray maps from to that preserve distance and, under suitable matrix conditions, dual containment, enabling the construction of high-performance linear codes. The authors then apply Euclidean and annihilator CSS constructions to produce MDS and almost-MDS quantum codes with explicit parameters, supported by numerous examples computed in MAGMA and SageMath. The work significantly broadens the toolkit for generating quantum codes with good parameters by leveraging ring-skew structures and Gray-mapped images.

Abstract

In this article, for a finite field and a natural number let denote the product ring Firstly, for an automorphism of a -derivation of and for a unit in we study -cyclic codes over In this direction, we give an algebraic characterization of a -cyclic code over , determine its generator polynomial, and find its decomposition over Secondly, we give a necessary and sufficient condition for a -cyclic code to be Euclidean dual-containing code over Thirdly, we study Gray maps and obtain several MDS and optimal linear codes over as Gray images of -cyclic codes over 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.
Paper Structure (12 sections, 27 theorems, 37 equations, 3 tables)

This paper contains 12 sections, 27 theorems, 37 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Automorphisms and derivations of
  4. -cyclic codes over
  5. -cyclic codes over
  6. -cyclic codes over
  7. Gray maps on $\mathcal{R}$
  8. Quantum codes from -cyclic codes over
  9. Dual-containing codes with respect to the Euclidean inner product
  10. Annihilator dual of a -cyclic code over
  11. Examples and Tables
  12. Conclusion

Key Result

Theorem 2.4

jian2022 Let $f(x),g(x)\in A[x;\theta,\delta_\theta],$ where the leading coefficient of $g(x)$ is a unit. Then there exists $q(x), r(x)\in A[x;\theta,\delta_\theta]$ such that where $r(x)=0$ or $\deg{r}(x)<\deg{g}(x).$

Theorems & Definitions (65)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Right Division Algorithm
  • Definition 2.5
  • Definition 2.6
  • Example 1
  • Definition 2.7
  • Lemma 3.1
  • proof
  • ...and 55 more