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

2025-01-03

AI Summary

This work addresses generalized cyclic codes—namely, $(Theta, Delta_Theta, mathbf{a})$-cyclic codes—in the vector space $mathcal{R} = mathbb{F}_q^l$ over a finite field $mathbb{F}_q$, where fundamental challenges persist: lack of a unified algebraic framework, unclear duality relations, and absence of systematic quantum code constructions. To resolve these, we first establish a unifying skew-polynomial ring framework; derive necessary and sufficient conditions for Euclidean and annihilator dual inclusions; and design an orthogonal-preserving Gray map that maintains code parameters when lifting codes from $mathcal{R}$ to $mathbb{F}_q$-linear codes. Leveraging this foundation, we systematically construct multiple families of MDS and almost-MDS quantum error-correcting codes. Key contributions include: a complete generating polynomial theory and decomposition theorem; the first unified model for $(Theta, Delta_Theta, mathbf{a})$-cyclic codes; an efficient duality testing algorithm; and novel quantum codes achieving optimal parameters.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

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 $mathbf{a}in mathcal{R}^ imes,$ 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.

Problem

• Quantum Coding
• Cyclic Codes
• Orthogonality Preservation

Innovation

• Cyclic Codes
• Quantum Coding
• Euclidean Duality