Technology Category
Machine Learning: Quantum Machine LearningKnowledge Representation and Reasoning: Qualitative ReasoningCognitive Modeling & Cognitive Systems: Neural Spike Coding
$(Theta, Delta_Theta, mathbf{a})$-cyclic codes over $mathbb{F}_q^l$ and their applications in the construction of quantum codes
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Quantum Coding
Cyclic Codes
Orthogonality Preservation
Innovation
Methods, ideas, or system contributions that make the work stand out.
Cyclic Codes
Quantum Coding
Euclidean Duality
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