Construction of Quantum Codes from (γ, Δ)-Cyclic Codes

Core Concepts

This paper proposes a new method to construct quantum codes from (γ, Δ)-cyclic codes over a class of finite commutative non-chain rings.

Abstract

The paper discusses the main algebraic properties of (γ, Δ)-cyclic codes over the ring Rq,s and provides a necessary and sufficient condition for these codes to contain their Euclidean duals.
Key highlights:

(γ, Δ)-cyclic codes over Rq,s are shown to be the direct sum of (θ, ℑ)-cyclic codes over Fq.
Necessary and sufficient conditions for both (γ, Δ)-cyclic and (θ, ℑ)-cyclic codes to contain their Euclidean duals are established.
Many new quantum codes are obtained by applying the dual containing criterion on the Gray images of these codes, with better parameters than those available in the literature.
The paper extends previous work on skew cyclic codes by considering both automorphisms and derivations, and proposes a fruitful application of (γ, Δ)-cyclic codes in the context of quantum code construction.

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Key Insights Distilled From

Construction of quantum codes from $(γ,Δ)$-cyclic codes

by Om Prakash,S... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01904.pdf

Deeper Inquiries

How can the proposed framework be extended to other types of non-commutative rings beyond Rq,s

The proposed framework of constructing quantum codes from (γ, Δ)-cyclic codes over Rq,s can be extended to other types of non-commutative rings by considering different automorphisms and derivations suitable for those rings. The key lies in adapting the definitions and properties of automorphisms and derivations to fit the specific structure of the non-commutative ring in question. By identifying the appropriate automorphisms and derivations for a given non-commutative ring, one can establish a similar framework for constructing quantum codes based on (γ, Δ)-cyclic codes in that context. This extension would involve analyzing the algebraic properties of the non-commutative ring, defining the necessary maps, and establishing the conditions for the codes to contain their duals.

What are the potential limitations or drawbacks of the (γ, Δ)-cyclic code approach for quantum code construction

One potential limitation of the (γ, Δ)-cyclic code approach for quantum code construction is the complexity that arises when dealing with non-commutative rings. Non-commutative rings introduce additional challenges compared to commutative rings, as the properties of automorphisms and derivations may behave differently in this context. The non-commutative nature of the rings can lead to more intricate calculations and analyses, making it harder to derive efficient quantum codes. Additionally, the application of (γ, Δ)-cyclic codes in quantum code construction may require a deeper understanding of the specific characteristics and behaviors of non-commutative rings, which could pose challenges in practical implementations.

Can the insights from this work be applied to improve the performance of quantum communication systems in practical scenarios

The insights from this work on constructing quantum codes from (γ, Δ)-cyclic codes can be applied to improve the performance of quantum communication systems in practical scenarios by enhancing the error-correcting capabilities of quantum codes. By leveraging the properties of (γ, Δ)-cyclic codes, researchers and engineers can design more robust quantum codes that are better equipped to handle errors and noise in quantum communication channels. This can lead to increased reliability and efficiency in quantum communication systems, ultimately improving the overall quality and security of quantum information transmission. Additionally, the development of advanced quantum codes based on (γ, Δ)-cyclic codes can contribute to the advancement of quantum cryptography, quantum key distribution, and other quantum communication protocols.