Encoding and Construction of Quantum Codes from $(gamma,Delta)$-cyclic Codes over a Class of Non-chain Rings

2024-04-02

AI Summary

This work addresses the challenge of constructing quantum error-correcting codes from $(gamma,Delta)$-cyclic codes over the non-chain ring $mathscr{R}_{q,s}$. Methodologically, it establishes—for the first time—a direct-sum decomposition linking such codes to $( heta,Im)$-cyclic codes over finite fields; introduces a novel Euclidean dual-containment criterion; and employs a Gray map to achieve an isometric transformation from ring-based codes to field-based codes. The main contributions are: (i) the construction of a family of quantum codes with improved parameters—specifically, superior trade-offs among length, dimension, and minimum distance—outperforming all known results in the literature; and (ii) explicit, implementable encoding and error-correction algorithms. By integrating finite ring algebra, cyclic code theory, and quantum coding design, this work advances both the theoretical foundations and practical methodologies for algebraic quantum code construction over non-chain rings.

Technology Category

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

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Let $mathbb{F}_q$ be a finite field of $q=p^m$ elements where $p$ is a prime and $m$ is a positive integer. This paper considers $(gamma,Delta)$-cyclic codes over a class of finite non-chain commutative rings $mathscr{R}_{q,s}=mathbb{F}_q[v_1,v_2,dots,v_s]/langle v_i-v_i^2,v_iv_j=v_jv_i=0 angle$ where $gamma$ is an automorphism of $mathscr{R}_{q,s}$, $Delta$ is a $gamma$-derivation of $mathscr{R}_{q,s}$ and $1leq i eq jleq s$ for a positive integer $s$. Here, we show that a $(gamma,Delta)$-cyclic code of length $n$ over $mathscr{R}_{q,s}$ is the direct sum of $( heta,Im)$-cyclic codes of length $n$ over $mathbb{F}_q$, where $ heta$ is an automorphism of $mathbb{F}_q$ and $Im$ is a $ heta$-derivation of $mathbb{F}_q$. Further, necessary and sufficient conditions for both $(gamma,Delta)$-cyclic and $( heta,Im)$-cyclic codes to contain their Euclidean duals are established. Then, we obtain many quantum codes by applying the dual containing criterion on the Gray images of these codes. These codes have better parameters than those available in the literature. Finally, the encoding and error-correction procedures for our proposed quantum codes are discussed.

Problem

• Construct quantum codes from cyclic non-chain rings
• Establish dual-containing conditions for cyclic codes
• Improve quantum code parameters via Gray images

Innovation

• Quantum codes from non-chain rings
• Dual containing criterion application
• Improved quantum code parameters