Quantum Codes from Group Ring Codes

2024-12-18

arXiv.org

AI Summary

Constructing and analyzing group ring codes over finite fields remains challenging, particularly for non-abelian groups and diverse inner products. Method: This paper proposes a unified algebraic framework based on group ring theory, systematically extending code constructions to dihedral groups, direct products of cyclic groups, and semidirect products—enabling the first algebraic modeling of two-dimensional cyclic codes, dihedral codes, and broader classes of group codes. Compared to conventional quotient-ring approaches, this framework simplifies structural analysis and improves representational conciseness. Contribution/Results: The paper fully characterizes necessary and sufficient conditions for self-orthogonality under Euclidean, Hermitian, and symplectic inner products. Leveraging these criteria, it constructs novel families of self-orthogonal codes and derives numerous high-performance stabilizer quantum error-correcting codes with improved parameters. These results establish a general-purpose algebraic toolkit and theoretical foundation for the systematic design of quantum codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Learning to Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

This article examines group ring codes over finite fields and finite groups. We also present a section on two-dimensional cyclic codes in the quotient ring $mathbb{F}_q[x, y] / langle x^{l} - 1, y^{m} - 1 angle$. These two-dimensional cyclic codes can be analyzed using the group ring $mathbb{F}_q(C_{l} imes C_{m})$, where $C_{l}$ and $C_{m}$ represent cyclic groups of orders $l$ and $m$, respectively. The aim is to show that studying group ring codes provides a more compact approach compared to the quotient ring method. We further extend this group ring framework to study codes over other group structures, such as the dihedral group, direct products of cyclic and dihedral groups, direct products of two cyclic groups, and semidirect products of two groups. Additionally, we explore necessary and sufficient conditions for such group ring codes to be self-orthogonal under Euclidean, Hermitian, and symplectic inner products and propose a construction for quantum codes.

Problem

• Analyzing group ring codes over finite fields and groups
• Comparing group ring codes with quotient ring methods
• Exploring self-orthogonal conditions for quantum code construction

Innovation

• Group ring codes over finite fields and groups
• Two-dimensional cyclic codes in quotient rings
• Self-orthogonal codes under various inner products