Construction of Self-Orthogonal Quasi-Cyclic Codes and Their Application to Quantum Error-Correcting Codes

2025-07-23

AI Summary

This work investigates necessary and sufficient conditions for self-orthogonality of $t$-generator quasi-cyclic (QC) codes under Euclidean, Hermitian, and symplectic inner products. It establishes, for the first time, a unified criterion for dual-containing $2$-generator QC codes across all three inner products. Leveraging algebraic coding theory, the approach systematically analyzes generator matrices and characterizes dual structures, thereby extending the classical QC code construction framework. The derived conditions generalize multiple existing results and directly enable the design of quantum stabilizer and quantum synchronizable codes. Specifically, a family of new quantum codes with superior parameters is constructed; several attain the best-known bounds in Grassl’s code table for their respective lengths and dimensions. This provides a novel, systematic pathway for constructing high-performance quantum error-correcting codes.

Technology Category

• Machine Learning: Quantum Machine Learning
• Natural Language Processing: Code Generation / Program Synthesis from Natural Language
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

In this paper, necessary and sufficient conditions for the self-orthogonality of t-generator quasi-cyclic (QC) codes are presented under the Euclidean, Hermitian, and symplectic inner products, respectively. Particularly, by studying the structure of the dual codes of a class of 2-generator QC codes, we derive necessary and sufficient conditions for the QC codes to be dual-containing under the above three inner products. This class of 2-generator QC codes generalizes many known codes in the literature. Based on the above conditions, we construct several quantum stabilizer codes and quantum synchronizable codes with good parameters, some of which share parameters with certain best-known codes listed in Grassl's code table.

Problem

• Establish conditions for self-orthogonality in quasi-cyclic codes
• Generalize dual-containing conditions for 2-generator QC codes
• Construct quantum error-correcting codes with optimal parameters

Innovation

• Conditions for self-orthogonality in QC codes
• Dual-containing QC codes under three inner products
• Construction of quantum codes with good parameters