New Quantum Stabilizer Codes from generalized Monomial-Cartesian Codes constructed using two different generalized Reed-Solomon codes

2025-12-18

AI Summary

This work addresses the limited flexibility in code length and minimum distance inherent in conventional quantum error-correcting code constructions. We propose a novel framework—Generalized Monomial Cartesian Codes (GMCC)—by integrating two families of Generalized Reed–Solomon (GRS) codes over finite fields to construct linear codes satisfying the Hermitian self-orthogonality condition. To our knowledge, this is the first establishment of a dual-GRS collaborative construction paradigm. We derive explicit sufficient conditions for Hermitian self-orthogonality of GMCCs and, leveraging the CSS/Hermitian quantum code construction method, obtain a new family of stabilizer quantum codes with tunable parameters. Our approach breaks the constraints imposed by single-GRS-based constructions, substantially enhancing design freedom in code length, dimension, and minimum distance. This advances the algebraic toolkit for quantum code construction and extends its theoretical and practical applicability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

In this work, we define Generalized Monomial Cartesian Codes (GMCC), which constitute a natural extension of generalized Reed-Solomon codes. We describe how two different generalized Reed-Solomon codes can be combined to construct one GMCC. We further establish sufficient conditions ensuring that the GMCC are Hermitian self-orthogonal, thus leading to new constructions of quantum codes.

Problem

• Constructs new quantum stabilizer codes from generalized monomial-cartesian codes
• Combines two generalized Reed-Solomon codes to build these codes
• Establishes conditions for Hermitian self-orthogonality to enable quantum code creation

Innovation

• Generalized Monomial Cartesian Codes extend Reed-Solomon codes
• Combine two different generalized Reed-Solomon codes to construct GMCC
• Ensure Hermitian self-orthogonality for new quantum stabilizer codes