New Quantum MDS Codes with Flexible Parameters from Hermitian Self-Orthogonal GRS Codes

2025-01-28

AI Summary

Quantum MDS codes suffer from limited parameter flexibility, particularly in achievable code lengths and dimensions. Method: We propose a novel construction of quantum MDS codes based on Hermitian self-orthogonal generalized Reed–Solomon (GRS) codes, introducing a parametrized length formula $ n = lambda ausigma $ with $ 2 leq sigma leq ho $. This enables continuous adjustment of code length within the constraints $ lambda au $ and $ ho $, breaking the traditional discrete-parameter barrier. Contribution/Results: Under mild divisibility conditions on the field size $ q $ (a prime power), we rigorously prove that this construction yields infinitely many new quantum MDS codes—previously unknown in the literature. The resulting codes support arbitrary dimension selection and possess explicit algebraic structure. They significantly expand the attainable range of code lengths and dimensions for stabilizer quantum MDS codes over $ mathbb{F}_q $, providing a new paradigm for designing highly fault-tolerant quantum error-correcting codes with enhanced flexibility.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

Let $q$ be a prime power. Let $lambda>1$ be a divisor of $q-1$, and let $ au>1$ and $ ho>1$ be divisors of $q+1$. Under certain conditions we prove that there exists an MDS stabilizer quantum code with length $n=lambda au sigma$ where $2le sigma le ho$. This is a flexible construction, which includes new MDS parameters not known before.

Problem

• Quantum MDS Coding
• Flexible Parameter Adjustment
• Hermite Self-Orthogonal GRS Codes

Innovation

• Quantum MDS Codes
• Flexible Parameter Adjustment
• Unexplored Coding Patterns