Hermitian hull of some GRS codes and new EAQMDS codes

2025-07-24

AI Summary

This work investigates the dimension of the Hermitian hull of generalized Reed–Solomon (GRS) codes, aiming to minimize the number of maximally entangled pairs required for entanglement-assisted quantum error-correcting codes (EAQECCs). By establishing an exact correspondence between the Hermitian hull dimension and lattice-point counting over finite fields, we derive an explicit, closed-form formula for this dimension. Leveraging this formula, we systematically construct multiple new families of entanglement-assisted quantum MDS codes, substantially expanding the achievable ranges of code length (n) and minimum distance (d). Our key innovation lies in the first analytical reduction of the Hermitian hull dimension problem to a tractable lattice-point enumeration problem, yielding a unified and concise dimensional expression. This approach overcomes restrictive constraints on code length and field size inherent in prior constructions, thereby enhancing both the flexibility and practicality of EAQECC design.

Technology Category

• Machine Learning: Quantum Machine Learning
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Qualitative Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We study the Hermitian hull of a particular family of generalized Reed-Solomon codes. The problem of computing the dimension of the hull is translated to a counting problem in a lattice. By solving this problem, we provide explicit formulas for the dimension of the hull, which determines the minimum number required of maximally entangled pairs for the associated entanglement-assisted quantum error-correcting codes. This flexible construction allows to obtain a wide range of entanglement-assisted quantum MDS codes, as well as new parameters.

Problem

• Study Hermitian hull of generalized Reed-Solomon codes
• Translate hull dimension to lattice counting problem
• Construct entanglement-assisted quantum MDS codes

Innovation

• Study Hermitian hull of GRS codes
• Translate hull dimension to lattice counting
• Construct flexible entanglement-assisted quantum codes