Hermitian Self-dual Generalized Reed-Solomon Codes

2026-02-06

AI Summary

This work addresses the long-standing open problem concerning the existence and explicit construction of Hermitian self-dual generalized Reed–Solomon (GRS) codes for code lengths \( n \leq q+1 \). By integrating tools from finite field algebra, Hermitian inner product theory, and structural analysis of GRS codes, we provide a complete classification of such codes. In particular, we confirm the conjecture that only two distinct families of Hermitian self-dual GRS codes exist within this parameter regime and present, for the first time, explicit constructions for both families. These results fully resolve the existence and construction questions for Hermitian self-dual GRS codes when \( n \leq q+1 \).

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Semi-Supervised Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

Maximum Distance Separable (MDS) self-dual codes are of significant theoretical and practical importance. Generalized Reed-Solomon (GRS) codes are the most prominent MDS codes. Correspondingly there have been many research on constructions of Euclidean self-dual MDS codes by using GRS codes. However, the study on Hermitian self-dual GRS codes is relatively limited. Since Hermitian self-dual GRS codes do not exist for $n>q+1$, this paper is devoted to an investigation of GRS codes in the case where $n\le q+1$. First, we prove that when $n\leq q+1$, there are only two classes of Hermitian self-dual GRS codes, confirming the conjecture in [13] and providing its proof simultaneously. Second, we present two explicit construction methods. Thus, the existence and construction of Hermitian self-dual GRS codes are fully solved.

Problem

• Hermitian self-dual
• Generalized Reed-Solomon codes
• MDS codes
• code existence

Innovation

• Hermitian self-dual
• Generalized Reed-Solomon codes
• MDS codes
• explicit construction
• code classification