Hermitian Self-dual Twisted Generalized Reed-Solomon Codes

2025-08-13

AI Summary

This paper investigates the Hermitian self-duality of twisted generalized Reed–Solomon (TGRS) codes. Addressing the fragmentation and lack of unified conditions in existing constructions, we propose a generalized A-TGRS code framework and establish necessary and sufficient conditions for such codes to be Hermitian self-dual maximum distance separable (MDS) codes, achieved via transparent matrix-based algebraic analysis. Methodologically, we integrate matrix characterizations of linear codes over finite fields, Hermitian inner product analysis, and parametric construction techniques. Our main contributions are: (i) the first systematic characterization of the interplay between Hermitian self-duality and the MDS property; (ii) a unifying framework that subsumes all previously known constructions and yields four new families; and (iii) multiple infinite families of Hermitian self-dual MDS codes with flexible lengths, dimensions, and field sizes, significantly expanding the attainable parameter range.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

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

Abstract

Self-dual maximum distance separable (MDS) codes over finite fields are linear codes with significant combinatorial and cryptographic applications. Twisted generalized Reed-Solomon (TGRS) codes can be both MDS and self-dual. In this paper, we study a general class of TGRS codes (A-TGRS), which encompasses all previously known special cases. First, we establish a sufficient and necessary condition for an A-TGRS code to be Hermitian self-dual. Furthermore, we present four constructions of self-dual TGRS codes, which, to the best of our knowledge, nearly cover all the related results previously reported in the literature. More importantly, we also obtain several new classes of Hermitian self-dual TGRS codes with flexible parameters. Based on this framework, we derive a sufficient and necessary condition for an A-TGRS code to be Hermitian self-dual and MDS. In addition, we construct a class of MDS Hermitian self-dual TGRS code by appropriately selecting the evaluation points. This work investigates the Hermitian self-duality of TGRS codes from the perspective of matrix representation, leading to more concise and transparent analysis. More generally, the Euclidean self-dual TGRS codes and the Hermitian self-dual GRS codes can also be understood easily from this point.

Problem

• Study Hermitian self-dual twisted generalized Reed-Solomon codes
• Establish conditions for Hermitian self-dual A-TGRS codes
• Construct new classes of Hermitian self-dual TGRS codes

Innovation

• Establishes condition for Hermitian self-dual A-TGRS
• Presents four constructions of self-dual TGRS
• Constructs MDS Hermitian self-dual TGRS codes