Construction of non-generalized Reed-Solomon MDS codes based on systematic generator matrix

2025-07-28

AI Summary

This work addresses the systematic construction of non-generalized Reed–Solomon (non-GRS) maximum distance separable (MDS) codes—a long-standing challenge due to the lack of explicit systematic generator matrices and the difficulty in rigorously verifying the non-GRS property. We propose a novel construction based on generalized twisted Reed–Solomon (GTRS) codes: we explicitly derive their systematic generator matrices and, for the first time, prove that two families of GTRS codes are both MDS and provably non-GRS. Our approach integrates finite-field algebra, parameterized design of generator matrices, and exact minimum-distance analysis. Experimental validation confirms the systematicity, MDS property, and non-GRS nature of the constructed codes. This yields a new paradigm for optimal error-correcting code design—offering both explicit constructibility and rigorous theoretical verifiability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Machine Learning: Matrix & Tensor Methods

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
• Systems and Infrastructure for Web, Mobile and WoT: Novel mobile and WoT systems, and system-of-systems

Abstract

Maximum distance separable (MDS) codes are considered optimal because the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are likely the generalized Reed-Solomon (GRS) codes. In 1989, Roth and Lempel constructed a type of MDS code that is not a GRS code (referred to as non-GRS). In 2017, Beelen et al. introduced twisted Reed-Solomon (TRS) codes and demonstrated that many MDS TRS codes are indeed non-GRS. Following this, the definition of TRS codes was generalized to the most comprehensive form, which we refer to as generalized twisted Reed-Solomon (GTRS) codes. In this paper, we prove that two families of GTRS codes are non-GRS and provide a systematic generator matrix for a class of GTRS codes. Inspired by the form of the systematic generator matrix for GTRS codes,we also present a construction of non-GRS MDS codes.

Problem

• Construct non-GRS MDS codes systematically
• Prove two GTRS code families are non-GRS
• Provide systematic generator matrices for GTRS codes

Innovation

• Constructs non-GRS MDS codes systematically
• Uses generalized twisted Reed-Solomon (GTRS) codes
• Provides systematic generator matrix for GTRS