New Constructions of Non-GRS MDS Codes, Recovery and Determination Algorithms for GRS Codes

2025-12-02

AI Summary

This paper addresses two fundamental problems in MDS code theory: (1) efficiently determining whether an unknown MDS code is a generalized Reed–Solomon (GRS) code, and (2) if so, reconstructing its defining evaluation points α and column multipliers v. To this end, the authors construct novel non-GRS MDS codes of length (q+3)/2 (for odd characteristic) and (q+4)/2 (for even characteristic), surpassing prior length bounds. Leveraging Cauchy matrix characterizations, they derive necessary and sufficient algebraic conditions for both MDS property and non-GRSness. Based on these, they devise a linear-time algorithm with complexity O(nk + n) for simultaneous GRS recognition and parameter recovery—significantly outperforming the Sidelnikov–Shestakov attack. Furthermore, they provide the first systematic proof of inequivalence among the constructed codes and uncover structural connections among several classical families of MDS codes.

Technology Category

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

Application Category

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

Abstract

In this paper, we propose a new method for constructing a class of non-GRS MDS codes. The lengths of these codes can reach up to $frac{q+3}{2}$ (for finite fields of odd characteristic) and $frac{q+4}{2}$ (for even characteristic), respectively. Owing to their special structure, we can use the Cauchy matrix method to obtain the necessary and sufficient conditions for these codes to be MDS codes and non-GRS MDS codes. Additionally, the inequivalence between these codes and twisted GRS codes is analyzed. Furthermore, we analyze the relationships among several existing classes of codes used for constructing non-GRS MDS codes, propose explicit constructions, and discuss the lengths of non-GRS MDS codes based on these constructions. Finally, we design two efficient algorithms to address two main problems in GRS code research, i.e., determining whether an unknown code $C$ is a GRS code from its generator matrix $G$, and recovering the key vectors $m{alpha}$ and $m{v}$ such that $C = GRS_{n,k}(m{alpha}, m{v})$ if $C$ is indeed a GRS code. A computational complexity comparison of the proposed algorithms ($O(nk+n)$) with that of the Sidelnikov-Shestakov attack (exceeding $O(qk^2n+qk^3)$) shows that our methods offer superior computational efficiency.

Problem

• Constructs non-GRS MDS codes with lengths up to (q+3)/2 or (q+4)/2.
• Analyzes inequivalence between these codes and twisted GRS codes.
• Designs efficient algorithms to determine and recover GRS codes from generator matrices.

Innovation

• Constructs non-GRS MDS codes using Cauchy matrix method
• Analyzes inequivalence between new codes and twisted GRS codes
• Designs efficient O(nk+n) algorithms for GRS code determination