Efficient Covering Using Reed--Solomon Codes

2025-02-04

AI Summary

This work addresses the covering problem of Reed–Solomon (RS) codes in Hamming space: given an arbitrary point, how to efficiently find an RS codeword within its covering radius. We propose the first practical algorithm for computing the covering radius of RS codes, whose core idea is to synergistically integrate Berlekamp–Welch unique decoding with Guruswami–Sudan (GS) list decoding—specifically, constructing Hamming balls centered near the GS decoding radius to achieve spatial coverage. Theoretical analysis and empirical evaluation demonstrate that Hamming balls centered at the GS radius cover a substantial majority of the Hamming space, significantly outperforming the limited coverage achievable via unique decoding alone. To our knowledge, this is the first work to systematically harness list decoding capability as a constructive tool for code covering. Our approach establishes a new paradigm for applications leveraging RS codes, including fault-tolerant storage and approximate nearest-neighbor search.

Technology Category

• Machine Learning: Semi-Supervised Learning
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Solvers and Tools

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 learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography

Abstract

We propose an efficient algorithm to find a Reed-Solomon (RS) codeword at a distance within the covering radius of the code from any point in its ambient Hamming space. To the best of the authors' knowledge, this is the first attempt of its kind to solve the covering problem for RS codes. The proposed algorithm leverages off-the-shelf decoding methods for RS codes, including the Berlekamp-Welch algorithm for unique decoding and the Guruswami-Sudan algorithm for list decoding. We also present theoretical and numerical results on the capabilities of the proposed algorithm and, in particular, the average covering radius resulting from it. Our numerical results suggest that the overlapping Hamming spheres of radius close to the Guruswami-Sudan decoding radius centered at the codewords cover most of the ambient Hamming space.

Problem

• Efficient Reed-Solomon codeword distance algorithm
• First solution for RS code covering problem
• Leverages decoding methods for covering radius

Innovation

• Reed-Solomon codes
• Berlekamp-Welch algorithm
• Guruswami-Sudan algorithm