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Efficient Covering Using Reed--Solomon Codes

Samin Riasat, Hessam Mahdavifar

TL;DR

The paper addresses the covering problem for Reed--Solomon (RS) and generalized RS (GRS) codes by introducing a puncturing-based algorithm that leverages off-the-shelf decoders, including Berlekamp--Welch and Guruswami--Sudan. It proves correctness of a covering procedure and analyzes complexity, average punctures, and the resulting average covering radius, comparing against worst-case bounds. Numerical results show that the GS-based approach achieves a decoding radius $\tau_{\mathrm{GS}} = n - 1 - \lfloor \sqrt{(k - 1)n} \rfloor$ that closely matches the best possible covering radius $\tau_{\max}$, with average coverage approaching MAP performance while requiring few punctures on average. The work suggests that Hamming spheres of radius near $\tau_{\mathrm{GS}}$ cover a large fraction of the ambient space, and it outlines future work to extend the framework to CP codes and Grassmann-space quantization. The findings have implications for efficient quantization and covering in high-dimensional finite-field spaces, with potential applications in storage, communications, and related coding-theoretic quantization tasks.

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.

Efficient Covering Using Reed--Solomon Codes

TL;DR

The paper addresses the covering problem for Reed--Solomon (RS) and generalized RS (GRS) codes by introducing a puncturing-based algorithm that leverages off-the-shelf decoders, including Berlekamp--Welch and Guruswami--Sudan. It proves correctness of a covering procedure and analyzes complexity, average punctures, and the resulting average covering radius, comparing against worst-case bounds. Numerical results show that the GS-based approach achieves a decoding radius that closely matches the best possible covering radius , with average coverage approaching MAP performance while requiring few punctures on average. The work suggests that Hamming spheres of radius near cover a large fraction of the ambient space, and it outlines future work to extend the framework to CP codes and Grassmann-space quantization. The findings have implications for efficient quantization and covering in high-dimensional finite-field spaces, with potential applications in storage, communications, and related coding-theoretic quantization tasks.

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.

Paper Structure

This paper contains 15 sections, 3 theorems, 16 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Reed--Solomon Code
  4. Covering Radius
  5. Sphere Packing Versus Sphere Covering
  6. Covering Algorithms for GRS Codes
  7. Improvement Using List Decoding
  8. Complexity Analysis
  9. Coverage Fraction
  10. Quantization Using CP Codes
  11. Numerical Results
  12. $\tau_{\mathop{\mathrm{GS}}\nolimits}$ Versus $\tau_{\max}$
  13. Average Number of Punctures
  14. Average Covering Radius
  15. Conclusion and Future Directions

Key Result

Theorem 1

Given an $[n, k, d]_{q}$ GRS code ${\mathcal{C}}$ and $y \in {\mathbb F}_{q}^{n}$, $\mathop{\mathrm{GRS-cover}}\nolimits({\mathcal{C}}, y)$ returns a codeword $c \in {\mathcal{C}}$ with $d_{{\rm H}}(y, c) \le d - 1$.

Figures (3)

  • Figure 1: Lower bound on fraction of space covered by Hamming spheres of radius $\tau \in (d / 2, d)$ for $[14, 2, d]_{17}$ GRS code
  • Figure 2: Comparison of $\tau_{\max}$ with $\tau_{\mathop{\mathrm{GS}}\nolimits}$ for $q = 47, n = 46$ and $1 \le k < n$
  • Figure 3: Comparison of simulated average covering radius of a $[6, k]_{7}$ GRS code for $1 \le k < 6$ using different algorithms

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Conjecture 1
  • Theorem 2
  • proof
  • Corollary 1
  • proof