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Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based Constructions

Samin Riasat, Hessam Mahdavifar

TL;DR

This work addresses covering problems in both Hamming and Grassmann spaces by introducing the average covering radius as a natural distortion measure and deriving explicit non-asymptotic random-coding bounds via one-shot rate-distortion theory. It develops a puncturing-based covering algorithm for generalized Reed--Solomon codes and extends the framework to character-Reed--Solomon (CRS) codes for Grassmann quantization, with a corresponding covering procedure that runs in polynomial time. Empirically, RS-based constructions often outperform random codebooks in the Hamming space, while CRS codes can asymptotically approach the random-coding bound in the one-dimensional Grassmann space at high rates, albeit with a gap due to constant-modulus constraints. The results illuminate the impact of algebraic structure on average-case covering performance and point to potential amplitude-adaptive structured codes for improved Grassmann-space covering, with implications for high-dimensional quantization and limited-feedback wireless systems.

Abstract

We study covering problems in Hamming and Grassmann spaces through a unified coding-theoretic and information-theoretic framework. Viewing covering as a form of quantization in general metric spaces, we introduce the notion of the average covering radius as a natural measure of average distortion, complementing the classical worst-case covering radius. By leveraging tools from one-shot rate-distortion theory, we derive explicit non-asymptotic random-coding bounds on the average covering radius in both spaces, which serve as fundamental performance benchmarks. On the construction side, we develop efficient puncturing-based covering algorithms for generalized Reed--Solomon (GRS) codes in the Hamming space and extend them to a new family of subspace codes, termed character-Reed--Solomon (CRS) codes, for Grassmannian quantization under the chordal distance. Our results reveal that, despite poor worst-case covering guarantees, these structured codes exhibit strong average covering performance. In particular, numerical results in the Hamming space demonstrate that RS-based constructions often outperform random codebooks in terms of average covering radius. In the one-dimensional Grassmann space, we numerically show that CRS codes over prime fields asymptotically achieve average covering radii within a constant factor of the random-coding bound in the high-rate regime. Together, these results provide new insights into the role of algebraic structure in covering problems and high-dimensional quantization.

Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based Constructions

TL;DR

This work addresses covering problems in both Hamming and Grassmann spaces by introducing the average covering radius as a natural distortion measure and deriving explicit non-asymptotic random-coding bounds via one-shot rate-distortion theory. It develops a puncturing-based covering algorithm for generalized Reed--Solomon codes and extends the framework to character-Reed--Solomon (CRS) codes for Grassmann quantization, with a corresponding covering procedure that runs in polynomial time. Empirically, RS-based constructions often outperform random codebooks in the Hamming space, while CRS codes can asymptotically approach the random-coding bound in the one-dimensional Grassmann space at high rates, albeit with a gap due to constant-modulus constraints. The results illuminate the impact of algebraic structure on average-case covering performance and point to potential amplitude-adaptive structured codes for improved Grassmann-space covering, with implications for high-dimensional quantization and limited-feedback wireless systems.

Abstract

We study covering problems in Hamming and Grassmann spaces through a unified coding-theoretic and information-theoretic framework. Viewing covering as a form of quantization in general metric spaces, we introduce the notion of the average covering radius as a natural measure of average distortion, complementing the classical worst-case covering radius. By leveraging tools from one-shot rate-distortion theory, we derive explicit non-asymptotic random-coding bounds on the average covering radius in both spaces, which serve as fundamental performance benchmarks. On the construction side, we develop efficient puncturing-based covering algorithms for generalized Reed--Solomon (GRS) codes in the Hamming space and extend them to a new family of subspace codes, termed character-Reed--Solomon (CRS) codes, for Grassmannian quantization under the chordal distance. Our results reveal that, despite poor worst-case covering guarantees, these structured codes exhibit strong average covering performance. In particular, numerical results in the Hamming space demonstrate that RS-based constructions often outperform random codebooks in terms of average covering radius. In the one-dimensional Grassmann space, we numerically show that CRS codes over prime fields asymptotically achieve average covering radii within a constant factor of the random-coding bound in the high-rate regime. Together, these results provide new insights into the role of algebraic structure in covering problems and high-dimensional quantization.
Paper Structure (26 sections, 12 theorems, 80 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 26 sections, 12 theorems, 80 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Motivation
  3. Related Work
  4. Our Contribution
  5. Preliminaries
  6. Reed--Solomon Code
  7. Chordal Distance and Subspace Code
  8. Covering Radius as a Measure of Distortion
  9. Average Covering Radius
  10. Average Distortion of Random Codes
  11. Covering Using GRS Codes
  12. Algorithm
  13. Improvement Using List Decoding
  14. Complexity Analysis
  15. Average Number of Punctures
  16. ...and 11 more sections

Key Result

Theorem 1

Let ${\mathcal{D}}_{{\mathcal{Y}}}$ denote the uniform distribution on the Hamming space ${\mathcal{Y}} = {\mathbb F}_{q}^{n}$. Then, the expected average covering radius of a random code ${\mathcal{C}}_{{\mathcal{Y}}}$ consisting of $M$ codewords drawn independently from ${\mathcal{D}}_{{\mathcal{Y where

Figures (6)

  • Figure 1: Comparison of simulated average covering radii of a $[6, k]_{7}$ GRS code and random codes for $1 \le k < 6$
  • Figure 2: Simulated average covering radius of a $[q - 1, \lfloor (q - 1) R \rfloor]_{q}$ GRS code given by $\mathop{\mathrm{GRS-cover}}\nolimits$ with $\mathop{\mathrm{GRS-decode}}\nolimits = \mathop{\mathrm{GS}}\nolimits$ and number of punctures until it succeeds for $5 \le q < 32$
  • Figure 3: Lower bound on fraction of space covered by Hamming spheres of radius $\tau \in (d / 2, d)$ for a $[14, 2, d]_{17}$ GRS code
  • Figure 4: $\tau_{\max}$ vs. $\tau_{\mathop{\mathrm{GS}}\nolimits}$ for $q = 47, n = 46$ and $1 \le k \le 37$
  • Figure 5: Comparison of simulated average covering radii of a $(6, k)_{7}$ CRS code and random codes for $1 \le k < 6$
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 1: Conway96Hessam22
  • Definition 2: Character
  • Definition 3: CP Code Hessam22
  • Definition 4: Average Covering Radius
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 20 more