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

TL;DR

This paper introduces a unified framework for covering problems in Hamming and Grassmann spaces, deriving bounds on average covering radius and proposing Reed--Solomon-based constructions that outperform random codes in average performance.

Contribution

It develops a unified coding-theoretic approach, introduces the average covering radius concept, and proposes Reed--Solomon-based algorithms for improved average covering in both spaces.

Findings

Reed--Solomon-based codes outperform random codes in average covering radius.

Structured codes show strong average performance despite poor worst-case guarantees.

CRS codes asymptotically approach the random-coding bound in high-rate regimes.

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.