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.

翻译：我们通过统一的编码理论与信息论框架研究汉明空间和格拉斯曼空间中的覆盖问题。将覆盖视为一般度量空间中的一种量化形式，我们引入平均覆盖半径的概念作为平均失真的自然度量，从而补充了经典的最坏情况覆盖半径。借助单次率失真理论的工具，我们推导出两个空间中平均覆盖半径的显式非渐近随机编码界限，这些界限构成了基本的性能基准。在构造方面，我们为汉明空间中的广义里德-所罗门码开发了高效的基于删余的覆盖算法，并将其扩展至一类新的子空间码——称为特征-里德-所罗门码——用于弦距离下的格拉斯曼量化。我们的结果表明，尽管这些结构化码在最坏情况覆盖保证方面表现较差，但它们展现出强大的平均覆盖性能。具体而言，汉明空间中的数值结果表明，基于RS的构造在平均覆盖半径方面通常优于随机码本。在一维格拉斯曼空间中，我们通过数值证明，在素数域上的CRS码在高码率区域渐近地达到与随机编码界限相差常数因子的平均覆盖半径。这些结果共同为覆盖问题和高维量化中代数结构的作用提供了新的见解。