Duality between Packings and Coverings of the Hamming Space

TL;DR

This paper explores the duality between packing and covering densities in binary codes, revealing that the prevalence of good packings or coverings in code classes is mutually exclusive, and characterizing maximal codes as either good packings or coverings.

Contribution

It establishes duality relationships between packing and covering problems in binary codes and characterizes maximal codes within this duality framework.

Findings

Almost all codes are either good packings or good coverings, but not both.

Maximal binary codes are either good packings or good coverings.

Duality relationships link the prevalence of packing and covering properties in code classes.

Abstract

We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems. Specifically, we prove that if almost all codes (in the class of linear or nonlinear codes) are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all codes are good coverings, then at most a vanishing fraction of codes are good packings. We also show that any specific maximal binary code is either a good packing or a good covering, in a certain well-defined sense.