Packing and covering in higher dimensions

TL;DR

This survey explores high-dimensional packing and covering problems, highlighting key developments like the Leech lattice and recent sphere packing solutions, and discusses bounds and structures of optimal arrangements.

Contribution

It provides a comprehensive overview of high-dimensional packing and covering, summarizing known results, methods for bounds, and structural insights into optimal configurations.

Findings

Recent solution of sphere packing in dimensions 8 and 24

Summary of upper bounds for packing densities

Discussion on the structure of optimal arrangements

Abstract

The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the Leech lattice and the linear programming bound, which culminated in the recent solution of the sphere packing problem in dimensions 8 and 24, were influenced by coding theory. After mentioning the known results concerning existence of economical packings and coverings we discuss the different methods yielding upper bounds for the density of packing congruent balls. We summarize the few results on upper bounds for the packing density of general convex bodies. The paper closes with some remarks on the structure of optimal arrangements.