Finite sphere packing and sphere covering

Generalized sphere-packing and sphere-covering bounds on the size of codes for combinatorial channels

Many of the classic problem of coding theory are highly symmetric, which makes it easy to derive spherepacking upper bounds and sphere-covering lower bounds on the size of codes. We discuss the generalizations of sphere-packing and sphere-covering bounds to arbitrary error models. These generalizations become especially important when the sizes of the error spheres are nonuniform. The best poss...

Sphere Packing Lecture

What is the most space-efficient way to stack spheres in three-dimensional space? Although the answer is obvious, a rigorous proof has eluded mathematicians for centuries, having only recently been found by Hales in 1998, who used an immense computerized proof-by-exhaustion. The more general problem of packing (n-1)-spheres into n-dimensional Euclidean (or other) space is still only poorly unde...

The sphere packing problem

Hales, T.C., The sphere packing problem, Journal of Computational and Applied Mathematics 44 (1992) 41-76. The sphere packing problem asks whether any packing of spheres of equal radius in three dimensions has density exceeding that of the face-centered-cubic lattice packing (of density IT/V%). This paper sketches a solution to this problem.

The Sphere-Packing Problem

A brief report on recent work on the sphere-packing problem. 1991 Mathematics Subject Classification: 52C17

Finite Sphere Packings and Sphere Coverings