The Sphere Packing Problem

. A brief report on recent work on the sphere-packing problem. 1991 Mathematics Subject Classification: 52C17 Keywords and Phrases: Sphere packings; lattices; quadratic forms; geometry of numbers 1 Introduction The sphere packing problem has its roots in geometry and number theory (it is part of Hilbert's 18th problem), but is also a fundamental question in information theory. The connection is via the sampling theorem. As Shannon observes in his classic 1948 paper [37] (which ushered in the age of digital communication), if f is a signal of bandwidth W hertz, with almost all its energy concentrated in an interval of T secs, then f is accurately represented by a vector of 2WT samples, which may be regarded as the coordinates of a single point in R n , n = 2WT . Nearly equal signals are represented by neighboring points, so to keep the signals distinct, Shannon represents them by n-dimensional `billiard balls', and is therefore led to ask: what is the best way to pack `billiard balls..