Section 1. An Introduction to Sphere Packing

In Euclidean n-space, En, how may disjoint, open, congruent n-spheres be located to maximize the fraction of the volume of En that the n-spheres cover? That is the sphere packing problem, which goes back to a book review that Gauss wrote in 1831, in which he pointed out that a problem concerning the minimal nonzero value assumed by a positive definite quadratic form in n variables, first considered by Lagrange in 1773, could be translated into a problem on packing spheres (cf. C. A. Rogers [105, pp. 1, 106]).

Though there might seem to be no connection between coding theory and sphere packing, John Leech in [72] and [74] used results in coding theory to obtain a packing of E24 that was much denser than any previously known. The present chapter will describe his packing and discuss how he happened to bridge the gap, so to speak, from codes to sphere packings.

The (23, 12) Golay code can be extended to a (24, 12) code by adding a 0 or 1 to each codeword so that all code-words have even weight. This (24, 12) Golay code thus has a minimum codeword distance of eight.