Packings and Kisses in High Dimensions

The world of mathematics is not confined to the three dimensions of the space that we inhabit. Mathematicians study sphere-packing problems in spaces of arbitrary dimension. Geometrical puzzles can be posed and solved in such spaces. Some practical challenges end up in such a form. With this chapter we present a short excursion into the outer space of high dimensions. The topic is explored in a very comprehensive way by Conway and Sloane in their book Sphere Packings, Lattices and Groups,1 which is considered by many to be the bible of this subject. Packings in many dimensions find applications in number theory, numerical solutions of integrals, string theory, theoretical physics and digital communications. In particular, some problems in the theory of communications, with a bearing on the optimal design of codes, can be expressed as the packing of d-dimensional spheres. Indeed, in signal processing it is convenient to divide the whole information into uniform pieces and associate each piece with a point in a d-dimensional space (a point in a d-dimensional space is simply a string of d real numbers {u1, u2, u3, . . . , ud}). To transmit and recover the information in the presence of noise one must ensure that these points are separated by a distance larger than that at which the additional noise would corrupt the signal. Each point (a piece of encoded information) can be seen as surrounded by a finite volume, a d-dimensional