ABSTRACT
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, nu-
merical 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
than the additional noise. The encoded recovered only if these balls are nonoverlap-which minimizes the energy necessary to transmit these balls in the closest possible packing. There-back to the greengrocer’s dilemma: How can we What is the maximum packing fraction for solid spheres in d dimensions?
