6 - Group Codes for the Gaussian Channel

Abstract

The packing problems of various types have long been of interest in mathematics. Perhaps the most investigated problem is that of packing spheres into n-dimensional Euclidean space. Recently, algebraic codes over finite fields have been used to construct few particularly dense lattice packings in En. This chapter discusses the problem related to the packing spherical caps on the unit sphere in En. It highlights the connection between this problem and the equal-energy signaling problem of communications. As the dimensionality of the problem increases and the number of caps packed onto the sphere increases, it becomes important to have a theory that can be used to predict the essential properties of the packing without actually constructing it. This is the role that group representation theory plays. The situation is somewhat similar to that of algebraic coding theory, where the techniques are provided to describe codes with a lower bound on the minimum distance without actually constructing the code. Unfortunately, the development of the application of group representation theory to constructing group codes is still in its infancy.