Extremal Problems in Coding Theory

This chapter is devoted to results of coding theory that are centered around the concept of a code as a packing of the corresponding metric space. We derive a few results in several rather diverse directions such as combinatorial bounds on codes and their invariants, properties of linear codes, error exponents, and applications of the polynomial method. A common goal of the problems considered is to establish bounds on natural combinatorial parameters of a code. The primary aim of this chapter is to explain the basic ideas that drive this part of coding theory. In particular, we do not strive to explain the best known result for each problem that we discuss. Our motivation is that, as in each living mathematical discipline, the current best results are often of an ad hoc nature and do not add to our understanding of the central ideas. Pointers to the literature that develops the subjects of this chapter are supposed to compensate for this.