Perfect Codes and Uniformly Packed Codes

Abstract

In this chapter we shall restrict ourselves to binary codes. To obtain insight into the methods and theorems of this part of coding theory this suffices. Nearly everything can be done (with a little more work) for arbitrary fields F _q . In the course of time many ways of studying perfect codes and related problems have been developed. The algebraic approach which will be discussed in the next section is perhaps the most elegant one. We start with a completely different method. We shall give an extremely elementary proof of a strong necessary condition for the existence of a binary perfect e-error-correcting code. The theorem was first proved by S. P. Lloyd (1957) (indeed for q = 2) using analytic methods. Since then it has been generalized by many authors (cf. [44]) but it is still referred to as Lloyd’s theorem. The proof in this section is due to D. M. Cvetković and J. H. van Lint (1977; cf. [17]).