chapter  Chapter 9
11 Pages

Asymptotic results

ByJuergen Bierbrauer

The Plotkin bound is a bound on codes of very large minimum distance. The proof is based on an elementary method, quadratic counting. This chapter discusses a classical bound on orthogonal arrays of strength 2 from the Plotkin bound. It argues that codes with very large distances have their own areas of application. The chapter shows one type of application. Shannon's channel coding theorem from 1948 is generally regarded as the starting point of coding theory. He uses elementary probabilistic arguments to prove a theorem, which says, loosely speaking, that good codes of large length always exist. The chapter provides a light treatment of the binary case, following van Lint. scenario for Shannon's theorem: The model for the information channel is the binary symmetric channel, with symbol error probability p. The strategy is block coding. The Gilbert-Varshamov bound proves the existence of certain good linear codes without actually constructing these codes. It is based on a simple counting argument.