Block Codes

Abstract

The main parameters of an error correcting block code, which we will often refer to simply as a code, are its length and minimum distance. In this chapter, we shall primarily be concerned with the relationship between the size of the code and these parameters. If we fix the length of the code, then we wish to maximise the minimum distance and the size of the code, which are contrary aims. If we fix the minimum distance too, then we simply consider the problem of maximising the size of the code. We shall prove the Gilbert–Varshamov lower bound, which is obtained by constructing block codes of a given length and minimum distance by applying the greedy algorithm. We will prove various upper bounds which will put limits on just how good a block code one can hope to find of a fixed length and minimum distance. Since Shannon’s theorem is an asymptotic result telling us what rates we can achieve with a code of arbitrarily long length, we shall for a large part of this chapter focus on sequences of codes whose length tends to infinity. If we use nearest neighbour decoding then, so that the probability we decode correctly does not tend to zero, we will be interested in finding sequences of codes for which both the transmission rate and the ratio of the minimum distance to the length are bounded away from zero. We set aside trying to answer the question of how these codes are implemented until later chapters in which we work with codes which have more structure.