In this chapter we introduce Coding Theory. This topic, also known as the theory of error-correcting codes, has its origin in communication theory. Applications are concerned with several situations in which ‘coded’ messages are transmitted over a so-called noisy channel that has the effect that symbols in ‘words’ of the message are sometimes changed to other symbols of the ‘alphabet’. The system is designed in such a way that the most likely error-patterns (at the receiver end) can be recognized and corrected. In this book these practical applications are of no concern. During the development of the discipline of coding theory it turned out that several results from design theory could be used to construct ‘good’ codes. Later, theorems from coding theory contributed considerably to design theory. These connections are what interests us here and therefore the subject will be introduced as an (abstract) area of mathematics.

In coding theory one considers a. set F of q distinct symbols which is called the alphabet. In practice q is generally 2 and F = F2. In most of the theory one takes q = pr (p prime) and F = Fq. The code is called a q-ary code (binary for q = 2, ternary for q = 3).

Using the symbols of F, one forms all n-tuples, that is, Fn, and calls these n-tuples words and n the word length. If F = Fq, we shall denote the set of all words by Fnq and interpret this as n,-dimensional vector space over the field F. Sometimes we omit the index and speak of the space Fn.