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.

The three subjects of this book all began life in the provinces of applicable mathematics. Design theory originated in statistics (its name reflects its initial use, in experimental design); codes in information transmission; and graphs in the modelling of networks of a very general kind (in the first instance, the bridges of Königsberg). All three have since become part of mainstream discrete mathematics.

We have not tried to write a textbook on three individual topics. Instead, our goal is more limited: we want to explore some of the ways in which the three topics have interacted with each other, with results and methods from one area being applied in another. Indeed, we believe that discrete mathematics is better defined by its methods than by its subject-matter, and our approach reflects this.

The book has its origins in the notes of two series of lectures given by the authors at Westfield College, London, at the invitation of Dan Hughes. The audience at those lectures consisted of design theorists, and our job was to show them that graphs and codes could be useful to them. The notes subsequently appeared in the London Mathematical Society Lecture Note Series in 1975, and in a considerably revised form in 1980. We tried then to make the notes accessible to a wider audience by adding an introductory chapter on design theory.

In the intervening decade, we have become aware that a number of students used the book as a textbook. Their task was not made easier by the ‘research notes’ style in which many assertions are left without proof.

INTRODUCTION

The theory of Schubert polynomials has its origins in algebraic geometry, and in particular in the enumerative geometry of the flag manifolds. The reader of this article will however detect no trace of geometry or reference to these origins. In recent years A. Lascoux and M.-P. Schiitzenberger have developed an elegant and purely combinatorial theory of Schubert polynomials in a long series of articles [L1]- [L3], [LS1]- [LS7]. It seems likely that this theory will prove to be a useful addition to the existing weaponry for attacking combinatorial problems relating to permutations and symmetric groups.

Most of the results expounded here occur somewhere in the publications of Lascoux and Schiitzenberger, though not always accompanied by proof, and I have not attempted to give chapter and verse at each point. For lack of space, many proofs have been omitted, especially in the earlier sections, but I hope I have retained enough to convey the flavour of the subject. Complete proofs will be found for example in [M2].

PERMUTATIONS

In this first section we shall review briefly, without proofs, some facts and notions relating to permutations that will be used later. Proofs may be found e.g. in [M2].

For each integer n ≥ 1, let Sn denote the symmetric group of degree n, namely the group of all permutations of the set [1, n] = {1,2,… n}. Each w ∈ Sn is a mapping of [1,n] onto itself; we shall write all mappings on the left of their arguments, so that the image of i ∈ [1, n] under w is w(i). We shall occasionally denote w by the sequence (w(1),…,w(n)).