Introduction

There are many known examples of 2-designs and, even with the restriction that they are symmetric, we have already constructed several infinite families. But t-designs with t > 2 are much rarer. Although (as we proved in Theorem 1.5) t-structures exist for arbitrarily high values of t, there are no known nontrivial 7-designs and, in fact, the first non-trivial 6-designs were not discovered until 1982 (see [6]). In this chapter we look at some of the important t-designs with t ≥ 3 and at various methods of constructing such designs.

Any t-design must be the extension of a (t – 1)-design and in Section 4.2 we prove Cameron's Theorem. This says that there are very few possibilities for starting with a symmetric design and extending it to a 3-design. This leads us to extending symmetric Hadamard 2-designs to get Hadamard 3-designs which, in turn, leads to the construction of the little Mathieu designs in Section 4.4.

Just as we used the 2-homogeneous groups ASL(2, q) for q ≡ 3 (mod 4) to construct the Paley designs ℒ(q), in Section 4.3 we use the 3-transitive groups PGL(2, q) to construct some 3-designs. We then, in Section 4.5, discuss some of Alltop's ideas for constructing (t + 1)-designs from t-ransitive groups and indicate how to use the 3-transitive groups PGL(4, 2n) to construct some 4-designs which, in fact, turn out to be 5-designs. Finally, in Section 4.6 we prove a generalisation of Fisher's Inequality for 2s-designs with s ≥ 1, which leads to the concept of tight designs. Both Sections 4.5 and 4.6 contain very little detail and any interested reader will have to consult the relevant references.

Introduction

In this chapter we continue the study of symmetric designs but in a somewhat more specific way than in Chapter 2. Section 3.2 contains a detailed discussion of the relation between projective and affine planes and develops some of the theory of non-desarguesian planes. (This latter development is primarily concerned with translation planes, quasifields and semifields. It has a different algebraic flavour than the rest of the book and, although the results are important, the proofs may be skipped if necessary.) Affine planes lead naturally to a discussion of latin squares in Section 3.3 followed by nets which are a very important class of 1-designs; in Section 3.4 one of the applications of nets discussed is the construction of a new infinite family of symmetric designs. Section 3.5 deals with Hadamard designs and Hadamard matrices and contains a construction of the Paley designs. Section 3.6 has a fairly detailed discussion of biplanes (symmetric designs with λ = 2). In Section 3.7 we study the special class of graphs called ‘strongly regular’ and develop their elementary theory (e.g. eigenvalues and multiplicities), as well as giving a number of infinite families. Such graphs enable us to construct some new symmetric designs, and in addition they will be used again later in the book (see Chapters 7 and 8). The connections between strongly regular graphs and design theory are among the most important examples of the fruitful relationship between graphs and designs.

The subject of design theory has grown out of several branches of mathematics, and has been increasingly influenced in recent years by developments in other areas. Its statistical origins are still evident in some of its standard terminology (thus ‘ν’ for the number of points in a structure comes from ‘varieties’). Today it has very fruitful connections with group theory, graph theory, coding theory and geometry; these ties have been two-way, by and large.

We have attempted in this book to lay the groundwork for an understanding of designs, with advanced undergraduate or postgraduate students in mind. Our aim is to prepare the reader to use designs in other fields or to enter the active field of designs themselves. Finite projective and affine geometries are central to design theory, and are introduced early in the book. Since classical geometry is a very large field, the student with a background in this subject will be at an advantage, but we have tried to present a treatment sufficiently self-contained to answer the needs of a reader with a reasonable knowledge of linear algebra. The subject of symmetric designs is also introduced early, and its important aspects (the Bruck–Ryser–Chowla Theorem, Singer groups and difference sets, Hadamard 2-designs, etc.) are developed. The first four chapters, covering basic definitions, geometry and symmetric designs, are designed to be part of any course based on the book.

The other four chapters can be studied more or less independently of one another.

Introduction

In general, 1-designs are less interesting than t-designs with t > 1; it is possible to construct them easily and there do not seem to be many deep theorems about them. But with certain extra properties imposed upon them, 1-designs can become complicated and important objects, in particular with crucial connections to group theory and geometry.

Among the most important 1-designs are generalised quadrangles, which are studied in Section 7.3 as special members of a class of 1-designs called Γα-geometries, introduced in Section 7.2. Using SR graphs we prove some elementary results about Γα-geometries, and we give some infinite families of examples. In particular, we develop two infinite families of generalised quadrangles, one classical (in the sense that it comes from a polarity of a projective geometry), the other not. The other classical generalised quadrangles involve deeper projective geometry and algebra, and we do not include them. (The surprisingly rich and complex theory of generalised quadrangles, and their connections to group theory as well as geometry, is beyond the scope of this book; some of the flavour of the subject is all that we can impart.) There are many unsolved problems (about existence, non-existence, and structure) in the area of Γα-geometries.

In Section 7.4 we study semisymmetric designs, which are 1-designs that generalise (and include) symmetric designs. Besides a number of examples, the section contains results about upper and lower bounds on the number of points in a semisymmetric design, and touches upon the many open questions in this area.