Quasi-residual designs can be useful in constructing symmetric designs. However, there are quasi-residual designs that cannot be embedded in a symmetric design. This may happen because the corresponding symmetric design does not exist. Another reason for nonembeddability could be the existence of several blocks whose intersection sizes do not allow them to be extended to blocks of a symmetric design. Also, a quasi-residual design may have a substructure preventing it from embeddability in a symmetric design.

Quasi-residuals of non-existing symmetric designs

Recall that a (v, b, r, k, λ)-design is called quasi-residual if r = k + λ. This condition is satisfied by any residual design of a symmetric (v + r, r, λ)-design. If a quasi-residual design is isomorphic to a residual of a symmetric design D, it is said to be embeddable inD. In this section we will construct several families of quasi-residual (v, b, r, k, λ)-designs for which a symmetric (v + r, r, λ)-design does not exist.

To obtain the first family of such designs, let D be the (v – 1)-fold multiple of the complete symmetric (v, v – 1, v – 2)-design. Then D is a (v, v(v – 1), (v – 1)2, v – 1, (v – 1)(v – 2))-design, so D is quasi-residual. If D is embeddable in a symmetric design, then the complement of this symmetric design is a projective plane of order v – 1. Nowthe Bruck–Ryser Theorem gives an infinite family of non-embeddable quasi-residual designs.

If the action of an automorphism group of a symmetric design on the block set is known, then the design can be constructed by finding one (base) block from each block orbit and then applying the automorphism group to obtain the remaining blocks. If a symmetric design admits an automorphism group such that all blocks of the design form a single orbit, then the group itself can be regarded as the point set of the design. The base block becomes a subset of the group and such subsets are called difference sets. The designs obtained from difference sets admit group invariant incidence matrices.

Group rings are a natural setting for investigating difference sets. The notion of a group of symmetries of a subset of a group ring will be crucial to constructing symmetric designs in subsequent chapters.

Group invariant matrices and group rings

A group invariant matrix is a matrix of order v whose columns can be obtained from the first column by applying all elements of a certain permutation group of order v to the entries of the first column. If we assume that the rows and columns of the matrix are indexed by the elements of the group, then this description leads us to the following definition.

Prototypes of many combinatorial designs come from finite projective geometries and finite affine geometries. Vector spaces over finite fields provide a natural setting for describing these geometries. Among the numerous incidence structures that can be constructed using affine and projective geometries are infinite families of symmetric designs, nets and Latin squares. Subspaces of a vector space over a finite field can be regarded as linear codes that will be used in later chapters for constructing other combinatorial structures, such as Witt designs and balanced generalized weighing matrices.

Finite fields

In this section we recall a few basic results on finite fields which will be used throughout this book.

For any prime p, the residue classes modulo p with the usual addition and multiplication form a finite field GF(p) of order p. These fields are called prime fields. Any finite field F of characteristic p contains GF(p) as a subfield. The field F then can be regarded as a finite-dimensional vector space over GF(p), and therefore, |F| = pn where n is the dimension of this vector space. Conversely, for any prime power q = pn, there is a unique (up to isomorphism) finite field of order q. This field is denoted by GF(q) and is often called the Galois field of order q. In general, the field GF(q) is isomorphic to (a unique) subfield of the field GF(r) if and only if r is a power of q.

An affine plane of order n has n2 + n lines, any two of which are either parallel or intersecting. The relation of parallelism on the set of lines is an equivalence relation, and so it partitions the set of lines into n + 1 parallel classes of cardinality n. Each point lies on exactly one line from each parallel class. The block set of the complement of an affine plane of order n can be partitioned into n + 1 classes so that each point is contained in exactly n – 1 blocks from each class. Similar partitions exist in affine geometries of higher dimension. In this chapter we study a more general notion of resolution of an incidence structure, i.e., a partition of the block set of the structure into classes so that each point is contained in a constant number of blocks from each class.

Bose's Inequality

The incidence structures on which we define the notion of resolution are pairwise balanced designs.

Definition 5.1.1. Let λ be a positive integer. A pairwise balanced design (PBD) of index λ is an incidence structure D = (X, B, I) such that X ≠ Ø, every x ∈ X is incident with more than λ blocks, and, for any distinct x, y ∈ X, there are precisely λ blocks that are incident with both x and y. If a PBD of index λ has constant replication number r, it is called an (r, λ)-design.

Incidence relations defining designs and incidence relations induced by designs can sometimes be expressed in terms of graphs. Such graphs usually have a high degree of regularity reflecting the regularity of the corresponding designs.

Strongly regular graphs

Let N be an incidence matrix of a symmetric (v, k, λ)-design. If N is symmetric with zeros on the diagonal, it serves as an adjacency matrix of a graph Γ of order v. This graph is regular of degree k, and for any distinct vertices x and y of Γ, there are exactly λ vertices which are adjacent to both x and y.

If N is a symmetric incidence matrix of a symmetric (v, k, λ)-design with ones on the diagonal, then N – I serves as an adjacency matrix of a regular graph of order v and degree k – 1. For any distinct vertices x and y of this graph, the number of vertices that are adjacent to both x and y is equal to λ – 2 if x and y are adjacent and is equal to λ otherwise.

The graphs we have just described are special cases of strongly regular graphs.

Design theory is a well-established branch of combinatorial mathematics. The origins of the subject can be traced back to statistics in the pioneering works of R. A. Fisher, F. Yates, and R. C. Bose. From the very beginning, one of the central objects of design theory has been symmetric designs. The prototype of a symmetric design is a finite projective plane, and the theory of symmetric designs borrows its methods and ideas from finite geometries, group theory, number theory, and linear algebra.

It is notoriously difficult to construct an infinite family of symmetric designs or even a single symmetric design. However, in recent years new ideas in constructing symmetric designs have been discovered and new infinite families have been found. The central role in these constructions is played by balanced generalized weighing matrices. These matrices generalize the notion of a symmetric design but until recently they were often regarded as a rather obscure combinatorial object. Now they seem to be a useful tool in unifying different construction methods that have been developed since the 1950s.

This book is primarily a research monograph which aims to give a unifying exposition of the theory of symmetric designs with emphasis on these new developments. The book covers the combinatorial aspects of the theory with particular attention to constructing symmetric designs and related objects. Recent results that have never previously appeared in book format are developed mainly in the last five chapters.