The work in this thesis centres mainly on the construction of various kinds of balanced designs. Chapter 1 contains some basic definitions and known results. In Chapter 2 the concept of a maximal arc in a finite protective plane is generalised to the case of an arbitrary balanced incomplete block design, and the "internal structure" of designs possessing maximal arcs is examined. Families of partially balanced designs are produced in some instances from designs with maximal arcs.
Chapters 3, 4 and 5 deal with different construction methods for balanced designs. In Chapter 3 all non-isomorphic balanced incomplete block designs with certain small parameters are found, and also small 3-designs on eight elements with blocks of size four.
In 1952 Tocher* introduced a new kind of balanced design, with constant block size, in which elements are allowed to occur more than once in a block; if each element may occur 0, 1, 2,•••, or n-1 times in a block the design is called a balanced n-ary design. With this definition a balanced incomplete block design is a balanced binary design. Some properties of balanced n-ary designs, including a form of the Bruck-Ryser-Chowla theorem, are given in section 4.1. In the remainder of Chapter 4 existing balanced incomplete block designs are used to construct balanced n-ary designs, while in Chapter 5, collections of cyclotomic classes in finite fields are used to produce n-ary supplementary difference sets, and the families of such sets that have constant block size generate balanced n-ary designs. Both of these constructions have been adapted in some instances to yield balanced incomplete block designs.
* Blocks of size 3 with repeated elements arose in: E. Hastings Moore, Concerning Triple Systems, Math. Ann. 43 (1893) 271-285 DOI:10.1007/BF01443649 However these are not balanced in the usual sense.