Special subsets of the block sets of designs KEN GRAY A combinatorial design is a way of choosing, from a given set of elements, a collection of subsets or blocks, satisfying certain properties. Such, properties might include for instance, the size of the blocks, the number of blocks which contain each element, the intersection sizes of the blocks, and so on. Substantial work has been done on the existence and classification of designs and the identification of their properties. Apart from

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... obvious pure mathematical interest, the search for designs with particular properties has been stimulated by their relevance to areas as diverse as the planning of experiments, coding theory and information security. Investigation of the underlying properties of designs has in turn led to the investigation of special subsets of the sets of blocks of a design. Such subsets include, for example, supplementary difference sets, trades and the orbits of automorphisms. Also of interest are the subsets of blocks which have common intersection properties, contain a common element or have each element occurring in the same number of blocks. Designs which have both the property that all the blocks have the same size and that every i-subset of the elements occurs in precisely the same number of blocks are known as ^-designs. This thesis focuses on the special subsets of the block sets of sucĥ -designs. The first two chapters provide necessary background for the convenience of the reader. In Chapter 3, a construction analogous to that of C.J. Colbourn and M.J. Colbourn is developed from the special sets of starter blocks of cyclic and quasi-cyclic designs. The properties of starter blocks are also utilised in Chapters 4 and 5, where the aim is to identify subdesigns, and hence partitions, of the ^-designs carried by the blocks of the unique [15, 11, 3] Hamming code; this follows on from work of E.P. Dawson. In these chapters the orbits of automorphisms are initially used to do this, but the consequential recognition of sets of blocks forming trades provides a viable alternative approach. Further, the use of trades also allows for construction of nonisomorphic designs and leads to consideration of block intersection and resolvability