Decomposition of complete designs

Abstract

Through six chapters, the concept of decomposing the complete design is demonstrated. Group divisible designs, symmetric designs, strongly regular graphs, and association schemes are examples of the combinatorial objects that complete designs are decomposed into. Reconstructing McFarland designs leads to the existence of sets of designs with disjoint incidence matrices whose sum is the complete design. The existence of infinite classes of symmetric association schemes follows from the decomposition. Applying a similar technique on the Spence designs provides sets of designs all sharing the same complete tripartite graphs. By appropriately splitting the designs a decomposition of the complete design is obtained leading to an infinite class of non-commutative association schemes. A final attempt is made to combine the constructed decomposition with specific classes of balanced generalized weighing matrices.