ALGEBRAIC METHODS IN THE THEORY OF COMBINATORIAL DESIGNS (INEQUALITIES, GROUP THEORY, T-WISE BALANCED)

A t-design or (Generalized Steiner System) S((lamda);t,k,v) is an incidence structure (X, B) with a v-set of points X and a b-set of blocks B such that, each block has exactly k points and any t points are contained in exactly (lamda) blocks. In Chapter II of this thesis the algebra of P(X) by P(X) matrices over the rationals left invariant under the natural action of a group G (LESSTHEQ) Sym(X) is introduced, and an epimorphism (tau) from this algebra onto the matrices, over the rationals, whose rows and columns are indexed by the G(VBAR)P(X)-orbits is discovered. This mapping carries the matrices of Wilson onto the matrices of Kramer and Mesner and therefore, enables us to generalize the t-design inequalities of Fisher, Wilson and Connor. An elementary proof of an important theorem of Livingstone and Wagner is also presented. In Chapter III the condition that every block has exactly k points is relaxed and blocks are allowed to have more than one size. Such designs have a connection with error-correcting codes. The central theme of this chapter is the classification of all transitive homogeneous S(1;3,{4,6},20) designs and answers a question posed by Ed Assmus