Constructions Of Generalized Anti-pasch BIBD With Block Size Four

A balanced incomplete block design (BIBD) of order v and block size k,(v,k,1)-BIBD, is a pair (V,B) where V is a v-set and B is a collection of k-subsets of V (blocks) such that any2-subset of V is contained in exactly one block.An (m, n) configuration in a BIBD is a subset of m blocks of B, whose union is an n-element subset of V. A Pasch configuration or a quadrilateral is a (4,6) configuration in a (v,3,1)-BIBD. A generalized Pasch configuration is a (k,k(k+1)/2) configura-tion of a(v,k,1)-BIBD. A (v,3,1)-BIBD is anti-Pasch if it does not contain a Pasch configuration, its existence spectrum was finally determined by Grannell et al in2000. A (v,k,1)-BIBD is generalized anti-Pasch if it does not contain a generalized Pasch configuration.In this thesis we modify the known constructions for BIBDs so that the resulting BIBDs have no generalized Pasch configuration. Some recursive constructions of gener-alized anti-Pasch (v,4,1)-BIBDs are then obtained. The auxiliary designs TD(4,n) in these recursive constructions are required to have no Pasch configuration after deleting any group. The odd orders have been completely determined, a product construction and a Wilson type construction are also given.