On minimal defining sets of full designs

Abstract

A defining set of a t-(v; k; ̧) design is a subset of the block set of the design which is not contained in any other design with the same parameters. A defining set is said to be minimal if none of its proper subsets is a defining set. A defining set is said to be smallest if no other defining set has a smaller cardinality. A t-(v; k; ̧) design D = (V; B) is called a full design if B is the collection of all possible k-subsets of V . Every simple t-design is contained in a full design and the intersection of a defining set of a full design with a simple t-design contained in it, gives a de ning set of the corresponding t-design. With this motivation, in this thesis, the full designs are studied when the block size is 3 and several families of non-isomorphic minimal de ning sets of full designs are given. Also, it is proven that there exists some sizes in the spectrum of the full design on v elements such that the number of non-isomorphic minimal de ning sets on each of that sizes goes to infinity as . Moreover, the lower bound on the size of the defining sets of the full designs is improved with finding the size of the smallest defining sets of the full designs on 8 and 9 points. Also, all smallest defining sets of the full designs on 8 and 9 points are classified.