Concerning the Generating Set of ...

The generating set of Z n is determined completely for n 8, with the exception of the (46,6,1) design. Some general results are presented for larger values of n. 1 Introduction A pairwise balanced design (or PBD) is a pair (V ,B) where 1. V is a finite set of points, 2. B is a collection of subsets of V called blocks, 3. every pair of distinct points of V occur in exactly one blocks. We use the notation PBD(v;K) when jV j = v and jBj 2 K for all B 2 B. We denote B(K)=fv : there exists a PBD(v,K)g. A set S of positive integers is PBD-closed if the existence of a PBD(v;S) implies that v belongs to S. Let K be a PBD-closed set. Let G be a set of positive integers such that K = B(G). Then G is a generating set for K. Let K be a PBD-closed set. An element x 2 K is essential in K if and only if x 62 B(Knfxg), that is, if and only if there does not exist a PBD(x;Knfxg). An element x 2 K is inessential if x is not essential. We denote by [a; b] the set of integers fx : a x bg, and by Z..