Uniform Orthogonal Group Divisible Designs with Block Size Three

The spectrum of orthogonal group divisible designs with block size three, and u groups each of size g, is studied. Existence is settled with few possible exceptions for each group size g. 1 Definitions and Background A pairwise balanced design (orPBD)isapair(X,A) such that X is a set of elements called points,and Ais a set of subsets (called blocks)ofX, each of cardinality at least two, such that every pair of points is in a unique block of A.Ifvis a positive integer and K is a set of positive integers, each of which is greater than or equal to two, then we say that (X,A)isa(v,K)-PBD if |X| = v,and|A|#Kfor every A #A. We denote by B(K) the set {v : #(v,K)-PBD}. A set is said to be PBD-closed if B(K)=K. APBDisresolvable if its blocks can be partitioned into parallel classes; a parallel class is a set of point-disjoint blocks whose union is the set of all points. The notation (v, K)-RPBD is used for a resolvable PBD. When K = {k},a(v,K)-PBD is a balanced incomplete block..