Pairwise Balanced Designs with Prime Power Block Sizes Exceeding 7

Abstract

In this paper, we construct pairwise balanced designs (PBDs) having block sizes which are prime powers exceeding 7. If we denote by P * the set of all prime powers exceeding 7 and by B ( P * ) the set of orders of PBDs of index unity having block sizes from P * , then it is shown that v ∈ .B ( P * ) for all v > 2206 and for many orders less than this value. We also give some applications to the construction of other types of combinatorial designs, such as conjugate orthogonal Latin squares and sets of mutually orthogonal Latin squares (MOLS). For example, it is proved that the spectrum of idempotent Latin squares with distinct and pair-wise orthogonal conjugates contains all orders v mentioned above, and some of the PBDs constructed in this paper can be used to obtain more MOLS of certain orders than previously known.