Pairwise balanced designs with odd block sizes exceeding five

Abstract In this paper we construct pairwise balanced designs (PBDs) having block sizes which are odd prime powers exceeding 5. Such a PBD contains an odd number of points. We show that this condition is sufficient when the number of points is at least 2129, with at most 103 possible exceptions below this value. This is accomplished, in part, by some interesting new recursive constructions for PBDs. Also, we give several applications to the construction of other types of combinatorial designs, such as Room squares, skew Room squares, Room cubes, separable orthogonal arrays, and perpendicular arrays. We prove the new result that there is a perpendicular array PA( n , 7) for all odd n ⩾2129.

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