Resolvable pairwise balanced designs

Abstract

This paper generalizes to pairwise balanced designs the concept of resolvability and affine resolvability introduced by Bose (Sankhya 6 (1942) 105–110) for balanced incomplete block designs. Analysing the vector space of linear polynomials in variables corresponding to the blocks, we extend Bose’s inequality and obtain a necessary and sufficient condition of affine resolvability for pairwise balanced designs. As a consequence, we extend the corresponding results of Shrikhande and Raghavarao (in: Contributions to Statistics, Pergamon Press, Oxford, 1964, pp. 471–480), Hughes and Piper (Geometriae Dedicata 5 (1976) 129–131) and Vanstone (J. Austral. Math. Soc. Series A 28 (1979) 471–478). Some examples considered in the paper show that affine resolvable pairwise balanced designs may represent an interesting regular structure other than balanced incomplete block designs. Some connections to λ-designs and pseudo-symmetric designs are also shown.