Existence of affine $\alpha$-resolvable PBIB designs with some constructions

Abstract

A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized to $\alpha$-resolvability and affine $\alpha$-resolvability in 1963. These arguments are mostly done for a class of balanced incomplete block designs. The present paper will make the combinatorial investigation on affine $\alpha$-resolvable partially balanced incomplete block designs with two associate classes. The characterization of parameters in a closed form will be given and then existence problems with construction methods will be discussed. Comprehensive and useful results on combinatorics are obtained. Several methods of construction are newly presented with some illustrations.