Extended Group Divisible Partially Balanced Incomplete Block Designs

Abstract

The class of partially balanced incomplete block designs (PBIB) with more than two associate classes has not yet been explored to a great extent. In fact, only a few $m$-associate class PBIB's $(m > 2)$ are known explicitly. One way to obtain such designs is certainly by generalizing the well-known PBIB's with two associate classes. Among these particularly the Group Divisible PBIB's lend themselves rather obviously to a generalization in this direction. Roy [8] and Raghavarao [7] have generalized the Group Divisible design of Bose and Connor [1] to $m$-associate class designs. The idea of another type of Group Divisible PBIB's with three associate classes, given by Vartak [11], was extended to an $m$-associate class design by Hinkelmann and Kempthorne [5] which they called an Extended Group Divisible PBIB (EGD/$m$-PBIB). In this paper we shall investigate the EGD/$m$-PBIB in some detail. The definition and parameters of this design are given in Section 2. In Section 3 we shall prove the uniqueness of its association scheme. For a design given by its incidence matrix $\mathbf{N}$, the properties of the matrix $\mathbf{NN}'$ will be explored in Section 4. The eigenvalues of $\mathbf{NN}'$, its determinant and its Hasse-Minkowski invariants $c_p$ are obtained, and non-existence theorems are given. These theorems are illustrated by examples. An example of an existent EGD/$m$-PBIB plan is given.