Generalized Right Angular Designs

Abstract

Investigations of association schemes for three or more associate class PBIB designs have been limited mainly to the works of Vartak [17], Raghavarao [5], Roy [9], Singh and Shukla [15], Raghavarao and Chandrasekhararao [8] and Tharthare [16]. In this article we generalize the right angular association scheme introduced by the author in [16] to study the combinatorial properties, construction and nonexistence of further class of four associate class designs. In Section 4, a simplified method of analysis of these designs is given, while Section 5 deals with the method of constructing $v \times s^m$ balanced asymmetrical factorial design in $vs^{m - 1}$ plots, where $s$ is a prime or a prime power. It can be seen that this balanced asymmetrical design is nothing but a particular case of the four associate class design introduced in this paper and hence its analysis can be completed by the method of Section 4. Some methods of construction discussed in Section 6 give a new way of arranging an $s^3$ factorial experiment in blocks of sizes different from $s$ and $s^2$, with better efficiency than the usual three dimensional lattice designs [4]. For convenience we denote PBIB designs with the generalized right angular association scheme by "GRAD".