Generalizing Clatworthy Group Divisible Designs

Abstract

Clatworthy described the eleven group divisible designs with three groups, block size four, and replication number at most 10. Each of these can be generalized in natural ways. In this dissertation neat constructions are provided for these new families of group divisible designs. In a previous paper the existence of one such design was settled. Here we essentially settle the existence of generalizations of eight of the remaining ten Clatworthy designs. In each case (namely, $\lambda_1 = 4$ and $\lambda_2 = 5$, $\lambda_1 = 4$ and $\lambda_2 = 2$, $\lambda_1 = 8$ and $\lambda_2 = 4$, $\lambda_1 = 2$ and $\lambda_2 = 1$, $\lambda_1 = 10$ and $\lambda_2 = 5$, $\lambda_1 = 6$ and $\lambda_2 = 3$, $\lambda_1 = 3$ and $\lambda_2 = 1$, and $\lambda_1 = 6$ and $\lambda_2 = 2$), we have proved that the necessary conditions found are also sufficient for the existence of such $GDD$'s with block size four and three groups, with one possible exception.