Generalizing Clatworthy group divisible designs

In this paper a neat construction is provided for three new families of group divisible designs that generalize some designs from Clatworthyʹs table of the only 11 designs with two associate classes that have block size four, three groups, and replication numbers at most 10. In each case (namely, λ 1 = 4 and λ 2 = 5 , λ 1 = 4 and λ 2 = 2 , and λ 1 = 8 and λ 2 = 4 ), we have proved that the necessary conditions found are also sufficient for the existence of such GDDʹs with block size four and three groups, with one possible exception.