Group divisible designs with block size four and group type for more small

Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of { 4 } -GDDs of type g u m 1 is on the case when g u is even, where the existence for small g has played a key role. In order to determine the spectrum for { 4 } -GDDs of type g u m 1 with g u being odd, we continue to investigate the small cases with g ∈ { 7 , 9 , 21 } in this paper. We show that, for each g ∈ { 7 , 9 , 21 } , the necessary conditions for the existence of a { 4 } -GDD of type g u m 1 are also sufficient. As the applications of these GDDs, we obtain a few pairwise balanced designs with minimum block size 4. Meanwhile, we also improve the existence result for frame self-orthogonal Mendelsohn triple systems of type h n by reducing an infinite class of possible exceptions, namely n = 9 and h ≡ 2 mod 6 , to eight undetermined cases.