Group divisible designs with block size four and group type g ^um¹ for more small g

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^um¹ is on the case when gu 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^um¹ with gu 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^um¹ 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 hⁿ by reducing an infinite class of possible exceptions, namely n = 9 and h 2 mod 6, to eight undetermined cases.