Group divisible designs with block size four and group type gum1gum1 for more small gg

Non-uniform group divisible designs are instrumental in the constructions for other types of designs. Most of the progress for the existence of {4}{4}-GDDs of type gum1gum1 is on the case when gugu is even, where the existence for small gg has played a key role. In order to determine the spectrum for {4}{4}-GDDs of type gum1gum1 with gugu being odd, we continue to investigate the small cases with g∈{7,9,21}g∈{7,9,21} in this paper. We show that, for each g∈{7,9,21}g∈{7,9,21}, the necessary conditions for the existence of a {4}{4}-GDD of type gum1gum1 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 hnhn by reducing an infinite class of possible exceptions, namely n=9n=9 and h≡2mod6h≡2mod6, to eight undetermined cases.