Group divisible designs with block size 4 and group sizes 2 and 5

In this paper we provide a $4$-GDD of type $2^2{5}^5$, thereby solving the existence question for the last remaining feasible type for a $4$-GDD with no more than $30$ points. We then show that $4$-GDDs of type $2^t{5}^s$ exist for all but a finite specified set of feasible pairs $(t,s)$.