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

TL;DR

This paper constructs a specific 4-group divisible design with block size 4 and group sizes 2 and 5, resolving the existence question for the last feasible type with up to 30 points and extending to almost all other pairs.

Contribution

It provides a construction for a 4-GDD of type 2^2 5^5 and establishes existence results for all but finitely many pairs of group sizes.

Findings

Constructed a 4-GDD of type 2^2 5^5

Proved existence of 4-GDDs of type 2^t 5^s for nearly all feasible pairs

Solved the existence problem for the last remaining feasible type with up to 30 points

Abstract

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 $(t,s)$.