Group divisible designs with block size 4 and group sizes 4 and 7

TL;DR

This paper investigates the existence conditions for group divisible designs with block size 4 and specific group sizes, establishing broad existence results with only finitely many exceptions.

Contribution

It provides new existence results for 4-GDDs with groups of sizes 4 and 7, covering almost all feasible parameter sets.

Findings

Existence of 4-GDDs of type 4^t 7^s for all but finitely many (t, s)

Identification of the finite set of feasible parameter exceptions

Extension of known results in combinatorial design theory

Abstract

In this paper, we consider the existence of group divisible designs (GDDs) with block size $4$ and group sizes $4$ and $7$. We show that there exists a 4-GDD of type $4^t 7^s$ for all but a finite specified set of feasible values for $(t, s)$.