Group divisible designs with three groups and block size four

We present new constructions and results on GDDs with three groups and block size four and also obtain new GDDs with two groups of size nine. We say a GDD with three groups is even, odd, or mixed if the sizes of the non-empty intersections of any of its blocks with any of the three groups is always even, always odd, or always mixed. We give new necessary conditions for these families of GDDs and prove that they are sufficient for these three types and for all group sizes except for the minimal case of mixed designs for group size 5t(t>1)5t(t>1). In particular, we prove that mixed GDDs allow a maximum difference between indices. We apply the constructions given to show that the necessary conditions are sufficient for all GDDs with three groups and group sizes two, three, and five, and also for group size four with two possible exceptions, a GDD(4,3,4;5,9)GDD(4,3,4;5,9) and a GDD(4,3,4;7,12)GDD(4,3,4;7,12).