New Constructions and Necessary Conditions for 4-GDDs With Block Size 5

Abstract

t-v,k,Λ designs are well-known objects of study. Here, we define t-group divisible designs (t-GDDs) and specifically study t-GDDs, where the number of groups are less than the block size. After a short survey of t-GDDs, we will present some examples of 4-GDDs with block size 5 and two groups and prove that the necessary conditions are sufficient for group sizes 3, 4, 5, 6, 7, 9, and 10. Though the necessary conditions for the existence of t-designs are well known, the actual spectrums for 4-v,5,Λ and for 4-GDD n,2,5;Λ1,Λ2 are obtained as a ready source for researchers interested in the constructions of such designs.