On 1-Sarvate–Beam designs

Hau Chan, Dinesh G. Sarvate
2011 Discrete Mathematics  
The solution to a set theory exercise, "Partition the set of positive integers {1, 2, . . . , v} into k subsets such that the sum of the elements in each subset is v(v + 1)/(2k) whenever v(v + 1)/(2k) is an integer", gives a construction of non-simple 1-SB designs. This raises a natural question of the existence of simple 1-SB designs. We show that the necessary conditions for the existence of simple 1-SB designs for block sizes 2, 3, 4, 5 and 6 are sufficient. Moreover, the technique exhibited
more » ... in the proof can be applied to block sizes greater than k = 6. We also show that simple t-SB(v, t + 1), 2-SB(v, 3) and 2-SB(v, 4) designs do not exist for any positive integers v and t. A natural question, "Can we obtain a construction for simple 1-SB designs similar to Billington's classical construction of simple 1-designs for any block size k?", remains open.
doi:10.1016/j.disc.2011.02.004 fatcat:hxynvf23cvfnnjwx2ea3gijiim