The District Leader Is 6 3 - Design Construction

t-wise balanced designs (t-(v, K, λ)) are very important in design theory. Fort = 2, much work has been done on pairwise balanced designs (PBD). For t ≥ 3, since it is difficult, not much is known. Two finite generating sets of 3-BD on k ≥ 4 and k ≥ 5, (?) k ∈ K, had been gotten by H. Hanani in 1971 and by Qiu-rong Wu in 1991 respectively. In this paper we investigate the finite generating sets of 3-BD on k ≥ 6, (?) k ∈ K, and get the following result: (?)v≥6, v∈ B₃(K₆,1 ), where K₆ = {6,7,....., 41,45,46,47, 51, 52, 53,83,84}\{22,26}.The above finite generating set is used to solve the existence of balanced 3-designs (i.e. 3-BD on fixed block size) on block size 6 and λ = 20 in this paper. Make use of the projective general linear group and the projective special linear group, we get some concrete constructions of balanced 3-designs on small orders and get the following result: (?) v ∈ K₆{35,39,40,45}, 3-(v, 6,20) exists.