All block designs with b = (kv)2 exist

Large sets of subspace designs

“…( For t = 2, the conjecture has been proven in [3]. Can anything be said about this conjecture in the q-analog case?…”

“…More precisely, an infinite series of halvings LS 3 [2](2, k, v) and LS 5 [2](2, k, v) with integers v ≥ 6, v ≡ 2 (mod 4) and 3 ≤ k ≤ v − 3, k ≡ 3 (mod 4) will be given. 1 The first step is the construction of the smallest members of both series (LS 3 [2](2, 3, 6) and LS 5 [2] (2,3,6)). In the first case, this large set is already known [8].…”

“…Based on computational results, in Section 5 two theorems about halvings with the parameters LS 3 [2] (2,3,6) and LS 5 [2] (2,3,6) are proven. In Section 6, the recursive construction method of Section 4 is applied to these two halvings.…”

“…By N/G ∼ = Z/2Z × Z/2Z and the correspondence theorem, the subgroups of N properly containing G are given by σ,φ 2 , σ 2 ,φ , σ 2 ,σφ,φ 2 and N. Again using the method of Kramer and Mesner, we checked computationally that there is no 2- (6,3,20) • 1 orbit of size 63:…”

Large sets of t-designs from groups

“…K h o s r o v s h a h i and S . A j o o d a n i -N a m i n i [1], [2], [3], [28], [29], greatly contributed to the repertory of recursive methods. Of strong impact has been the work of R e i n h a r d L a u e and his group of researchers at Bayreuth [9], [10], [11], [41], [42], [43], particularly in the direct construction of t-designs and large sets.…”