More on halving the complete designs

Large sets of t-designs from groups

“…Since there are It is known that, up to isomorphism, there is a unique plane of order 2. Thus the symmetric group S 7 acts transitively on all Fano planes on V = [1,7]. Since the automorphism group of a Fano plane is P SL 3 Observe next that PG(2,3), the projective plane of order 3, is a 2-(13,4,1) design, with 13 lines, each incident with 4 points.…”

“…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.…”

Constructing regular self-complementary uniform hypergraphs

“…Hence a t-subset-regular k-hypergraph X of order n is a t-(n, k, λ) design in which λ is equal to the t-valency of X. A large [2](t, k, n) in which the t-designs are isomorphic. Hence results regarding sucient conditions on the order a t-subset regular selfcomplementary k-hypergraph imply the corresponding results for the order of a LS [2](t, k, n).…”

“…Theorem 1.1 was originally stated in the language of large sets of t-designs. Moreover, for t ∈ {1, 2}, large sets LS [2](t, k, n) have been constructed for all pairs of integer n and k satisfying condition (1) of Theorem 1.1 [1,2,3,4,5,6,7,8,14]. However, it is important to note that these existence results do not imply that condition (1) of Theorem 1.1 is sucient for t ∈ {1, 2}, since there is no guarantee that two designs in the large sets constructed in these papers are isomorphic.…”

“…A large [2](t, k, n) in which the t-designs are isomorphic. Hence results regarding sucient conditions on the order a t-subset regular selfcomplementary k-hypergraph imply the corresponding results for the order of a LS [2](t, k, n). Theorem 1.1 was originally stated in the language of large sets of t-designs.…”