Large sets of t-designs from groups

Abstract

ABSTRACT. This paper addresses questions related to the existence and construction of large sets of t- (v, k, λ) designs. It contains material from my talk in the Combinatorial Designs Conference in honor of Alex Rosa's 70th birthday, which took place in beautiful Bratislava, in July, 2007. Naturally, only a small number of "highlight" topics could be included, and for the most part these involve the use of symmetry, that is, it is assumed that the particular designs or large sets of designs, are invariant under a prescribed group of automorphisms. I present almost no proofs, but give references so that the reader can find a much wider repertory of theorems and constructions in the literature. For completeness, I include the statement of a few recursive constructions. The latter are extremely important on their own right, and deserve extensive attention elsewhere. I hope the reader becomes interested in the intriguing open problems posed at the end of the paper and succeeds in solving some of them.

Some Properties from Construction of Finite Projective Planes of Order 2 and 3

“…In other words, a large set is a partition of the set of all k-subsets of v-set into Steiner systems S (t, k, v). For finite projective planes, according to [1], there is only known a large set of projective plane of order 3 or Steiner system S(2, 4, 13) [2], while a large set of projective plane of order 2 or Steiner system S(2, 3, 7) do not exist and the existence of large set of projective plane of order 4 or Steiner system S(2, 5, 21) is still unsettled problem.…”