The number of designs with geometric parameters grows exponentially

Abstract

It is well-known that the number of 2-designs with the parameters of a classical point-hyperplane design PG n−1 (n, q) grows exponentially. Here we extend this result to the number of 2-designs with the parameters of PG d (n, q), where 2 ≤ d ≤ n − 1. We also establish a characterization of the classical geometric designs in terms of hyperplanes and, in the special case d = 2, also in terms of lines. Finally, we shall discuss some interesting configurations of hyperplanes arising in designs with geometric parameters. Mathematics Subject Classifications (2000) 05B05 · 51E20 PreliminariesIt is well-known that the number of symmetric 2-designs with the parameters of a classical point-hyperplane design PG n−1 (n, q) grows exponentially; a similar result holds for affine 2-designs with the parameters of a classical point-hyperplane design AG n−1 (n, q). These results were originally established by the first author [6], whose bounds were subsequently improved in several papers [8][9][10].In this paper, we shall establish a corresponding result for the number of 2-designs with the parameters of PG d (n, q), where d is in the range 2 ≤ d ≤ n − 2. We also obtain a Communicated by Ron Mullin.To Spyros Magliveras on the occasion of his 70th birthday.

Recent results on designs with classical parameters

“…We remark that Theorem 5.2 was claimed in [30] only for 2 ≤ d ≤ n − 2, but that the proof given there also applies for the case d = n − 1. We also note that the result holds likewise for d = 1 by Theorem 3.3; of course, the notion of a hyperplane used there is a considerably weaker concept than the one employed in this section, so that Theorem 5.2 would not be particularly interesting for d = 1.…”

“…Recently, some further cases were established: the case d = 2 in [30]; the cases d = 3 and d = 4 in [27]; and the case d = n − 2 under an additional assumption on the intersection size of blocks in [28]. Finally, Metsch [39] obtained the general result using an elegant recursive argument.…”

“…This method was introduced by Tonchev and the present author in [29,30], but the idea of distorting a classical design in some appropriate manner is by no means new and has been used successfully before, see e.g. [13,26,32].…”

“…Then it is easy to prove the following result. Theorem 2.2 is obtained in [30] by estimating the number of non-isomorphic designs obtained from Lemma 5.1. To do so, one needs the following characterization of the classical design P G d (n, q) in terms of the hyperplanes just introduced.…”

“…In [30], a design with the parameters of P G d (2d, q) is called pseudo-geometric if it admits a hyperplane and has the same intersection numbers as the classical design. We shall refer to the specific examples of pseudo-geometric designs constructed with the help of a polarity as polarity designs.…”