Greg Gamble
| This paper appeared as: | UWA Research Report 6 (1992) |
| and in a slightly abbreviated form in: | J. Geom. 51 (1994), 36-49. |
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Abstract
Let the 3-design be the pair D = (X, B) where X = (Zp)d with p prime and d in N, and B = BG with G = AGLd(p) = Aut D and B a subset of X. A necessary condition for a non-trivial design D to exist is that:q1(k) = (k(k - 1)(k - 2)(p - 2)) / (6(pd-2)) in N,where 2 < p, k = |B|. Let K = {k | q1(k) in N and 3 <= k <= pd / 2}. The size of K is found in terms of p and d, and also a method for determining all the elements of K is given. In particular, for G = AGL7(3), K is shown to contain 12 elements, two of which are 115 and 116. Block-transitive 3-designs are shown to exist for these values of k by constructing a k-element subset of X with certain properties.
(Greg Gamble) http://www.csee.uq.edu.au/~gregg