A new bound on the number of designs with classical affine parameters

The hyperplanes in the affine geometry AG(d, q) yield an affine resolvable design with parameters 2-(q(d), q(d-1), q(d-1)-1/q-1). Jungnickel [3] proved an exponential lower bound on the number of non-isomorphic affine resolvable designs with these parameters for d greater than or equal to 3. The bound of Jungnickel was improved recently [5] by a factor of q(d2)+d-6/2(q-1)(d-2) for any d greater than or equal to 4. In this paper, a construction of 2-(q(d), q(d-1), q(d-1)-1/q-1) designs based on group divisible designs is given that yields at least (q(d-1)+q(d-2)+...+1)!(q-1)/| PGammaL(d,q)|| AGammaL(d,q)| non-isomorphic designs for any d greater than or equal to 3. This new bound improves the bound of [5] by a factor of [GRAPHICS] For any given q and d, It was previously known [2,11] that there are at least 8071 non-isomorphic 2-(27,9,4) designs. We show that the number of non-isomorphic 2-(27,9,4) is at least 245,100,000.