The number of designs with classical parameters grows exponentially (Q798323)

The number of non-isomorphic block designs S(2,k;v) grows very fast with v. \textit{R. M. Wilson} [Math. Z. 135, 303-313 (1974; Zbl 0269.05005)] proved that it is at least exp \(\alpha\) v\({}^ 2\) with \(\alpha >0\) for large v. For affine and symmetric designs other methods are necessary. The author shows that the number of non-isomorphic affine designs with the parameters of affine spaces \(AG_{d-1}(d,q)\) [the blocks are the hyperplanes] grows exponentially with q. The same holds for symmetric designs with the parameters of projective spaces \(PG_{d-1}(d,q)\). Similar estimates hold for biaffine divisible designs and symmetric nets (for details see the paper). In the formula on p. 170, row 3 from below, read \((q-1)^ 2\) instead of \((q-q)^ 2\).