The number of designs with geometric parameters grows exponentially
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 a parts per thousand currency sign d a parts per thousand currency sign 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.
design hyperplane projective geometry