Characterizing Geometric Designs

We conjecture that the classical geometric 2-designs formed by the points and d-dimensional subspaces of the projective space of dimension n over the field with q elements, where 2 <= d <= n-1, are characterized among all designs with the same parameters as those having line size q+1. The conjecture is known to hold for the case d=n-1 (the Dembowski-Wagner theorem) and also for d=2 (a recent result established by Tonchev and the present author). Here we extend this result to the cases d=3 and d=4. The general case remains open and appears to be difficult