Characterizing Geometric Designs

We conjecture that the classical geometric 2-designs PGd(n, q), 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 seems to be difficult. 1 – Introduction In this note, we are concerned with the problem of characterizing the classical geometric designs PGd(n, q), where d is in the range 2 ≤ d ≤ n− 2, among all designs with the same parameters. For the convenience of the reader, we first recall basic facts about these designs. Let Π denote PG(n, q), the n-dimensional projective space over the field GF (q) with q elements. Then the points and d-spaces of Π form a 2-(v, k, λ) design D = PGd(n, q) with parameters