Combinatorial geometries representable over GF(3) and GF(q). I. The number of points

Abstract

Let q be a prime power not divisible by 3. We show that the number of points (or rank-1 flats) in a combinatorial geometry (or simple matroid) of rank n representable over GF(3) and GF(q) is at most n2. When q is odd, this bound is sharp and is attained by the Dowling geometries over the cyclic group of order 2. © 1990 Springer-Verlag New York Inc.