On blocking sets of inversive planes

Let S be a blocking set in an inversive plane of order q. It was shown by Bruen and Rothschild [1] that |S| >= 2q for q >= 9. We prove that if q is sufficiently large, C is a fixed natural number and |S| = 2q + C, then roughly 2/3 of the circles of the plane meet S in one point and 1/3 of the circles of the plane meet S in four points. The complete classification of minimal blocking sets in inversive planes of order q <= 5 and the sizes of some examples of minimal blocking sets in planes of order q <= 37 are given. Geometric properties of some of these blocking sets are also studied.