Minimal blocking sets of size [formula omitted] of [formula omitted], q an odd prime, do not exist

Abstract

It is known that every blocking set of Q ( 4 , q ) , q > 2 even, with less than q 2 + 1 + q points contains an ovoid, and hence Q ( 4 , q ) has no minimal blocking set B with q 2 + 1 < | B | < q 2 + 1 + q . In contrast to this, it is even not known whether or not Q ( 4 , q ) , q odd, has minimal blocking sets of size q 2 + 2 . In this paper, the non-existence of a minimal blocking set of size q 2 + 2 of Q ( 4 , q ) , q an odd prime, is shown. Strong geometrical information is obtained using an algebraic description of W ( 3 , q ) . Geometrical and combinatorial arguments complete the proof.