On the smallest minimal blocking sets of Q(2n,q), for q an odd prime

We characterize the smallest minimal blocking sets of Q(2n,q), q an odd prime, in terms of ovoids of Q(4,q) and Q(6,q). The proofs of these results are written for q=3,5,7 since for these values it was known that every ovoid of Q(4,q) is an elliptic quadric. Recently, in Ball et al. (Des. Codes Cryptogr., to appear), it has been proven that for all q prime, every ovoid of Q(4,q) is an elliptic quadric. Since as many proofs as possible were written for general q, using the classification result of De Beule and Metsch (J. Combin. Theory Ser. A, 106 (2004) 327-333) on the smallest blocking sets of Q(6,q), q>3 prime, the results for Q(2n,q), n⩾4, q=5,7, are also valid for q prime, q>7. The case q=3 is treated separately since this is the only value for q an odd prime for which Q(6,q) has an ovoid. We end the article by discussing the possibilities and remaining problems to obtain the characterization for general q odd.