Maximal partial spreads of T2(O) and T3(O)

Assuming a partial spread of T2(O) or T3(O), with deficiency δ, is maximal and using results on minihypers, which are closely related to blocking sets in PG(2,q), we obtain lower bounds for δ. If q is even, using extendability of arcs in PG(2,q), we prove that a maximal partial spread of T2(O) which does not cover (∞) does not exist if δ≤q−1. This improves a theorem of Tallini (Proceedings of the First International Conference on Blocking Sets (Giessen, 1989) 201 (1991) 141) for T2(O)≅Q(4,q), and, furthermore, this result is sharp since partial spreads with deficiency δ=q are constructed.