Maximal partial ovoids and maximal partial spreads in hermitian generalized quadrangles

Maximal partial ovoids and maximal partial spreads of the hermitian generalized quadrangles H (3, q 2 ) and H (4, q 2 ) are studied in great detail. We present improved lower bounds on the size of maximal partial ovoids and maximal partial spreads in the hermitian quadrangle H (4, q 2 ). We also construct in H (3, q 2 ), q =2 2 h +1 , h ≥ 1, maximal partial spreads of size smaller than the size q 2 +1 presently known. As a final result, we present a discrete spectrum result for the deficiencies of maximal partial spreads of H (4, q 2 ) of small positive deficiency δ. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 101–116, 2008