A spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q)Q(4,q), qq even

This article presents a spectrum result on maximal partial ovoids of the generalized quadrangle Q(4,q)Q(4,q), qq even. We prove that for every integer kk in an interval of, roughly, size [q2/10,9q2/10][q2/10,9q2/10], there exists a maximal partial ovoid of size kk on Q(4,q)Q(4,q), qq even. Since the generalized quadrangle W(q)W(q), qq even, defined by a symplectic polarity of PG(3,q)PG(3,q) is isomorphic to the generalized quadrangle Q(4,q)Q(4,q), qq even, the same result is obtained for maximal partial ovoids of W(q)W(q), qq even. As equivalent results, the same spectrum result is obtained for minimal blocking sets with respect to planes of PG(3,q)PG(3,q), qq even, and for maximal partial 1-systems of lines on the Klein quadric Q+(5,q)Q+(5,q), qq even.