  A {\em maximal partial ovoid} of a generalized quadrangle is a maximal set of
points no two of which are collinear. The problem of determining the smallest
size of a maximal partial ovoid in quadrangles has been extensively studied in
the literature. In general, theoretical lower bounds on the size of a maximal
partial ovoid in a quadrangle of order $(s,t)$ are linear in $s$. In this
paper, in a wide class of quadrangles of order $(s,t)$ we give a construction
of a maximal partial ovoid of size at most $s \cdot \mbox{polylog}(s)$, which
is within a polylogarithmic factor of theoretical lower bounds. The
construction substantially improves previous quadratic upper bounds in
quadrangles of order $(s,s^2)$, in particular in the well-studied case of the
elliptic quadrics $Q^-(5,s)$.
