  We investigate what we call generalized ovoids, that is families of totally
isotropic subspaces of finite classical polar spaces such that each maximal
totally isotropic subspace contains precisely one member of that family. This
is a generalization of ovoids in polar spaces as well as the natural $q$-analog
of a subcube partition of the hypercube (which can be seen as a polar space
with $q=1$). Our main result proves that a generalized ovoid of $k$-spaces in
polar spaces of large rank does not exist. More precisely, for $q=p^h$, $p$
prime, and some positive integer $k$, a generalized ovoid of $k$-spaces in a
polar space $\mathcal{P}$ with rank $r \geq r_0(k, p)$ in a vector space
$V(n,q)$ does not exist.
