  Grassmannian $\mathcal{G}_q(n,k)$ is the set of all $k$-dimensional subspaces
of the vector space $\mathbb{F}_q^n.$ Recently, Etzion and Zhang introduced a
new notion called covering Grassmannian code which can be used in network
coding solutions for generalized combination networks. An
$\alpha$-$(n,k,\delta)_q^c$ covering Grassmannian code $\mathcal{C}$ is a
subset of $\mathcal{G}_q(n,k)$ such that every set of $\alpha$ codewords of
$\mathcal{C}$ spans a subspace of dimension at least $\delta +k$ in
$\mathbb{F}_q^n.$ In this paper, we derive new upper and lower bounds on the
size of covering Grassmannian codes. These bounds improve and extend the
parameter range of known bounds.
