  This paper considers the quantization problem on the Grassmann manifold with
dimension n and p. The unique contribution is the derivation of a closed-form
formula for the volume of a metric ball in the Grassmann manifold when the
radius is sufficiently small. This volume formula holds for Grassmann manifolds
with arbitrary dimension n and p, while previous results are only valid for
either p=1 or a fixed p with asymptotically large n. Based on the volume
formula, the Gilbert-Varshamov and Hamming bounds for sphere packings are
obtained. Assuming a uniformly distributed source and a distortion metric based
on the squared chordal distance, tight lower and upper bounds are established
for the distortion rate tradeoff. Simulation results match the derived results.
As an application of the derived quantization bounds, the information rate of a
Multiple-Input Multiple-Output (MIMO) system with finite-rate channel-state
feedback is accurately quantified for arbitrary finite number of antennas,
while previous results are only valid for either Multiple-Input Single-Output
(MISO) systems or those with asymptotically large number of transmit antennas
but fixed number of receive antennas.
