  In this paper, a downlink communication system, in which a Base Station (BS)
equipped with M antennas communicates with N users each equipped with K receive
antennas ($K \leq M$), is considered. It is assumed that the receivers have
perfect Channel State Information (CSI), while the BS only knows the partial
CSI, provided by the receivers via feedback. The minimum amount of feedback
required at the BS, to achieve the maximum sum-rate capacity in the asymptotic
case of $N \to \infty$ and different ranges of SNR is studied. In the fixed and
low SNR regimes, it is demonstrated that to achieve the maximum sum-rate, an
infinite amount of feedback is required. Moreover, in order to reduce the gap
to the optimum sum-rate to zero, in the fixed SNR regime, the minimum amount of
feedback scales as $\theta(\ln \ln \ln N)$, which is achievable by the Random
Beam-Forming scheme proposed in [14]. In the high SNR regime, two cases are
considered; in the case of $K < M$, it is proved that the minimum amount of
feedback bits to reduce the gap between the achievable sum-rate and the maximum
sum-rate to zero grows logaritmically with SNR, which is achievable by the
"Generalized Random Beam-Forming" scheme, proposed in [18]. In the case of $K =
M$, it is shown that by using the Random Beam-Forming scheme and the total
amount of feedback not growing with SNR, the maximum sum-rate capacity is
achieved.
