  The paper establishes the optimal generalized degrees of freedom (GDOF) of
3-user $M \times N$ multiple-input multiple-output (MIMO) Gaussian interference
channel (GIC) in which each transmitter has $M$ antennas and each receiver has
$N$ antennas. A constraint of $2M \leq N$ is imposed so that random coding with
message-splitting achieves the optimal GDOF. Unlike symmetric case, two cross
channels to unintended receivers from each transmitter can have different
strengths, and hence, well known Han-Kobayashi common-private message splitting
would not achieve the optimal GDOF. Instead, splitting each user's message into
three parts is shown to achieve the optimal GDOF. The capacity of the
corresponding deterministic model is first established which provides
systematic way of determining side information for converse. Although this
deterministic model is philosophically similar to the one considered by Gou and
Jafar, additional constraints are imposed so that capacity description of the
deterministic model only contains the essential terms for establishing the GDOF
of Gaussian case. Based on this, the optimal GDOF of Gaussian case is
established with $\mathcal{O}(1)$ capacity approximation. The behavior of the
GDOF is interestingly different from that of the corresponding symmetric case.
Regarding the converse, several multiuser outer bounds which are suitable for
asymmetric case are derived by non-trivial generalization of the symmetric
case.
