  This paper considers linear precoding for constant channel-coefficient
$K$-User MIMO Gaussian Interference Channel (MIMO GIC) where each
transmitter-$i$ (Tx-$i$), requires to send $d_i$ independent complex symbols
per channel use that take values from fixed finite constellations with uniform
distribution, to receiver-$i$ (Rx-$i$) for $i=1,2,\cdots,K$. We define the
maximum rate achieved by Tx-$i$ using any linear precoder, when the
interference channel-coefficients are zero, as the signal to noise ratio (SNR)
tends to infinity to be the Constellation Constrained Saturation Capacity
(CCSC) for Tx-$i$. We derive a high SNR approximation for the rate achieved by
Tx-$i$ when interference is treated as noise and this rate is given by the
mutual information between Tx-$i$ and Rx-$i$, denoted as $I[X_i;Y_i]$. A set of
necessary and sufficient conditions on the precoders under which $I[X_i;Y_i]$
tends to CCSC for Tx-$i$ is derived. Interestingly, the precoders designed for
interference alignment (IA) satisfy these necessary and sufficient conditions.
Further, we propose gradient-ascent based algorithms to optimize the sum-rate
achieved by precoding with finite constellation inputs and treating
interference as noise. Simulation study using the proposed algorithms for a
3-user MIMO GIC with two antennas at each node with $d_i=1$ for all $i$, and
with BPSK and QPSK inputs, show more than 0.1 bits/sec/Hz gain in the ergodic
sum-rate over that yielded by precoders obtained from some known IA algorithms,
at moderate SNRs.
