  In the scalar dirty multiple-access channel, in addition to Gaussian noise,
two additive interference signals are present, each known non-causally to a
single transmitter. It was shown by Philosof et al. that for strong
interferences, an i.i.d. ensemble of codes does not achieve the capacity
region. Rather, a structured-codes approach was presented, that was shown to be
optimal in the limit of high signal-to-noise ratios, where the sum-capacity is
dictated by the minimal ("bottleneck") channel gain. In this paper, we consider
the multiple-input multiple-output (MIMO) variant of this setting. In order to
incorporate structured codes in this case, one can utilize matrix
decompositions that transform the channel into effective parallel scalar dirty
multiple-access channels. This approach however suffers from a "bottleneck"
effect for each effective scalar channel and therefore the achievable rates
strongly depend on the chosen decomposition. It is shown that a recently
proposed decomposition, where the diagonals of the effective channel matrices
are equal up to a scaling factor, is optimal at high signal-to-noise ratios,
under an equal rank assumption. This approach is then extended to any number of
transmitters. Finally, an application to physical-layer network coding for the
MIMO two-way relay channel is presented.
