  We propose a novel space-division based network-coding scheme for
multiple-input multiple-output (MIMO) two-way relay channels (TWRCs), in which
two multi-antenna users exchange information via a multi-antenna relay. In the
proposed scheme, the overall signal space at the relay is divided into two
subspaces. In one subspace, the spatial streams of the two users have nearly
orthogonal directions, and are completely decoded at the relay. In the other
subspace, the signal directions of the two users are nearly parallel, and
linear functions of the spatial streams are computed at the relay, following
the principle of physical-layer network coding (PNC). Based on the recovered
messages and message-functions, the relay generates and forwards network-coded
messages to the two users. We show that, at high signal-to-noise ratio (SNR),
the proposed scheme achieves the asymptotic sum rate capacity of MIMO TWRCs
within 1/2log(5/4) = 0.161 bits per user-antenna for any antenna configuration
and channel realization. We perform large-system analysis to derive the average
sum-rate of the proposed scheme over Rayleigh-fading MIMO TWRCs. We show that
the average asymptotic sum rate gap to the capacity upper bound is at most
0.053 bits per relay-antenna. It is demonstrated that the proposed scheme
significantly outperforms the existing schemes.
