  Characterizing the capacity region of multi-source wireless relay networks is
one of the fundamental issues in network information theory. The problem is,
however, quite challenging due to inter-user interference when there exist
multiple source--destination (S--D) pairs in the network. By focusing on a
special class of networks, we show that the capacity can be found. Namely, we
study a layered linear binary field network with time-varying channels, which
is a simplified model reflecting broadcast, interference, and fading natures of
wireless communications. We observe that fading can play an important role in
mitigating inter-user interference effectively for both single-hop and
multi-hop networks. We propose new encoding and relaying schemes with
randomized channel pairing, which exploit such channel variations, and derive
their achievable rates. By comparing them with the cut-set upper bound, the
capacity region of single-hop networks and the sum capacity of multi-hop
networks can be characterized for some classes of channel distributions and
network topologies. For these classes, we show that the capacity region or sum
capacity can be interpreted as the max-flow min-cut theorem.
