  Lattice coding techniques may be used to derive achievable rate regions which
outperform known independent, identically distributed (i.i.d.) random codes in
multi-source relay networks and in particular the two-way relay channel. Gains
stem from the ability to decode the sum of codewords (or messages) using
lattice codes at higher rates than possible with i.i.d. random codes. Here we
develop a novel lattice coding scheme for the Two-way Two-relay Channel: 1 <->
2 <-> 3 <-> 4, where Node 1 and 4 simultaneously communicate with each other
through two relay nodes 2 and 3. Each node only communicates with its
neighboring nodes. The key technical contribution is the lattice-based
achievability strategy, where each relay is able to remove the noise while
decoding the sum of several signals in a Block Markov strategy and then
re-encode the signal into another lattice codeword using the so-called
"Re-distribution Transform". This allows nodes further down the line to again
decode sums of lattice codewords. This transform is central to improving the
achievable rates, and ensures that the messages traveling in each of the two
directions fully utilize the relay's power, even under asymmetric channel
conditions. All decoders are lattice decoders and only a single nested lattice
codebook pair is needed. The symmetric rate achieved by the proposed lattice
coding scheme is within 0.5 log 3 bit/Hz/s of the symmetric rate capacity.
