  We consider the problem of two transmitters wishing to exchange information
through a relay in the middle. The channels between the transmitters and the
relay are assumed to be synchronized, average power constrained additive white
Gaussian noise channels with a real input with signal-to-noise ratio (SNR) of
snr. An upper bound on the capacity is 1/2 log(1+ snr) bits per transmitter per
use of the medium-access phase and broadcast phase of the bi-directional relay
channel. We show that using lattice codes and lattice decoding, we can obtain a
rate of 1/2 log(0.5 + snr) bits per transmitter, which is essentially optimal
at high SNRs. The main idea is to decode the sum of the codewords modulo a
lattice at the relay followed by a broadcast phase which performs Slepian-Wolf
coding with structured codes. For asymptotically low SNR's, jointly decoding
the two transmissions at the relay (MAC channel) is shown to be optimal. We
also show that if the two transmitters use identical lattices with minimum
angle decoding, we can achieve the same rate of 1/2 log(0.5 + snr). The
proposed scheme can be thought of as a joint physical layer, network layer code
which outperforms other recently proposed analog network coding schemes.
