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We consider a communication system where two transmitters wish to exchange information through a central relay. The transmitter and relay nodes exchange data over 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 multiple access phase and broadcast phase of the bidirectional relay channel. We show that, using lattice codes and lattice decoding, we can obtain a rate of 1/2 log(1/2 + snr) bits per transmitter, which is essentially optimal at high SNR. 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. 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(1/2 + snr …