  In this paper, we consider a Gaussian two-way relay channel (GTRC), where two
sources exchange messages with each other through a relay. We assume that there
is no direct link between sources, and all nodes operate in full-duplex mode.
By utilizing nested lattice codes for the uplink (i.e., MAC phase), and
structured binning for the downlink (i.e., broadcast phase), we propose two
achievable schemes. Scheme 1 is based on compute and forward scheme of [1]
while scheme 2 utilizes two different lattices for source nodes based on a
three-stage lattice partition chain. We show that scheme 2 can achieve capacity
region at the high signal-to-noise ratio (SNR). Regardless all channel
parameters, the achievable rate of scheme 2 is within 0.2654 bit from the
cut-set outer bound for user 1. For user 2, the proposed scheme achieves within
0.167 bit from the outer bound if channel coefficient is larger than one, and
achieves within 0.2658 bit from the outer bound if channel coefficient is
smaller than one. Moreover, sum rate of the proposed scheme is within 0.334
bits from the sum capacity. These gaps for GTRC are the best gap-to-capacity
results to date.
