  We present a natural and low-complexity technique for achieving the capacity
of the Gaussian relay network in the high SNR regime. Specifically, we propose
the use of end-to-end structured lattice codes with the amplify-and-forward
strategy, where the source uses a nested lattice code to encode the messages
and the destination decodes the messages by lattice decoding. All intermediate
relays simply amplify and forward the received signals over the network to the
destination. We show that the end-to-end lattice-coded amplify-and-forward
scheme approaches the capacity of the layered Gaussian relay network in the
high SNR regime. Next, we extend our scheme to non-layered Gaussian relay
networks under the amplify-and-forward scheme, which can be viewed as a
Gaussian intersymbol interference (ISI) channel. Compared with other schemes,
our approach is significantly simpler and requires only the end-to-end design
of the lattice precoding and decoding. It does not require any knowledge of the
network topology or the individual channel gains.
