  Lattice codes are known to achieve capacity in the Gaussian point-to-point
channel, achieving the same rates as independent, identically distributed
(i.i.d.) random Gaussian codebooks. Lattice codes are also known to outperform
random codes for certain channel models that are able to exploit their
linearity. In this work, we show that lattice codes may be used to achieve the
same performance as known i.i.d. Gaussian random coding techniques for the
Gaussian relay channel, and show several examples of how this may be combined
with the linearity of lattices codes in multi-source relay networks. In
particular, we present a nested lattice list decoding technique, by which,
lattice codes are shown to achieve the Decode-and-Forward (DF) rate of single
source, single destination Gaussian relay channels with one or more relays. We
next present two examples of how this DF scheme may be combined with the
linearity of lattice codes to achieve new rate regions which for some channel
conditions outperform analogous known Gaussian random coding techniques in
multi-source relay channels. That is, we derive a new achievable rate region
for the two-way relay channel with direct links and compare it to existing
schemes, and derive another achievable rate region for the multiple access
relay channel. We furthermore present a lattice Compress-and-Forward (CF)
scheme for the Gaussian relay channel which exploits a lattice Wyner-Ziv
binning scheme and achieves the same rate as the Cover-El Gamal CF rate
evaluated for Gaussian random codes. These results suggest that
structured/lattice codes may be used to mimic, and sometimes outperform, random
Gaussian codes in general Gaussian networks.
