Lattice codes for the Gaussian relay channel: Decode-and-Forward and   Compress-and-Forward
  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. 
Yiwei Song and Natasha Devroye
cs.IT math.IT