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Lattice codes are known to achieve capacity in the Gaussian point-to-pointchannel, achieving the same rates as independent, identically distributed(i.i.d.) random Gaussian codebooks. Lattice codes are also known to outperformrandom codes for certain channel models that are able to exploit theirlinearity. In this work, we show that lattice codes may be used to achieve thesame performance as known i.i.d. Gaussian random coding techniques for theGaussian relay channel, and show several examples of how this may be combinedwith the linearity of lattices codes in multi-source relay networks. Inparticular, we present a nested lattice list decoding technique, by which,lattice codes are shown to achieve the Decode-and-Forward (DF) rate of singlesource, single destination Gaussian relay channels with one or more relays. Wenext present two examples of how this DF scheme may be combined with thelinearity of lattice codes to achieve new rate regions which for some channelconditions outperform analogous known Gaussian random coding techniques inmulti-source relay channels. That is, we derive a new achievable rate regionfor the two-way relay channel with direct links and compare it to existingschemes, and derive another achievable rate region for the multiple accessrelay channel. We furthermore present a lattice Compress-and-Forward (CF)scheme for the Gaussian relay channel which exploits a lattice Wyner-Zivbinning scheme and achieves the same rate as the Cover-El Gamal CF rateevaluated for Gaussian random codes. These results suggest thatstructured/lattice codes may be used to mimic, and sometimes outperform, randomGaussian codes in general Gaussian networks.

Lattice codes for the Gaussian relay channel: Decode-and-Forward and Compress-and-Forward