List decoding for nested lattices and applications to relay channels

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We demonstrate a decoding scheme for nested lattice codes which is able to decode a list of a particular size which contains the transmitted codeword with high probability. This list decoder is analogous to that used in random coding arguments in achievability schemes of relay channels, and allows for the effective combination of information from the relay and source node. Using this list decoding result, we demonstrate 1) that lattice codes may achieve the capacity of the physically degraded AWGN relay channel, 2) an achievable rate region for the two-way relay channel with direct links using lattice codes, and 3) that we may improve the constant gap to capacity for specific cases of the two-way relay channel with direct links.