Nested lattice codes have played an important role in network information theory. However, their achievability proofs are often involved, even for the case of the additive white Gaussian noise (AWGN) channel. In sharp contrast, their finite-field counterparts, nested linear codes, enjoy much simpler achievability proofs. In this paper, we present a simple and direct proof that nested lattice codes achieve the AWGN channel capacity. In particular, we make use of an intriguing connection between nested lattice codes and nested linear codes, which allows us to keep the proof as simple as that for nested linear codes.
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