Construction of Capacity-Achieving Lattice Codes: Polar Lattices

In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity 1 2 log(1+ SNR) of the additive white Gaussian-noise (AWGN) channel for any signal-to-noise ratio (SNR). Our construction follows the multilevel approach of Forney et al., where on each level we construct a capacityachieving polar code. The component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We prove that polar lattices are AWGN-good, in the sense that the error probability (for infinite lattice decoding) vanishes for any fixed volume-to-noise ratio (VNR) greater than 2πe. Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. The proposed polar lattices permit low-complexity multistage successive cancellation decoding. Both the construction and shaping are explicit, and the overall complexity of encoding and decoding is O(N logN) for any fixed target error probability.