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 $\frac{1}{2}\log(1+\text{SNR})$ 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 capacity-achieving 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 $2\pi e$. 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 $O(N\log N)$ for any fixed target error probability.