Construction of Capacity-Achieving Lattice Codes: Polar Lattices

Abstract

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 Gaussiannoise (AWGN) channel. Our construction follows the multilevel approach of Forney et al., where we construct a capacity-achieving polar code on each level. The component polar codes are shown to be naturally nested, thereby, fulfilling the requirement of the multilevel lattice construction. We prove that the polar lattices are AWGN-good. Furthermore, using the technique of source polarization, we propose discrete Gaussian shaping over the polar lattice to satisfy the power constraint. 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.