Construction of Capacity-Achieving Lattice Codes: Polar Lattices

TL;DR

This paper introduces polar lattices, a new class of capacity-achieving lattice codes for the AWGN channel, constructed from nested polar codes with efficient encoding and decoding.

Contribution

The paper presents a novel explicit construction of polar lattices that achieve channel capacity and are AWGN-good, with efficient multilevel encoding and shaping techniques.

Findings

Polar lattices are proven to be AWGN-good.

The construction achieves capacity with $O(N \, \log N)$ complexity.

Discrete Gaussian shaping enables power constraint satisfaction.

Abstract

In this paper, we propose a new class of lattices constructed from polar codes, namely polar lattices, to achieve the capacity $\frac{1}{2}\log(1+\SNR)$ of the additive white Gaussian-noise (AWGN) channel. Our construction follows the multilevel approach of Forney \textit{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 polar lattices are \emph{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.