Secrecy-good polar lattices with optimal shaping for the Gaussian wiretap channels

Polar lattices have been proved to be able to achieve the strong secrecy capacity of the Mod-Λ_s additive white Gaussian noise (AWGN) wiretap channel. In this work, we propose an explicit shaping scheme and extend polar lattice coding to the genuine Gaussian wiretap channel. This shaping technique is based on discrete lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. The construction of polar codes for an asymmetric channel can be converted to that for a related symmetrized channel, and it turns out that this symmetrized channel is equivalent to a scaled Λ/Λ´ channel in lattice coding in terms of polarization. By employing the asymmetric polar coding technique, we construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously.