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.

In this paper, an explicit scheme of wiretap coding based on polar lattices is proposed to achieve the secrecy capacity of the additive white Gaussian noise (AWGN) wiretap channel. First, polar lattices are used to construct secrecy-good lattices for the mod-ΛsGaussian wiretap channel (GWC). Then, we propose an explicit shaping scheme to remove this mod-Λsfront end and extend polar lattices to the genuine GWC. The shaping technique is based on the lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. By employing the asymmetric polar coding technique, we construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. As a result, the encoding complexity for the sender and the decoding complexity for the legitimate receiver are both O(N log N log (log N)) . The proposed scheme is proven to be semantically secure.

Polar lattices have been proved to be able to achieve the strong secrecy capacity of the Mod-Λsadditive 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.

Polar lattices, which are constructed from polar codes, are provably good for the additive white Gaussian noise (AWGN) channel. In this work, we propose a new polar lattice construction that achieves the secrecy capacity under the strong secrecy criterion over the mod-Λ Gaussian wiretap channel. This construction leads to an AWGN-good lattice and a secrecy-good lattice simultaneously. The design methodology is mainly based on the equivalence in terms of polarization between the Λ/Λ' channel in lattice coding and the equivalent channel derived from the chain rule of mutual information in multilevel coding.

In this paper, we propose the explicit construction of a new class of lattices based on polar codes, which are provably good for the additive white Gaussian noise (AWGN) channel. We follow the multilevel construction of Forney et al. (i.e., Construction D), where the code on each level is a capacity-achieving polar code for that level. The proposed polar lattices are efficiently decodable by using multistage decoding. Performance bounds are derived to measure the gap to the generalized capacity at given error probability. A design example is presented to demonstrate the performance of polar lattices.

We employ polar codes as the building blocks of Construction D to construct lattices for the additive white Gaussian noise (AWGN) channel. The construction of these component polar codes is based on the idea of Pedarsani et al. for binary-input memoryless symmetric (BMS) channels. Our lattice construction takes the advantage of the performance gain of polar codes over Reed-Muller codes. Simulation results show the lattices constructed from polar codes outperform the benchmark Barnes-Wall lattices, which are constructed from Reed-Muller codes.

Secrecy gain has been proposed as a criterion to design wiretap lattice codes. Its analysis relies on the theta series of a lattice, which is closely related to the union bound on the decoding error probability. In this paper, theta series of trellis codes are computed by using a modified transfer function originally for the union bound. We find that the low SNR region of the union bound dominates the secrecy gain. Numerical results show that trellis codes can achieve a similar or higher secrecy gain with less complexity compared to unimodular lattice codes.