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 capacityachieving 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.
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 work, 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. Firstly, polar lattices are used to construct secrecy-good lattices for the mod-Λs Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-Λs front end and extend polar lattices to the genuine Gaussian wiretap channel. 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.In simple terms, the secrecy capacity is defined as the maximum achievable rate under both the reliability and strong secrecy conditions. When W and V are both symmetric, and W is degraded with respect to V , the secrecy capacity is given by C(V ) − C(W ) [3], where C(·) denotes the channel capacity. DRAFT Polar codes [7] have shown their great potential in solving the wiretap coding problem. The polar coding scheme proposed in [8], combined with the block Markov coding technique [9], was proved to achieve the strong secrecy capacity when W and V are both binary-input symmetric channels, and W is degraded with respect to V . More recently, polar wiretap coding has been extended to general wiretap channels (not necessarily degraded or symmetric) in [10] and [11]. For continuous channels such as the GWC, there also has been notable progress in wiretap lattice coding. On the theoretical aspect, the existence of lattice codes achieving the secrecy capacity to within 1 2 nat under the strong secrecy as well as semantic security criterion was demonstrated in [6]. On the practical aspect, wiretap lattice codes were proposed in [12] and [13] to maximize the eavesdropper's decoding error probability. June 13, 2018 DRAFT 3 A. Our contribution Polar lattices, the counterpart of polar codes in the Euclidean space, have already been proved to be additive white Gaussian noise (AWGN)-good [14] and further to achieve the AWGN channel capacity with lattice Gaussian shaping [15] 1 . Motivated by [8], we will propose polar lattices to achieve both strong secrecy and reliability over the mod-Λ s GWC. Conceptually, this polar lattice structure can be regarded as a secrecy-good lattice Λ e nested within an AWGN-good lattice Λ b (Λ e ⊂ Λ b ). Further, we will propose a Gaussian shaping scheme over Λ b and Λ e , using the multilevel asymmetric polar coding technique. As a result, we will accomplish the design of an explicit lattice coding scheme which achieves the secrecy capacity of the GWC with semantic security.• The first technical contribution of this paper is the explicit construction of secrecy-good polar latt...
Polar lattices have been proved to be able to achieve the strong secrecy capacity of the Mod-Λs 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.
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.
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