Construction of lattices for communications and security

In this thesis, we propose a new class of lattices based on polar codes, namely polar lattices. Polar lattices enjoy explicit construction and provable goodness for the additive white Gaussian noise (AWGN) channel, \textit{i.e.}, they are \emph{AWGN-good} lattices, in the sense that the error probability (for infinite lattice coding) vanishes for any fixed volume-to-noise ratio (VNR) greater than $2\pi e$. Our construction is based on the multilevel approach of Forney \textit{et al.}, where on each level we construct a capacity-achieving polar code. We show the component polar codes are naturally nested, thereby fulfilling the requirement of the multilevel lattice construction. We present a more precise analysis of the VNR of the resultant lattice, which is upper-bounded in terms of the flatness factor and the capacity losses of the component codes. The proposed polar lattices are efficiently decodable by using multi-stage decoding. Design examples are presented to demonstrate the superior performance of polar lattices. However, there is no infinite lattice coding in the practical applications. We need to apply the power constraint on the polar lattices which generates the polar lattice codes. We prove polar lattice codes can achieve the capacity \frac{1}{2}\log(1+\SNR) of the power-constrained AWGN channel with a novel shaping scheme. The main idea is that by implementing the lattice Gaussian distribution over the AWGN-good polar lattices, the maximum error-free transmission rate of the resultant coding scheme can be arbitrarily close to the capacity \frac{1}{2}\log(1+\SNR). The shaping technique is based on discrete lattice Gaussian distribution, which leads to a binary asymmetric channel at each level for the multilevel lattice codes. Then it is straightforward to employ multilevel asymmetric polar codes which is a combination of polar lossless source coding and polar channel coding. The construction of polar codes for an asymmetric channel can be converted to that for a related symmetric channel, and it turns out that this symmetric channel is equivalent to an minimum mean-square error (MMSE) scaled $\Lambda/\Lambda'$ channel in lattice coding in terms of polarization, which eventually simplifies our coding design. Finally, we investigate the application of polar lattices in physical layer security. Polar lattice codes are proved to be able to achieve the strong secrecy capacity of the Mod-$\Lambda$ AWGN wiretap channel. The Mod-$\Lambda$ assumption was due to the fact that a practical shaping scheme aiming to achieve the optimum shaping gain was missing. In this thesis, we use our shaping scheme and extend polar lattice coding to the Gaussian wiretap channel. By employing the polar coding technique for asymmetric channels, we manage to construct an AWGN-good lattice and a secrecy-good lattice with optimal shaping simultaneously. Then we prove the resultant wiretap coding scheme can achieve the strong secrecy capacity for the Gaussian wiretap channel.Open Acces

Construction of Capacity-Achieving Lattice Codes: Polar Lattices

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.Comment: full version of the paper to appear in IEEE Trans. Communication

On Achievable Rate Regions of the Asymmetric AWGN Two-Way Relay Channel

This paper investigates the additive white Gaussian noise two-way relay channel, where two users exchange messages through a relay. Asymmetrical channels are considered where the users can transmit data at different rates and at different power levels. We modify and improve existing coding schemes to obtain three new achievable rate regions. Comparing four downlink-optimal coding schemes, we show that the scheme that gives the best sum-rate performance is (i) complete-decode-forward, when both users transmit at low signal-to-noise ratio (SNR); (ii) functional-decode-forward with nested lattice codes, when both users transmit at high SNR; (iii) functional-decode-forward with rate splitting and time-division multiplexing, when one user transmits at low SNR and another user at medium--high SNR.Comment: to be presented at ISIT 201