1,132 research outputs found

    Construction of lattices for communications and security

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    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 2πe2\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 Λ/Λ\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-Λ\Lambda AWGN wiretap channel. The Mod-Λ\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

    A Study on the Capacity of Gaussian Channels: From a Lattice Code Perspective

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    As one of the most significant classes of structure codes, lattice codes are related to various geometric and coding problems, such as sphere packing and covering, quantization, signaling for the additive white Gaussian noise (AWGN) channel, Wyner-Ziv coding, dirty-paper coding, etc. In this thesis, we are especially interested in the construction of lattice codes for the AWGN channel, since from the classical channel coding theory, the capacity-achieving codebooks for the AWGN channel may possess little or no structure, making them ill-suited for applications. Specifically, we investigate the employment of lattice codes into the one-input-two-output AWGN channel to achieve its capacity. For the one-input-two-output AWGN channel, the receiver decodes jointly with its two observations which are the outputs of the transmitted signal going through two independent AWGN channels. An angle-decoding scheme is proposed, and we prove that under such decoding scheme, the capacity of the one-input-two-output AWGN channel can be achieved using lattice codes, where the bounding region of the lattice code is an n-dimensional ball to preserve the structure and symmetry of the underlying lattices, instead of a “thin” spherical shell as in previous studies. Moreover, to further preserve the lattice symmetry and to reduce complexity, the nested lattice code with lattice decoding is incorporated into the one-input-two- output AWGN channel, as under the lattice decoding scheme, the receiver decodes to the nearest lattice point, neglecting the effects of bounding regions. In contrast, minimum-distance decoding or the proposed angle-decoding aims to find the nearest codeword inside the bounding region. We first transform the one-input-two-output AWGN channel into a modulo-lattice additive noise (MLAN) channel with vanishing information loss, and then apply the nested lattice code on the MLAN channel. Furthermore, we extend the nested lattice code to a general single-input-multiple-output AWGN channel and prove it to be capacity achieving

    Achieving Secrecy Capacity of the Gaussian Wiretap Channel with Polar Lattices

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    In this work, an explicit wiretap coding scheme 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\Lambda_s Gaussian wiretap channel. Then we propose an explicit shaping scheme to remove this mod-Λs\Lambda_s 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 logN log(logN)). The proposed scheme is proven to be semantically secure.Comment: Submitted to IEEE Trans. Information Theory, revised. This is the authors' own version of the pape

    Construction of Capacity-Achieving Lattice Codes: Polar Lattices

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    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(NlogN)O(N\log N) for any fixed target error probability.Comment: full version of the paper to appear in IEEE Trans. Communication

    Approaching Gaussian Relay Network Capacity in the High SNR Regime: End-to-End Lattice Codes

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    We present a natural and low-complexity technique for achieving the capacity of the Gaussian relay network in the high SNR regime. Specifically, we propose the use of end-to-end structured lattice codes with the amplify-and-forward strategy, where the source uses a nested lattice code to encode the messages and the destination decodes the messages by lattice decoding. All intermediate relays simply amplify and forward the received signals over the network to the destination. We show that the end-to-end lattice-coded amplify-and-forward scheme approaches the capacity of the layered Gaussian relay network in the high SNR regime. Next, we extend our scheme to non-layered Gaussian relay networks under the amplify-and-forward scheme, which can be viewed as a Gaussian intersymbol interference (ISI) channel. Compared with other schemes, our approach is significantly simpler and requires only the end-to-end design of the lattice precoding and decoding. It does not require any knowledge of the network topology or the individual channel gains
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