An information reconciliation method for continuous-variable quantum key distribution with Gaussian modulation that is based on non-binary low-density parity-check (LDPC) codes is presented. Sets of regular and irregular LDPC codes with different code rates over the Galois fields @math , @math , @math , and @math have been constructed. We have performed simulations to analyze the efficiency and the frame error rate using the sum-product algorithm. The proposed method achieves an efficiency between @math and @math if the signal-to-noise ratio is between @math dB and @math dB.
While LDPC codes over alphabets with more than two elements have already been introduced in the classic work by Gallagher @cite_31 , Davey and MacKay first reported that non-binary LDPC codes can outperform their binary counterparts under the message-passing algorithm over the BSC and the binary input additive white Gaussian noise channel (BI-AWGNC) @cite_32 . This behavior is attributed to the fact that the non-binary graph contains in general much fewer cycles than the corresponding binary graph @cite_13 . Motivated by this fact, non-binary LDPC codes have been used in @cite_38 to improve the efficiency of information reconciliation in DV QKD.
@cite_38 @cite_31 @cite_13 @cite_32
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We show that for low-density parity-check (LDPC) codes whose Tanner graphs have sufficient expansion, the linear programming (LP) decoder of Feldman, Karger, and Wainwright can correct a constant fraction of errors. A random graph will have sufficient expansion with high probability, and recent work shows that such graphs can be constructed efficiently. A key element of our method is the use of a dual witness: a zero-valued dual solution to the decoding linear program whose existence proves decoding success. We show that as long as no more than a certain constant fraction of the bits are flipped by the channel, we can find a dual witness. This new method can be used for proving bounds on the performance of any LP decoder, even in a probabilistic setting. Our result implies that the word error rate of the LP decoder decreases exponentially in the code length under the binary-symmetric channel (BSC). This is the first such error bound for LDPC codes using an analysis based on "pseudocodewords." Recent work by Koetter and Vontobel shows that LP decoding and min-sum decoding of LDPC codes are closely related by the "graph cover" structure of their pseudocodewords; in their terminology, our result implies that that there exist families of LDPC codes where the minimum BSC pseudoweight grows linearly in the block length We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of ldquoprobabilistic expansionrdquo in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting. We study the information reconciliation (IR) scheme for quantum key distribution (QKD) protocols. The IR for the QKD can be seen as the asymmetric Slepian-Wolf problem, which low-density parity-check (LDPC) codes can solve with efficient algorithms, i.e., the belief propagation. However, the LDPC codes are needed to be chosen properly from a collection of codes optimized for multiple key rates, which leads to complex decoder devices and performance degradation for unoptimized key rates. Therefore, it is desired that establish an IR scheme with a single LDPC code which supports multiple rates. To this end, in this paper, we propose an IR scheme with a rate-compatible non-binary LDPC code. Numerical results show the proposed scheme achieves IR efficiency comparable to the best know conventional IR scheme with lower decoding error rates. This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source-channel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signal-to-noise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on space-filling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on space-filling curves operates within 1.1 dB of Shannon's rate-distortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the rate-distortion bound. The second scheme is based on low-density parity-check (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution from this observation, namely erasure-channel, Gaussian-capacity, and reciprocal-channel approximations. The reciprocal-channel approximation, based on dualizing LDPC codes, provides a very accurate model of density evolution for the AWGN channel. We also develop another approximation method, Gaussian approximation, which enables us to visualize infinite-dimensional density evolution and optimization of LDPC codes. We also develop other tools to better understand density evolution. Using these tools, we design some LDPC codes that approach the Shannon limit extremely closely. For multilevel AWGN channels, we design a rate 1 2 code that has a threshold within 0.0063 dB of the Shannon limit of the noisiest level. For binary-input AWGN channels, our best rate 1 2 LDPC code has a threshold within 0.0045 dB of the Shannon limit. Simulation results show that we can achieve within 0.04 dB of the Shannon limit at a bit error rate of 10−6 using a block length of 10 7. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)