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 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. A low-density parity-check code is a code specified by a parity-check matrix with the following properties: each column contains a small fixed number j 3 of l's and each row contains a small fixed number k > j of l's. The typical minimum distance of these codes increases linearly with block length for a fixed rate and fixed j . When used with maximum likelihood decoding on a sufficiently quiet binary-input symmetric channel, the typical probability of decoding error decreases exponentially with block length for a fixed rate and fixed j . A simple but nonoptimum decoding scheme operating directly from the channel a posteriori probabilities is described. Both the equipment complexity and the data-handling capacity in bits per second of this decoder increase approximately linearly with block length. For j > 3 and a sufficiently low rate, the probability of error using this decoder on a binary symmetric channel is shown to decrease at least exponentially with a root of the block length. Some experimental results show that the actual probability of decoding error is much smaller than this theoretical bound. Having trouble deciding which coding scheme to employ, how to design a new scheme, or how to improve an existing system? This summary of the state-of-the-art in iterative coding makes this decision more straightforward. With emphasis on the underlying theory, techniques to analyse and design practical iterative coding systems are presented. Using Gallager's original ensemble of LDPC codes, the basic concepts are extended for several general codes, including the practically important class of turbo codes. The simplicity of the binary erasure channel is exploited to develop analytical techniques and intuition, which are then applied to general channel models. A chapter on factor graphs helps to unify the important topics of information theory, coding and communication theory. Covering the most recent advances, this text is ideal for graduate students in electrical engineering and computer science, and practitioners. Additional resources, including instructor's solutions and figures, available online: www.cambridge.org 9780521852296. Gallager's (1962) low-density binary parity check codes have been shown to have near-Shannon limit performance when decoded using a probabilistic decoding algorithm. We report the empirical results of error-correction using the analogous codes over GF(q) for q>2, with binary symmetric channels and binary Gaussian channels. We find a significant improvement over the performance of the binary codes, including a rate 1 4 code with bit error probability <10 sup -5 at E sub b N sub 0 =0.2 dB.