In this paper, we propose an efficient method to reduce error floors in quantum error correction using non-binary low-density parity-check (LDPC) codes. We identify and classify cycle structures in the parity-check matrix where estimated noise becomes trapped, and develop tailored decoding methods for each cycle type. For Type-I cycles, we propose a method to make the difference between estimated and true noise degenerate. Type-II cycles are shown to be un-correctable, while for Type-III cycles, we utilize the fact that cycles in non-binary LDPC codes do not necessarily correspond to codewords, allowing us to estimate the true noise. Our method significantly improves decoding performance and reduces error floors.

This paper investigates the construction and analysis of permutation codes under the Chebyshev distance. Direct product group permutation (DPGP) codes, independently introduced by Kløve et al. and Tamo et al., represent the best-known class of permutation codes in terms of both size and minimum distance, while also allowing for algebraic and efficient encoding and decoding. In contrast, this study focuses on recursively extended permutation (REP) codes, proposed by Kløve et al. as a recursive alternative. We analyze the properties of REP codes and prove that, despite their distinct construction principles, optimal REP codes achieve exactly the same size and minimum distance as the best DPGP codes under the Chebyshev metric. This surprising equivalence uncovers a deep connection between two structurally dissimilar code families and establishes REP codes as a structurally flexible yet equally powerful alternative to DPGP codes. In addition, we present efficient encoding and decoding algorithms for REP codes, including a sequential encoder withO(nlogn) complexity and a bounded-distance decoder withO(nlog2n) complexity.

In a standard group test setup, the result of each pooled test is either positive or negative. Polymerase chain reaction (PCR) is a method of making billions of copies of DNA from a small amount of DNA sample. Recently, the COVID-19 pandemic has dramatically impacted public health worldwide. Massive population PCR testing allows the isolation of infected individuals and the pandemic control. Quantitative PCR (q-PCR) test provides more information about infected samples than the standard group test. In this paper, we model the q-PCR and demonstrate group q-PCR testing with sparse test matrices and by belief propagation.

Quasi-cyclic low-density parity-check (QC-LDPC) codes are one of the most important code classes of LDPC codes. They have two drawbacks: lack of randomness and limited girth lead to a degraded decoding performance in the waterfall and error floor regions, respectively. To tackle these problems, we present a new class of LDPC codes, named linear permutation polynomial (LPP) codes, whose parity-check matrix consists of permutation matrices based on LPPs. The girth of regular QC-LDPC codes is upper bounded by 12, while LPP codes break this limit. We demonstrate that LPP codes have error performance almost equivalent to random ones.

In this paper, we deal with coding for synchronous errors. In [1], we introduced a synchronously erroneous finite state Markov channel model whose SIR is computable. Numerical experiments demonstrated spatially-coupled codes approach the SIR of the channel. However, at high-rate region, there is a gap to the SIR even if very long code length was used. In this paper, for the channel, we apply density evolution analysis [2] and the extended version for FSMC [3]. The results show that the threshold of protograph coupled codes is degraded at high-rate region, while random coupled codes approach the SIR even in the high-rate region.

Due to many small cycles in factor graphs, detection and decoding over two-dimensional inter-symbol interference (2D-ISI) channels has been considered difficult. It is reported that generalized belief propagation (GBP) detection for 2D-ISI performs much better than BP. However, conventional studies for GBP for 2D-ISI were restricted to the separate uses of detection and decoding. Joint usage of detection and decoding were proposed in [1] but hard-decisions are inserted between detector and decoder. This makes a gap to the optimal performance. In this paper, we study full-joint use of GBP for detector and decoder for 2D-ISI, i.e., we use only soft-messages. Furthermore, we clarify a necessary condition with respect to the region graph that messages converge to a GBP fixed point. We found most of regions graph proposed in the literature do not satisfy the convergence condition. This prevents the full-joint use of GBP for 2D-ISI. Moreover, we propose a method to construct region graphs which satisfy the condition. Numerical experiments show that the proposed joint detector and decoder perform better than the conventional separate detector and decoder [2] in a simple 2D-ISI channels.

Fukushima et al. proved that spatially-coupled codes, without pilot symbols and any optimization for the channels, universally achieve the symmetric information rate (SIR) of generalized erasure channels with memory. We expect that the universality is also valid for fading channels. The receiver performs joint iterative channel estimation and decoding where factor-graphs-based BCJR channel estimator for finite-state Markov channels and the LDPC decoder are considered. We demonstrate that the reliable transmission is possible at a rate close to the SIR.

In this paper, we deal with coding for synchronization errors. In conventional studies, to combat such errors, periodic synchronization markers are inserted or specifier and watermark codes are concatenated. These codes enable estimation of synchronous errors, but do not have ability to correct random errors. Low-density parity-check codes are usually concatenate to correct random errors. Due to the lack of dependence of information, periodic synchronization marker insertion prevents codes to approach the capacity. Recently, it is observed that spatially-coupled codes universally approach the symmetric information rate (SIR) of arbitrary finite state Markov channels. We introduce a synchronously erroneous finite state Markov channel model whose SIR is computable. Numerical experiments demonstrate spatially-coupled codes approach the SIR of the channel.

This paper investigates the belief propagation decoding of spatially-coupled MacKay-Neal (SC-MN) codes over erasure channels with memory. We show that SC-MN codes with bounded degree universally achieve the symmetric information rate (SIR) of arbitrary erasure channels with memory. We mean by universality the following sense: the sender does not need to know the whole channel statistics but needs to know only the SIR, while the receiver estimates the transmitted codewords from channel statistics and received words. The proof is based on the potential function.

Despite their ability to reach within the channel capacity in shorter codeblock lengths, non-binary LDPC codes have a higher decoding complexity that poses non-trivial barriers to their generalized adoption at algorithmic and compute-intensive levels. In this work, we propose a programmable FFT-SPA decoder that delivers high decoding throughput at low power consumptions, while retaining a design flexibility at the system level which surpasses typical VLSI descriptions, guaranteeing quick retargeting and prototyping of variants of this family of signal processing algorithms with effective decoding throughputs of up to 1 Mbit/s and potential throughputs of dozens of Mbit/s.

Obata et al. proved that spatially-coupled (SC) MacKay-Neal (MN) codes achieve the capacity of BEC. However, the SC-MN codes have many variable nodes of degree two and have higher error floors. In this paper, we prove that SC-MN codes with no variable nodes of degree two achieve the capacity of BEC.

Raptor codes are known as precoded rateless codes that achieve the capacity of BEC. However the maximum degree of Raptor codes needs to be unbounded to achieve the capacity. In this paper we prove that spatially-coupled precoded rateless codes achieve the capacity with bounded degree under BP decoding.

We study LDPC codes for the channel with input x ∈ Fqmand output y = x + z ∈ Fqm. The aim of this paper is to evaluate decoding performance of qm-ary non-binary LDPC codes for large m. We give density evolution and decoding performance evaluation for regular non-binary LDPC codes and spatially-coupled (SC) codes. We show the regular codes do not achieve the capacity of the channel while SC codes do.

Digital Video Broadcasting-Satellite-Second Generation (DVB-S2) is the current European standard for satellite broadcast and broadband communications. It relies on high-order modulations up to 32 amplitude/phase-shift keying (APSK) in order to increase the system spectral efficiency. Unfortunately, as the modulation order increases, the receiver becomes more sensitive to physical-layer impairments and, notably, to the distortions induced by the power amplifier and the channelizing filters aboard the satellite. The predistortion of a nonlinear satellite channel has been studied for many years. However, the performance of existing predistortion algorithms generally becomes poor when high-order modulations are used on a nonlinear channel with long memory. In this paper, we investigate a new iterative method that predis-torts the blocks of transmitted symbols to minimize the Euclidian distance between the transmitted and received symbols. We also propose approximations to relax the predistorter complexity while keeping its performance acceptable.

It is well known that non-binary LDPC codes outperform the BER performance of binary LDPC codes for the same code length. The superior BER performance of non-binary codes comes at the expense of more complex decoding algorithms that demand higher computational power. In this paper, we propose parallel signal processing algorithms for performing the FFT-SPA and the corresponding decoding of non-binary LDPC codes over GF(q). The constraints imposed by the complex nature of associated subsystems and kernels, in particular the Check Nodes, present computational challenges regarding multicore systems. Experimental results obtained on GPU for a variety of GF(q) show throughputs in the order of 2 Mbps, which is far above from the minimum throughput required, for example, for real-time video applications that can benefit from such error correcting capabilities.

In this paper, we propose a decoding algorithm which lowers decoding erasure rates in the error floor regions for non-binary low-density parity-check codes transmitted over the binary erasure channels. This decoding algorithm is a combination with belief propagation (BP) decoding and maximum a posteriori (MAP) decoding on zigzag cycles, which cause decoding erasures in the error floor region. We show that MAP decoding on the zigzag cycles is realized by means of a message passing algorithm. A simulation result shows that the decoding erasure rates in the error floor regions by the proposed decoding algorithm are lower than those by the BP decoder.

In this paper, we derive the average weight distributions for the irregular non-binary cluster low-density parity-check (LDPC) code ensembles. Moreover, we give the exponential growth rate of the average weight distribution in the limit of large code length. We show that there exist (2, dc)-regular non-binary cluster LDPC code ensembles whose normalized typical minimum distances are strictly positive.

Convolutional (or spatially-coupled) low-density parity-check (LDPC) codes have now been shown to approach capacity for a variety of problems. Yet, most of these results require sequences of regular LDPC ensembles with increasing variable and check degrees. Previously, Kasai and Sakaniwa showed empirically that, for the BEC, this limitation can be overcome by using spatially-coupled MacKay-Neal (MN) and Hsu-Anastasopoulos (HA) ensembles. In this paper, we prove this analytically for (k, 2, 2)-MN and (2, k, 2)-HA ensembles when k is at least 3. The proof is based on the simple approach to threshold saturation, introduced by Yedla et al., which relies on potential functions. The key step is verifying the non-negativity of a potential function associated with the uncoupled system. Along the way, we derive the potential function general multi-edge type (MET) LDPC ensembles and establish a duality relationship between dual ensembles of MET LDPC codes.

Spatially-coupled (SC) codes are constructed by coupling many regular low-density parity-check codes in a chain. The decoding chain of SC codes aborts when facing burst erasures. This problem cannot be overcome by increasing the chain length. In this paper, we introduce multi-dimensional (MD) SC codes to circumvent it. Numerical results show that two-dimensional SC codes are more robust against the burst erasures than one-dimensional SC codes. Furthermore, we consider designing multidimensional SC codes with smaller rateloss.

Raptor codes are rateless codes that achieve the capacity on the binary erasure channels. However the maximum degree of optimal output degree distribution is unbounded. This leads to a computational complexity problem both at encoders and decoders. Aref and Urbanke investigated the potential advantage of universal achieving-capacity property of proposed spatially-coupled (SC) low-density generator matrix (LDGM) codes. However the decoding error probability of SC-LDGM codes is bounded away from 0. In this paper, we investigate SC-LDGM codes concatenated with SC low-density parity-check codes. The proposed codes can be regarded as SC Hsu-Anastasopoulos rateless codes. We derive a lower bound of the asymptotic overhead from stability analysis for successful decoding by density evolution. The numerical calculation reveals that the lower bound is tight. We observe that with a sufficiently large number of information bits, the asymptotic overhead and the decoding error rate approach 0 with bounded maximum degree.

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding. For fixed numbers of BP iterations, the bit error probability approaches a limit as the blocklength tends to infinity, and the limit is obtained via density evolution. The finite-blocklength correction behaves like α(ε,t)/n+Θ(n-2) as the blocklengthntends to infinity where α(ε,t) denotes a specific constant determined by the code ensemble considered, the numbertof iterations, and the erasure probability ε of the BEC. In this paper, we derive a set of recursive formulas which allows the evaluation of the constant α(ε,t) for standard irregular ensembles. The dominant difference α(ε,t)/ncan be considered as effects of cycle-free and single-cycle structures of local graphs. Furthermore, it is confirmed via numerical simulations that estimation of the bit error probability using α(ε,t) is accurate even for small blocklengths.

Recently, it has been proved that both MMSE and MF detectors are near optimal detection for large scale MIMO systems, e.g., MIMO systems with hundreds of antennas. In order to attain near capacity region with reasonable complexity, low-complexity detector with soft-output generation is necessary for coded large MIMO systems. We show in this paper that the non-binary LDPC codes and well-known soft-output MMSE detector can be utilized to significantly reduce the gap to capacity. We also propose a novel soft-output MF-based detector for the non-binary LDPC coded large MIMO systems. With this proposed detector, capacity approaching performance, i.e., the gap to capacity of 1.6 dB, can be achieved with ultra low-complexity detection, e.g., just 0.28% of MMSE detection. Moreover, use of the proposed scheme in large MIMO systems is found to be robust to the presence of imperfect channel estimation and spatial fading correlation which are both the realistic scenarios for large MIMO systems.

We study LDPC codes for the channel with 2m-ary input x ϵ F2mand output y = x + z ϵ F2m. The receiver knows a subspace V ⊂ F2mfrom which z = y - x is uniformly chosen. Or equivalently, the receiver receives an affine subspace y-V where x lies. We consider a joint iterative decoder involving the channel detector and the LDPC decoder. The decoding system considered in this paper can be viewed as a simplified model of the joint iterative decoder over non-binary modulated signal inputs e.g., 2m-QAM. We evaluate the performance of binary spatially-coupled MacKay-Neal codes by density evolution. The iterative decoding threshold is seriously degraded by increasing m. EXIT-like function curve calculations reveal that this degradation is caused by wiggles and can be mitigated by increasing the randomized window size. The resultant iterative decoding threshold values are very close to the Shannon limit.

In this paper, we compare the decoding error rates in the error floors for non-binary low-density parity-check (LDPC) codes over the general linear group with those for non-binary LDPC codes over finite field transmitted over the q-ary memoryless symmetric channel under belief propagation decoding. To analyze non-binary LDPC codes defined over both general linear group GL(m, F2) and finite field F2m, we investigate non-binary LDPC codes defined over GL(m3, F2m4). We propose a method to lower the error floors for non-binary LDPC codes. In this analysis, we see that the optimized non-binary LDPC codes defined over general linear group have the same decoding performance in the error floors as those defined over finite field. The non-binary LDPC codes defined over general linear group have more choices of the labels in the edges which satisfy the condition for the optimization.

In this paper, we have investigated the application of non-binary LDPC codes to spatial multiplexing MIMO systems with a large number of low power antennas. We demonstrate that such large MIMO systems incorporating with low-complexity MMSE detector and non-binary LDPC codes can achieve low probability of bit error at near MIMO capacity. The new proposed non-binary LDPC coded system also performs better than other coded large MIMO systems known in the present literature. For instance, non-binary LDPC coded BPSK-MIMO system with 600 transmit/receive antennas performs within 3.4 dB from the capacity while the best known turbo coded system operates about 9.4 dB away from the capacity. Based on the simulation results provided in this paper, the proposed non-binary LDPC coded large MIMO system is capable of supporting ultra high spectral efficiency at low bit error rate.

From an information-theoretic point of view, it is well known that the capacity of relay channel comprising of three terminals is much more than that of two terminal direct channel especially for low SNRs. Previously invented relay coding strategies have not been designed to achieve this enormous capacity occurring in the low SNR region. In this paper, we propose a simple coding strategy for a relay channel with low SNRs or, equivalently, for a very noisy relay channel. The multiplicative repetition is utilized to design this simple coding strategy. We claim that the proposed strategy is simple since the destination and the relay can decode with almost the same computational complexity by sharing the same structure of decoder. An appropriate static power allocation which yields the maximum throughput close to the optimal one in low SNRs is suggested. Under practical constraints such as equal time-sharing etc., the asymptotic performance of this simple strategy is within 0.5 dB from the achievable rate of relay channel. Furthermore, the performance at short code lengths enjoys a relaying gain by approximately 1.4 dB.

While it is known that spatially-coupled low-density lattice codes (SC-LDLC) have better decoding performance than conventional (non-coupled) LDLC lattices, in this paper it is shown that their encoding complexity is also lower. Since nonzero elements are mainly in lower triangular entries of the sparse inverse generator matrix of SC-LDLC, iterative encoding with the Gauss-Seidel method performs well. The convergence speed of iterative encoding is evaluated by both the mean square error (MSE) and the symbol error rate between a given integer vector b and the inversely generated integer vector from the codeword of b. Numerical experiments show that the convergence of encoding for SC-LDLC is 3 times faster than that of the conventional LDLC, at an MSE of 10-10for dimension n = 10000.

Spatially-coupled low-density parity-check codes attract much attention due to their capacity-achieving performance and a memory-efficient sliding-window decoding algorithm. On the other hand, the encoder needs to solve large linear equations to terminate the encoding process. In this paper, we propose modified spatially-coupled codes. The modified (dl, dr, L) codes have less rate-loss, i.e., higher coding rate, and have the same threshold as (dl, dr, L) codes and are efficiently terminable by using an accumulator.

We study fountain codes transmitted over the binary-input symmetric-output channel. For channels with small capacity, receivers in fountain coding systems needs to collects many channel outputs to recover information bits. Since a collected channel output yields a check node in the decoding Tanner graph, the channel with small capacity leads to large decoding complexity. In this paper, we introduce a novel fountain coding scheme with non-binary LDPC codes. The decoding complexity of the proposed fountain code does not depend on the channel. Numerical experiments show that the proposed codes exhibit better performance than conventional fountain codes, especially for moderate number of information bits.

In this paper, we consider quantum error correction over depolarizing channels with nonbinary low-density parity-check codes defined over Galois field of size 2p. The proposed quantum error correcting codes are based on the binary quasi-cyclic Calderbank, Shor, and Steane (CSS) codes. The resulting quantum codes outperform the best known quantum codes and surpass the performance limit of the bounded distance decoder. By increasing the size of the underlying Galois field, i.e., 2p, the error floors are considerably improved.

The scaling law developed by Amraoui et al. is a powerful technique to estimate the block erasure probabilities of finite- length low-density parity-check (LDPC) codes. Solving a system of differential equations called covariance evolution, one can obtain the scaling parameter. However, the covariance evolution has not been analytically solved. In this paper, we present the analytical solution of the covariance evolution for irregular LDPC code ensembles.

For decoding non-binary low-density parity-check (LDPC) codes, logarithm-domain sum-product (Log-SP) algorithms were proposed for reducing quantization effects of SP algorithm in conjunction with FFT. Since FFT is not applicable in the logarithm domain, the computations required at check nodes in the Log-SP algorithms are computationally intensive. What is worse, check nodes usually have higher degree than variable nodes. As a result, most of the time for decoding is used for check node computations, which leads to a bottleneck effect. In this paper, we propose a Log-SP algorithm in the Fourier domain. With this algorithm, the role of variable nodes and check nodes are switched. The intensive computations are spread over lower-degree variable nodes, which can be efficiently calculated in parallel. Furthermore, we develop a fast calculation method for the estimated bits and syndromes in the Fourier domain.

In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We clarify a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the BP decoder. We expurgate non-binary LDPC code ensemble to analyze and to lower the error floor by using the above condition. Finally, we show upper and lower bounds of the error floors of the expurgated LDPC code ensemble over the MBIOS channel.

For designing low-density parity-check (LDPC) codes for quantum error-correction, we desire to satisfy the conflicting requirements below simultaneously. 1) The row weights of parity-check “should be large”: The minimum distances are bounded above by the minimum row weights of parity-check matrices of constituent classical codes. Small minimum distance tends to result in poor decoding performance at the error-floor region. 2) The row weights of parity-check matrices “should not be large”: The performance of the sum-product decoding algorithm at the water-fall region is degraded as the row weight increases. Recently, Kudekar et al. showed spatially-coupled (SC) LDPC codes exhibit capacity-achieving performance for classical channels. SC LDPC codes have both large row weight and capacity-achieving error-floor and water-fall performance. In this paper, we propose a new class of quantum LDPC codes based on spatially coupled quasi-cyclic LDPC codes. The performance outperforms that of quantum “non-coupled” quasi-cyclic LDPC codes.

In this paper, we propose a construction method for two-level quantum error-correcting codes via non-binary LDPC codes over an extended field of order 2p, p an integer p >; 1. The proposed quantum error-correcting codes are based on binary quasi-cyclic LDPC codes which have almost achieved a “Bounded Distance Decoding (BDD)” limit but have not surpassed the limit yet. Quantum codes constructed from the proposed method surpass the BDD limit. Furthermore the codes outperform the efficiently-decodable state-of-the-art quantum codes.

Kudekar et al. recently proved that for transmission over the binary erasure channel (BEC), spatial coupling of LDPC codes increases the BP threshold of the coupled ensemble to the MAP threshold of the underlying LDPC codes. One major drawback of the capacity-achieving spatially coupled LDPC codes is that one needs to increase the column and row weight of parity-check matrices of the underlying LDPC codes.

We consider spatially coupled code ensembles. A particular instance are convolutional LDPC ensembles. It was recently shown that, for transmission over the memoryless binary erasure channel, this coupling increases the belief propagation threshold of the ensemble to the maximum a-posteriori threshold of the underlying component ensemble. This paved the way for a new class of capacity achieving low-density parity check codes. It was also shown empirically that the same threshold saturation occurs when we consider transmission over general binary input memoryless channels. In this work, we report on empirical evidence which suggests that the same phenomenon also occurs when transmission takes place over a class of channels with memory. This is confirmed both by simulations as well as by computing EXIT curves.

We consider spatially coupled code ensembles over a multiple access channel. Convolutional LDPC ensembles are one instance of spatially coupled codes. It was shown recently that, for transmission over the binary erasure channel, this coupling of individual code ensembles has the effect of increasing the belief propagation threshold of the coupled ensembles to the maximum a-posteriori threshold of the underlying ensemble. In this sense, spatially coupled codes were shown to be capacity achieving. It was observed, empirically, that these codes are universal in the sense that they achieve performance close to the Shannon threshold for any general binary-input memoryless symmetric channels. In this work we provide further evidence of the threshold saturation phenomena when transmitting over a class of multiple access channel. We show, by density evolution analysis and EXIT curves, that the belief propagation threshold of the coupled ensembles is very close to the ultimate Shannon limit.

The fixed points of the belief propagation decoder for non-binary low-density parity-check (LDPC) codes are referred to as stopping constellations. In this paper, we give the stopping constellation distributions for the irregular non-binary LDPC code ensembles defined over the general linear group. Moreover, we derive the exponential growth rate of the average number of the stopping constellation distributions in the limit of large code length.

We consider spatially-coupled LDPC codes for the three terminal erasure relay channel. It is observed that BP threshold value of spatially-coupled LDPC codes, in particular spatially-coupled MacKay-Neal code, is close to the theoretical limit for the relay channel. Empirical results suggest that spatially-coupled LDPC codes have great potential to achieve theoretical limit of a general relay channel.

We propose nonbinary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular nonbinary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,.... Surprisingly, such simple low-rate nonbinary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.

Rate-compatible asymmetric Slepian-Wolf coding with non-binary LDPC codes of moderate code length is presented.The proposed encoder and decoder use only one single mother code.With the proposed scheme, better compressed rate and lower error rate than those ofconventional scheme are achieved with even smaller source length.

This paper provides a performance analysis of the regular (c, d) LDPC code ensemble of codelength n with parity-check matrices defined over the general linear group GL(2m). The transmission is assumed to take place over the binary erasure channel with erasure probability ε. In this work, the scaling approximation of the block erasure rate is generalized to the non-binary case, and the scaling parameter α of the approximation is derived. The proposed estimation is then compared with numerical results, showing that it predicts well the slope of the block erasure rate vs. channel erasure probability.

We propose non-binary LDPC codes concatenated with multiplicative repetition codes. To the best of the authors' knowledge, for the transmissions over the memoryless binary-input output-symmetric channels, 2m-ary the (2,dc)-regular LDPC code for m ~ 8 and dc≥ 3 is the best code so far among codes with moderate code length. By multiplicatively repeating the 2m-ary (2,3)-regular LDPC code of rate 1/3, we construct rate-compatible codes of lower rates 1/6,1/9,1/12,.... Surprisingly, such simple low-rate codes outperform the best low-rate binary codes so far.

In this paper, we analyze (2, k)-regular non-binary low-density parity-check codes over the binary erasure channels. We propose a method to improve the error floors by optimizing labels in zigzag cycles in the Tanner graph. We analyze the error floors for codes designed by the proposed optimization method and show that the error floors are decreasing in the size of Galois field.

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.

Communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) codes and belief propagation (BP) decoding is considered. The average bit error probability of an irregular LDPC code ensemble after a fixed number of iterations converges to a limit, which is calculated via density evolution, as the blocklength n tends to infinity. The difference between the bit error probability with blocklength n and the large-blocklength limit behaves asymptotically like ¿/n, where the coefficient ¿ depends on the ensemble, the number of iterations and the erasure probability of the BEC. In, ¿ is calculated for regular ensembles. In this paper, ¿ for irregular expurgated ensembles is derived. It is demonstrated that convergence of numerical estimates of ¿ to the analytic result is significantly fast for irregular unexpurgated ensembles.

For a (lambda(x); rho(x)) standard irregular LDPC code ensemble, the growth rate of the average weight distribution for small relative weight omega is given by log(lambda'(0)rho'(1))omega + O(omega2) in the limit of code length n. If lambda'(0)rho'(1) < 1, there exist exponentially few code words of small linear weight, as n tends to infinity. It is known that the condition coincides with the stability condition of density evolution over the erasure channels with the erasure probability 1. In this paper, we show that this is also the case with multi-edge type LDPC (MET-LDPC) codes. MET-LDPC codes are generalized structured LDPC codes introduced by Richardson and Urbanke. The parameter corresponding lambda'(0)rho'(1) appearing in the conditions for MET-LDPC codes is given by the spectral radius of the matrix defined by extended degree distributions.

The covariance evolution is a system of differential equations with respect to the covariance of the number of edges connecting to the nodes of each residual degree. Solving the covariance evolution, we can derive distributions of the number of check nodes of residual degree 1, which helps us to estimate the block error probability for finite-length LDPC code. Amraoui et al. resorted to numerical computations to solve the covariance evolution. In this paper, we give the analytical solution of the covariance evolution.

We consider communication over the binary erasure channel (BEC) using low-density parity-check (LDPC) code and belief propagation (BP) decoding. Furthermore, a gap between the bit error probability after finite number of iterations for finite block length n and that for infinite block length is asymptotically α/n, where α denotes a speci..c constant determined by a degree distribution, a number of iterations and erasure probability. Our main result is to derive an ef..cient algorithm for calculating α for regular ensembles.

Multi-Edge type LDPC codes are introduced by Richardson and Urbanke, and they show examples of their ensembles has better performance than other known ensembles. Orlitsky et al. derived the condition for irregular LDPC code ensembles with minimum distance linearly increasing in code length. We derive the condition corresponding to Orlitsky’s condition for two-edge type LDPC code ensembles which is simple example of Multi-Edge type LDPC code ensembles.

Luby et al. derived evolution of degree distributions in residual graphs for irregular LDPC code ensembles. Evolution of degree distributions in residual graphs is an important characteristic which is used for finite-length analysis of the expected block and bit error probabilities over the binary erasure channel. In this paper, we derive detailed evolution of degree distributions in residual graphs for irregular LDPC code ensembles with joint degree distributions.

Upper bounds of minimum distance distributions of Gallger codes and irregular LDPC codes were derived by Callage and Di, respectively. Di's bounds are tight for irregular LDPC codes which have variable nodes of degree two, however, it is not tight for irregular LDPC codes which do not. In this paper, we derive tight lower and upper bounds of minimum distance distributions of irregular LDPC code ensembles without variable nodes of degree two.

The support weight distribution of a code is the number of unique subspaces of the code with specified dimension and support weight. In this paper, we formulate the average second support weight distribution and its asymptotic exponent of regular LDPC code ensembles

In this paper, we have derived the upper bound of the average block error probability of a given detailedly represented irregular low-density parity-check (LDPC) code ensemble under maximum likelihood decoding.

In this paper, we explicitly formulate the average stopping set distributions and their asymptotic exponents of two instances of two-edge type LDPC code ensembles. Further we investigate the relation between the asymptotic exponents of those two code ensembles

This work presents an ensemble of irregular low-density parity check (LDPC) codes based on an ensemble of bipartite graphs, which formulated the average weight distribution. An analysis of the error performance of LDPC codes over a binary erasure channel together with an iterative decoding algorithm based on belief propagation has been clarified that a notion of stopping sets and their distribution with the asymptotic expression are derived. An irregular LDPC code ensemble that exhibits better performance in the sense of threshold is obtained by density evolution and explicitly formulates weight and stopping set distributions are defined.