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In this paper, we define a window code to be the portion of a Spatially-coupled low-density parity check (SC-LDPC) code seen by a single iteration of a windowed decoder. We consider the design of SC-LDPC codes for windowed decoding via optimization of the window code. In particular, because iterative decoding is optimal on codes with cycle-free graph representations, we ask fundamental questions about the construction and parameters of cycle-free window codes. We show that it is possible to have an SC-LDPC code with cycles and with cycle-free window codes. We consider the relationship between the distance of the window code and the distance of the SC-LDPC code. Further, we show that SC-LDPC codes with MDS window codes exist, and all such codes are asymptotically bad. This work gives insight into the tradeoffs between window code parameters and performance of the SC-LDPC code. [ABSTRACT FROM AUTHOR]

Among numerous dynamic scheduling strategies for low‐density parity‐check codes, many of them are of high complexity due to repetitive computation and ordering of belief residuals. To address these challenges, two novel approaches are proposed: the derived informed variable‐to‐check residual belief propagation and the derived variable‐node and variable‐to‐check‐edge residual belief propagation for low‐density parity‐check codes. Simulation results show that the derived dynamic scheduling exhibits a substantial gain of approximately 0.15 dB in signal‐to‐noise ratio over the non‐derived one with considerably lower decoding complexity when targeting a bit‐error rate of 10−4. [ABSTRACT FROM AUTHOR]

The residual belief propagation (RBP) algorithm, which is the most classic informed dynamic scheduling strategy, achieves outstanding performance in error correction and can drastically accelerate convergence speed. However, the greedy algorithmic property of this iterative decoding will inevitably cause loss of decoding performance. To address this, a novel algorithm called the partial average bundle residual belief propagation (PABRBP) is proposed in this paper. According to the construction characteristics of a base matrix of protograph‐LDPC codes, informed dynamic scheduling (IDS) strategies are applied to an edge bundle of base matrices for the first time. This edge bundle of the base matrix can be applied to a corresponding cyclic permutation matrix. Furthermore, the update level of each bundle is determined by the value of the Partially Average Bundle Residual (PABR). Therefore, the edge message with the maximum residual in each bundle is updated in order, and the process of iterative decoding is less likely to become trapped in a local optimum. Additionally, the generation of silent nodes is reduced as much as possible. To further improve the PABRBP decoding performance for medium and long codes over the fading channel, the adjusted compensation term is periodically introduced. Analysis and simulation results show that PABRBP demonstrates a notable convergence quality and decoding performance improvement over the fading channels compared to existing state‐of‐art IDS algorithms. [ABSTRACT FROM AUTHOR]

In this article, a graph-theoretic method (taking advantage of constraints among sets associated with the corresponding parity-check matrices) is applied for the construction of a double low-density parity-check (D-LDPC) code (also known as LDPC code pair) in a joint source-channel coding (JSCC) system. Specifically, we pre-set the girth of the parity-check matrix for the LDPC code pair when jointly designing the two LDPC codes, which are constructed by following the set constraints. The constructed parity-check matrices for channel codes comprise an identity submatrix and an additional submatrix, whose column weights can be pre-set to be any positive integer numbers. Simulation results illustrate that the constructed D-LDPC codes exhibit significant performance improvement and enhanced flexible frame length (i.e., adaptability under various channel conditions) compared with the benchmark code pair. [ABSTRACT FROM AUTHOR]

It is well known that some harmful objects in the Tanner graph of low-density parity-check (LDPC) codes have a negative impact on their error correction performance under iterative message-passing decoding. Depending on the channel and the decoding algorithm, these harmful objects are different in nature and can be stopping sets, trapping sets, absorbing sets, or pseudocodewords. Differently from LDPC block codes, the design of spatially coupled LDPC codes must take into account the semi-infinite nature of the code, while still reducing the number of harmful objects as much as possible. We propose a general procedure, based on edge spreading, enabling the design of good quasi-cyclic spatially coupled LDPC (QC-SC-LDPC) codes. These codes are derived from quasi-cyclic LDPC (QC-LDPC) block codes and contain a considerably reduced number of harmful objects with respect to the original QC-LDPC block codes. We use an efficient way of enumerating harmful objects in QC-SC-LDPCCs to obtain a fast algorithm that spans the search space of potential candidates to select those minimizing the multiplicity of the target harmful objects. We validate the effectiveness of our method via numerical simulations, showing that the newly designed codes achieve better error rate performance than codes presented in previous literature. [ABSTRACT FROM AUTHOR]

We propose a distributed quasi-cyclic low-density parity-check (QC-LDPC) coded spatial modulation (D-QC-LDPCC-SM) scheme with source, relay and destination nodes. At the source and relay, two distinct QC-LDPC codes are used. The relay chooses partial source information bits for further encoding, and a distributed code corresponding to each selection is generated at the destination. To construct the best code, the optimal information bit selection algorithm by exhaustive search in the relay is proposed. However, the exhaustive-based search algorithm has large complexity for QC-LDPC codes with long block length. Then, we develop another low-complexity information bit selection algorithm by partial search. Moreover, the iterative decoding algorithm based on the three-layer Tanner graph is proposed at the destination to carry out joint decoding for the received signal. The recently developed polar-coded cooperative SM (PCC-SM) scheme does not adopt a better encoding method at the relay, which motivates us to compare it with the proposed D-QC-LDPCC-SM scheme. Simulations exhibit that the proposed exhaustive-based and partial-based search algorithms outperform the random selection approach by 1 and 1.2 dB, respectively. Because the proposed D-QC-LDPCC-SM system uses the optimized algorithm to select the information bits for further encoding, it outperforms the PCC-SM scheme by 3.1 dB. [ABSTRACT FROM AUTHOR]

A key component of the joint source-channel coding (JSCC) scheme based on double low-density parity-check (D-LDPC) codes is the introduction of a linking matrix between the source LDPC code and channel LDPC code, by which the decoding information including the source redundancy and channel state information can be transferred iteratively. However, the linking matrix is a fixed one-to-one mapping, i.e., an identity matrix in a conventional D-LDPC code system, which may not take full advantage of the decoding information. Therefore, this paper introduces a general linking matrix, i.e., a non-identity linking matrix, connecting the check nodes (CNs) of the source LDPC code and the variable nodes (VNs) of the channel LDPC code. Further, the encoding and decoding algorithms of the proposed D-LDPC coding system are generalized. A joint extrinsic information transfer (JEXIT) algorithm is derived for calculating the decoding threshold of the proposed system with a general linking matrix. In addition, several general linking matrices are optimized with the aid of the JEXIT algorithm. Finally, the simulation results demonstrate the superiority of the proposed D-LDPC coding system with general linking matrices. [ABSTRACT FROM AUTHOR]

In this paper, a joint group shuffled scheduling decoding (JGSSD) algorithm for a joint source-channel coding (JSCC) scheme based on double low-density parity-check (D-LDPC) codes is presented. The proposed algorithm considers the D-LDPC coding structure as a whole and applies shuffled scheduling to each group; the grouping relies on the types or the length of the variable nodes (VNs). By comparison, the conventional shuffled scheduling decoding algorithm can be regarded as a special case of this proposed algorithm. A novel joint extrinsic information transfer (JEXIT) algorithm for the D-LDPC codes system with the JGSSD algorithm is proposed, by which the source and channel decoding are calculated with different grouping strategies to analyze the effects of the grouping strategy. Simulation results and comparisons verify the superiority of the JGSSD algorithm, which can adaptively trade off the decoding performance, complexity and latency. [ABSTRACT FROM AUTHOR]

Recently, a mesh model-based merging (M 3 ) method and four basic graph models were proposed to construct the double protograph low-density parity-check (P-LDPC) code pair of the joint source channel coding (JSCC). Designing the protograph (mother code) of the P-LDPC code with both a good waterfall region and lower error floor is a challenge, and few works have existed until now. In this paper, the single P-LDPC code is improved to further verify the availability of the M 3 method, and its structure is different from the channel code in the JSCC. This construction technique yields a family of new channel codes with lower power consumption and higher reliability. The structured design and better performance demonstrate that the proposed code is hardware-friendly. [ABSTRACT FROM AUTHOR]

This study presents two reconfigurable turbo/low-density parity-check (LDPC) decoding kernels for the two powerline communication standards, HomePlug and G.hn. Two architectures are presented, both of which use the radix-4 double-binary enhanced max-log maximum a posteriori probability algorithm with next-iteration initialization in turbo decoding. In LDPC decoding, the two architectures employ the normalized min-sum and the layered radix-4 forward and backward algorithms. The two algorithms cause differences in the architecture and throughput rate. Consequently, the proposed decoding kernels have different architectures when combined with the turbo decoding algorithm, and the two proposed decoding kernels each have their own advantages and disadvantages in terms of throughput and area cost. To make the features of two kernels more evident, we have implemented the proposed decoding kernels that lead to significant throughput gains and better area efficiency compared with other studies. The proposed decoding kernels can be operated in all modes specified in the HomePlug and G.hn standards using a 40-nm complementary metal-oxide-semiconductor (CMOS) process. Moreover, the proposed decoding kernels provide different solutions to achieve the expected throughput rates of the G.hn and HomePlug standards. [ABSTRACT FROM AUTHOR]

In this paper, we focus on decoding nonbinary low-density parity-check (LDPC) codes in Galois fields of characteristic two via the proximal alternating direction method of multipliers (proximal-ADMM). By exploiting Flanagan/Constant-Weighting embedding techniques and the decomposition technique based on three-variables parity-check equations, two efficient proximal-ADMM decoders for nonbinary LDPC codes are proposed. We show that both of them are theoretically guaranteed convergent to some stationary point of the decoding model and either of their computational complexities in each proximal-ADMM iteration scales linearly with LDPC code’s length and the size of the considered Galois field. Moreover, the decoder based on the Constant-Weight embedding technique satisfies the favorable property of codeword symmetry. Simulation results demonstrate their effectiveness in comparison with state-of-the-art LDPC decoders. [ABSTRACT FROM AUTHOR]

In this paper, we propose a modified adaptive belief propagation (ABP) algorithm, which is referred to as the random ABP (R-ABP) algorithm, for decoding short channel codes adopted in the fifth generation (5G) New Radio (NR) wireless systems. Based on the cyclic redundancy check (CRC) concatenation structure of 5G channel codes, we can take partial or even all CRC bits into iterative R-ABP decoding to improve the error correction performance, while the error detection capability can still be guaranteed by the remaining CRC bits and a proposed threshold-based acceptance criterion. We call this improved R-ABP decoding the threshold-and-CRC-aided R-ABP (TCA-R-ABP) decoding. The simulation results show that our proposed TCA-R-ABP algorithm outperforms the state-of-the-art decoding algorithms for many 5G low-density parity-check (LDPC) and polar codes. Moreover, the proposed TCA-R-ABP algorithm leads to a unified decoder for LDPC and polar codes, which has the potential to be a lower-complexity decoder over the combination of the belief propagation (BP) and CRC-aided successive cancellation list (CA-SCL) decoders. [ABSTRACT FROM AUTHOR]

Most synchronization error correction codes deal with random independent insertion and deletion errors without correlation. In this paper, we propose a probabilistic channel model with correlated insertion and deletion (CID) errors to capture the data dependence applicable to the bit-patterned media recording (BPMR) system. We also investigate the error performance and decoding complexity of a concatenated LDPC-marker code over the CID channel. Furthermore, we modify the forward backward decoding algorithm to make it suitable for the CID channel, and elaborate it based on a two-dimensional state transition diagram. Compared with the conventional marker coding scheme dealing with random errors, the concatenated LDPC-marker code takes into account the dependence between synchronization errors, improves the error performance, and reduces the decoding complexity. The BER performance of the concatenated LDPC-marker code is improved by more than 50% on average, and the decoding time is reduced by nearly 35% when the LDPC code (n = 4521, k = 3552) and the marker code (Nm = 2, Nc = 30) are used over the CID channel. [ABSTRACT FROM AUTHOR]

In the low-error rate regime, message-passing (MP) decoding for low-density parity-check (LDPC) codes is known to have performance degradation due to trapping sets (TSs), which often limits the use of LDPC codes for applications with low target error rates like storage devices. This work proposes a novel deep-learning based decoding algorithm which is tailored for breaking TSs. In particular, when MP decoding fails due to TSs, there exist pairs of unsatisfied check nodes (CNs) which are connected through paths only with error variable nodes (VNs), i.e., VNs with erroneous hard-decision results. The proposed algorithm efficiently identifies the paths with error VNs between unsatisfied CNs with the aid of deep-learning techniques. Then, the decoding failures are resolved by repeating the MP decoding after re-initializing the channel outputs for the error VNs in the identified paths. In addition, by analyzing the behaviors of the deep-learning based algorithm, we propose a low-complexity algorithm, called adaptive-error-path (AEP) detector. Simulation results show that the proposed algorithms efficiently break the TSs and significantly improve the error-floor performance in the low error-rate regime. [ABSTRACT FROM AUTHOR]

Designs and methods for nested lattice codes using Construction D’ lattices for coding and convolutional code lattices for shaping are described. Two encoding methods and a decoding algorithm for Construction D’ coding lattices that can be used with shaping lattices for power-constrained channels are given. We construct nested lattice codes with good coding properties, high shaping gain, and low-complexity encoding and decoding. Convolutional code generator polynomials for Construction A lattices with the greatest shaping gain are given, as a result of an extensive search. It is shown that rate 1/3 convolutional codes provide a more favorable performance-complexity trade-off than rate 1/2 convolutional codes. Tail-biting convolutional codes have higher shaping gain than that of zero-tailed convolutional codes. A design for quasi-cyclic low-density parity-check (QC-LDPC) codes to form Construction D’ lattices which have efficient encoding and indexing is presented. The resulting QC-LDPC Construction D’ lattices are evaluated using four shaping lattices: the $E_{8}$ lattice, the $BW_{16}$ lattice, the Leech lattice and our best-found convolutional code lattice, showing a shaping gain of approximately 0.65 dB, 0.86 dB, 1.03 dB and 1.25 dB at dimension 2304. [ABSTRACT FROM AUTHOR]

The informed dynamic scheduling (IDS) strategies, in which the edge message with the maximum message‐residual is updated preferentially, achieve remarkable error‐correction performance when applied to low‐density parity‐check (LDPC) codes. However, the IDS strategies incur inferior convergence in iterative decoding owing to the greedy problem, which is called the update‐relayed trend here. In order to solve the greediness, two locally informed dynamic scheduling algorithms based on the law of large numbers are proposed. The proposed decoding algorithms use random select of check nodes over a predefined update range (RSPUR) which effectively suppresses the propagation of the update‐relayed trend and accordingly restrains the forming of multi‐update cycles. Moreover, the decoding algorithm is further improved based on random select of check nodes over an adjustable update range (RSCAR). The update ranges are selected based on the law of large numbers. Therefore, the computational resources can be allocated more equitably by increasing iterations. Simulation results show that both the proposed algorithms achieve excellent performance in terms of throughput and convergence with low decoding complexity over the Additive White Gaussian Noise (AWGN) and the fading channels compared to the previous IDS strategies. Hence, the proposed algorithms behave excellently over the wireless channels. [ABSTRACT FROM AUTHOR]

Serial decoding algorithms of low-density parity-check (LDPC) code converge efficiently with low errors. Previously, a serial decoding algorithm, named a shuffled belief-propagation (SBP), was applied in turbo equalization of bit-patterned magnetic recording (BPMR) systems. With the SBP algorithm, an LDPC decoder converged twice as fast as one using conventional BP algorithms. We further improved the convergence speed of SBP by updating the messages in an adaptive order, which played a flexible role throughout decoding. We proposed two adaptive-serial algorithms for LDPC codes in turbo equalization. One updated the messages using the extrinsic loglikelihood ratio (LLR) and the result of the parity-check equation checking. The second contained an additional rule that tracked the LLR sign changes in each iteration. Both algorithms converged faster and with lower bit error rates (BERs) than the SBP and previous adaptive-serial algorithms in a BPMR system with media noise. [ABSTRACT FROM AUTHOR]

This letter focuses on improving the error correction performance of the extended min‐sum (EMS) decoding algorithm. Introducing factor/offset modifications in the check node update process of the EMS algorithm can improve the performance. Based on this, the authors propose a novel threshold‐based modified EMS algorithm that introduces the factor/offset in a targeted manner by setting a threshold for the posterior probability. Simulation results show that an additional performance gain of the order of 0.08 dB is obtained with the proposed method without significantly increasing the complexity of the method. [ABSTRACT FROM AUTHOR]

We study the stability of low-density parity-check codes under blockwise or bitwise maximum a posteriori decoding, where transmission takes place over a binary-input memoryless output-symmetric channel. Our study stems from the consideration of constructing universal capacity-achieving codes under low-complexity decoding algorithms, where universality refers to the fact that we are considering a family of channels with equal capacity. Consider, e.g., the right-regular sequence by Shokrollahi and the heavy-tail Poisson sequence by Luby et al. Both sequences are provably capacity-achieving under belief propagation decoding when transmission takes place over the binary erasure channel. In this paper we show that many existing capacity-achieving sequences of low-density parity-check codes are not universal under belief propagation decoding. We reveal that the key to showing this non-universality result is determined by the stability of the underlying codes. More concretely, for an ordered and complete channel family and a sequence of low-density parity-check code ensembles, we determine a stability threshold associated with them, which gives rise to a sufficient condition under which the sequence is not universal under belief propagation decoding. Moreover, we show that the same stability threshold applies to blockwise or bitwise maximum a posteriori decoding as well. We demonstrate how the stability threshold can determine an upper bound on the corresponding blockwise or bitwise maximum a posteriori threshold, revealing the operational significance of the stability threshold. [ABSTRACT FROM AUTHOR]

A new approach for designing bilayer and multi-layer LDPC codes is proposed and studied in the asymptotic regime. The ensembles are defined through individual uni-variate degree distributions, one for each layer. We present a construction that: 1) enables low-complexity decoding for high-SNR channel instances, 2) provably approaches capacity for low-SNR instances, 3) scales linearly (in terms of design complexity) in the number of layers. For the setup where decoding the second layer is significantly more costly than the first layer, we propose an optimal-cost decoding schedule and study the trade-off between code rate and decoding cost. [ABSTRACT FROM AUTHOR]

Layered decoding (LD) facilitates a partially parallel architecture for performing belief propagation (BP) algorithm for decoding low-density parity-check (LDPC) codes. Such a schedule for LDPC codes has, in general, reduced implementation complexity compared to a fully parallel architecture and higher convergence rate compared to both serial and parallel architectures, regardless of the codeword length or code-rate. In this paper, we introduce a modified shuffling method which shuffles the rows of the parity-check matrix (PCM) of a quasi-cyclic LDPC (QC-LDPC) code, yielding a PCM in which each layer can be produced by the circulation of its above layer one symbol to the right. The proposed shuffling scheme additionally guarantees the columns of a layer of the shuffled PCM to be either zero weight or single weight. This condition has a key role in further decreasing LD complexity. We show that due to these two properties, the number of occupied look-up tables (LUTs) on a field programmable gate array (FPGA) reduces by about 93% and consumed on-chip power by nearly 80%, while the bit error rate (BER) performance is maintained. The only drawback of the shuffling is the degradation of decoding throughput, which is negligible for low values of E b / N 0 until the BER of 1e−6. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new method to design low-density parity-check Hadamard (LDPC-Hadamard) codes — a type of ultimate-Shannon-limit approaching channel codes. The technique is based on applying Hadamard constraints to the check nodes in a generalized protograph-based LDPC code, followed by lifting the generalized protograph. We name the codes formed protograph-based LDPC Hadamard (PLDPC-Hadamard) codes. We also propose a modified Protograph Extrinsic Information Transfer (PEXIT) algorithm for analyzing and optimizing PLDPC-Hadamard code designs. The proposed algorithm further allows the analysis of PLDPC-Hadamard codes with degree- 1 and/or punctured nodes. We find codes with decoding thresholds ranging from −1.53 dB to −1.42 dB. At a BER of 10−5, the gaps of our codes to the ultimate-Shannon-limit range from 0.40 dB (for rate = 0.0494) to 0.16 dB (for rate = 0.003). Moreover, the error performance of our codes is comparable to that of the traditional LDPC-Hadamard codes. Finally, the BER performances of our codes after puncturing are simulated and compared. [ABSTRACT FROM AUTHOR]

Power line communication (PLC) technology is actually among the most renowned technologies for home environments due to their low-cost installation opportunities. In this study, the bit error rate (BER) performances of binary low-density parity check (LDPC) coded orthogonal frequencydivision multiplexing (OFDM) systems have been considered over indoor PLC channels. Performances comparison of diverse soft and hard decision LDPC decoder schemes such as Min-Sum (MS), weighted bit flipping (WBF), gradient descent bit-flip (GDBF), noisy gradient descent bit-flip (NGDBF) and its few variants including the single-bit NGDBF (S-NGDBF), multi-bit NGDBF (M-NGDBF) and smoothed-multi-bit NGDBF (SM-NGDBF) decoders were examined in the modeled network. To evaluate the BER performance analyses three different PLC channel scenarios were generated by using new and more realistic PLC channel model proposal were also employed. All of the simulations performed in Canete's PLC channel model showed that remarkable performance improvement can be achieved by using short-length LDPC codes. Especially, the improvements are striking when the MS or SM-NGDBF decoding algorithms are employed on the receiver side. [ABSTRACT FROM AUTHOR]

Alternating direction method of multipliers (ADMM) is an efficient implementation of linear programming (LP) decoding for low-density parity-check (LDPC) codes. By adding penalty terms to the objective function of the LP decoding model, ADMM variable node (VN) penalized decoding can suppress the non-integral solutions and improve the frame error rate (FER) performance in the low signal-to-noise ratio (SNR) region. In this paper, we propose a novel ADMM check node (CN) penalized decoding algorithm. Codeword solutions which satisfy all parity-check equations will have smaller penalty values than non-codeword solutions, including the non-integral solutions. We discuss the required properties of CN-penalty functions, propose a few functions that satisfy those properties, and study their performance/complexity trade-offs. We also investigate the convergence properties of the proposed algorithm and prove that its performance is independent of the transmitted codeword. Using Monte Carlo simulations and instanton analysis, we then demonstrate that the proposed CN-penalized decoder outperforms ADMM VN penalized decoders in both waterfall and error floor regions. This comes at the expense of some increase in the decoding complexity. [ABSTRACT FROM AUTHOR]

With low-density parity-check (LDPC) codes and polar codes selected as the standard codes for the fifth generation (5 G) enhanced Mobile Broad Band scenario (eMBB), a decoding architecture that can simultaneously support the decoding of control and data plane becomes necessary at the terminal side to meet the raising requirement for 5 G network deployment. Due to the special structure of LDPC codes according to the Release 15 (R15) 5 G standard, a straight-forward extension of the existing reconfigurable scheme is in general difficult. Therefore in this paper, a unified decoding architecture is proposed that can be reconfigured to either LDPC codes or polar codes. Due to various differences between the two codes such as parity-check matrices, codeword lengths and iterative methods, a joint decoding algorithm is introduced by reordering basic decoding operations to the unified add-comparison-add pattern for both codes. Then, a pipelined structure of reconfigurable decoding unit (RDU) is presented correspondingly which is fully compatible with all decoding patterns of the R15 standard. And finally, a reconfigurable decoder is proposed with multiple levels of parallelism, and the reconfiguration scheme is introduced to improve the hardware utilization and decoding efficiency. The proposed decoder is implemented in FPGA and ASIC, respectively, and has achieved state-of-the-art performance in throughput and area efficiency compared to LDPC-only and polar-only decoders. [ABSTRACT FROM AUTHOR]

Summary: Information theory coding is an impressive and most celebrated field of research that has spawned numerous extremely important solutions to the intractable problems of secure data communications. Recent advancements in error control coding methods have seen a huge surge in using low‐density parity‐check (LDPC) code‐based decoding algorithms to solve imperative issues related to reliable data transmission and reception. Till date, extensive research efforts have been consistently being made on LDPC codes which focus on algorithm‐driven and hardware‐realization‐based approaches. The main intension of this research work is to provide an extensive systematic elucidation on the recent advancements in LDPC decoding algorithms. In addition, a thorough performance evaluation and analysis of several outstanding LDPC decoding techniques is presented. Finally, conclusions are drawn by summarizing the important research findings, interesting open problems, current challenges and broader perspectives for future directions of research. [ABSTRACT FROM AUTHOR]

Modern navigation satellite communication has the characteristic of high transmitting rate. To avoid bit errors in data transmission, low density parity check (LDPC) codes are widely recognized as efficient ways for navigation communication. Conventionally, the LDPC decoding is applied for additive white Gaussian noise (AWGN) channel and degrades severely while facing the impulsive noise. However, navigation communication often suffers from impulsive interference due to the occurrence of high amplitude "spikes". At this time, the conventional Gaussian noise assumption is inadequate. The impulsive component of interference has been found to be significant which influences the reliability of transmitted information. Therefore the LDPC decoding algorithms for AWGN channel are not suitable for impulsive noise environments. Consider that LDPC codes over GF(q) perform better than binary LDPC in resisting burst errors for current navigation system, it is necessary to conduct research on LDPC codes over GF(q). In this paper, an optimized initialization by calculating posterior probabilities of received symbols is proposed for non-binary LDPC decoding on additive white Class A noise (AWAN) channel. To verify the performance of the proposed initialization, extensive experiments are performed in terms of convergence, validity, and robustness. Preliminary results demonstrate that the decoding algorithm with the optimized initialization for non-binary LDPC codes performs better than the competing methods and that of binary LDPC codes on AWAN channel. [ABSTRACT FROM AUTHOR]

Area efficient and high speed forward error correcting codes decoder are the demand of many high speed next generation communication standards. This paper explores a low complexity decoding algorithm of low density parity check codes, called the min-sum iterative construction a posteriori probability (MS-IC-APP), for this purpose. We performed the error performance analysis of MS-IC-APP for a (648,1296) regular QC-LDPC code and proposed an area and throughput optimized hardware implementation of MS-IC-APP. We proposed to use the layered scheduling of MS-IC-APP and performed other optimizations at architecture level to reduce the area and to increase the throughput of the decoder. Synthesis results show 6.95 times less area and 4 times high throughput as compared to the standard min-sum decoder. The area and throughput are also comparable to the improved variants of hard-decision bit-flipping (BF) decoders, whereas, the simulation results show a coding gain of 2.5 over the best implementation of BF decoder in terms of error performance. [ABSTRACT FROM AUTHOR]

We propose a two-dimensional modified min-sum algorithm for the LDPC codes in the fifth generation (5G) networks standard to approach the error performance of the sum-product algorithm (SPA). In the proposed decoding algorithm, we adopt a partial self-correction method followed by message amplification to improve the reliability of the variable-to-check (V2C) messages. To further approach the performance of the maximum likelihood decoding for 5G short LDPC codes, we propose an enhanced quasi-maximum likelihood (EQML) decoding method. The proposed decoding method performs multiple rounds of decoding tests once the first decoding attempt fails, where the decoder inputs of the selected unreliable variable nodes are modified in each decoding test. A novel node selection method based on the sign fluctuation of V2C messages is proposed for the EQML decoding method. We also present a partial pruning stopping (PPS) rule to reduce the decoding complexity by deactivating part of the decoding tests once a valid codeword is found. A lower bound on the error performance is also derived by using the semi-analytical method. Simulation results show that the EQML decoding method outperforms the SPA with the same decoding complexity and other QML decoding methods, and it approaches the Polyanskiy-Poor-Verdú bound within 0.4 dB. [ABSTRACT FROM AUTHOR]

Racetrack memories (RMs) are prone to alignment faults called position errors (PEs), which manifest as insertions and deletions of stored data bits. Conventional coding schemes for PEs demonstrate the promising results by employing low-density parity-check (LDPC) codes with an iterative detection and decoding algorithm. However, the computational complexity of detection is relatively high. In this article, for channels with PEs, we present a new coding scheme that can effectively decode corrupted data, even if the detection is executed only once (i.e., a non-iterative detection and decoding scenario). The proposed code consists of a concatenation of an inner 2-D marker code, which is specialized for PEs to mitigate the effect of insertion and deletion (ID) errors, and an outer irregular LDPC code. We also provide tractable design methodologies for these constituent codes. First, we identify the 2-D-marker code structures that offer higher achievable information rates in a non-iterative scenario and then optimize irregular LDPC codes to ensure good decoding properties. Through asymptotic-performance analysis and finite-length simulation, we confirm the effectiveness of the proposed coding scheme. Ultimately, the proposed coding scheme has the capability to reduce code rate loss and to provide excellent decoding performance under a non-iterative scenario, which also helps in understanding the reliability of RM when a low-complexity decoding algorithm is used to correct ID errors caused by PEs. [ABSTRACT FROM AUTHOR]

Polar codes are known as the first capacity-achievable codes with low encoding and decoding complexity. The sequential decoding nature of traditional polar decoding algorithms such as successive cancellation (SC) results in high decoding latency, which is not suitable for services that require high reliability and low latency. Deep learning for decoding, referred to as neural network decoder (NND), has shown strong competitiveness because of its non-iterative and full-parallel feature. Whereas, the bit-error-rate (BER) performance of NND is still not satisfactory. In this paper, we first propose a residual learning denoiser (RLD) for polar codes. The RLD can remarkably improve the signal-to-noise ratio (SNR) and reduce the symbol-error-rate (SER) of received symbols. In order to decode polar codes more efficiently, we subsequently propose a residual neural network decoder (RNND) for polar codes. Different from the traditional pure-NND (PNND) which directly uses neural network for decoding received symbols, the proposed RNND concatenates a RLD used for denoising and a NND used for decoding. We provide a novel multi-task learning (MTL) strategy to jointly optimize the denoiser and decoder, and find that the denoising gain of Joint-RLD is more significant than the Independent-RLD. Numerical results show that the proposed RNND outperforms its counterpart PNND with regard to the BER performance. In addition, the optimal RNND(MLP-MLP) approaches the traditional SC decoding performance, while saving more than one hundred times of computation time. Eventually, scalability of the RNND to longer polar codes as well as LDPC codes further demonstrates the superiority of our proposed scheme. [ABSTRACT FROM AUTHOR]

Compute-and-forward (CAF) relaying is an effective way to increase bandwidth efficiency of wireless two-way relay channels. Design of error-correcting codes and their decoding algorithms suitable for CAF relaying schemes remains an important issue to be studied. In this paper, we will analyze an asymptotic behavior of LDPC codes over two-way relay channels based on density evolution (DE). Because of the asymmetric characteristics of the channel, we use the population dynamics DE combined with DE formulas for asymmetric channels to obtain the belief propagation (BP) thresholds. Additionally, we also evaluate the asymptotic performance of spatially coupled LDPC codes for two-way relay channels. The results indicate that the spatially coupled codes yield improvements in the BP threshold compared with corresponding uncoupled codes for two-way relay channels. [ABSTRACT FROM AUTHOR]

Low-Density Parity-Check (LDPC) codes are widely used from hard-disk systems to satellite communications. Sliding-Window Belief Propagation (SWBP) is an effective decoding algorithm of LDPC codes for time-varying channels and demonstrates near-optimal performance in many experiments. However, to adaptively find the best window size, SWBP may need very long computing time. Inspired by Graphics Processing Unit and Compute Unified Device Architecture, in this paper we propose a novel method to address the issue of SWBP's computing complexity. Different from sequential SWBP, we simultaneously compute the metrics of different window sizes in parallel, which enables us to quickly find the best window size. We use coalesced memory access to accelerate reading and writing processes. Registers and shared memory are also considered in our program to reduce memory latency. On the GTX 1080Ti platform, experimental results show that parallel SWBP can achieve about 14 × to 118 × speedup ratio for different regular LDPC codes, and about 8 × to 120 × speedup ratio for different irregular LDPC codes, respectively. According to the trend of our experiments, we strongly believe that, as the length of LDPC codes increases, a higher speedup ratio can be obtained. [ABSTRACT FROM AUTHOR]

Non-binary low-density parity-check (NB-LDPC) codes are known to offer several advantages over their binary counterparts, but the higher complexity, and the resource-hungry nature of decoding algorithms have so far restricted their practical usage. In this paper, we propose a new decoding algorithm for NB-LDPC codes over finite fields of characteristic 2, based on a novel binary expansion of the ${Q}$ -ary Tanner graph. While it offers substantial complexity gains, simulation results demonstrate that the performance loss of the new algorithm, in comparison to the best known decoder, is quite small. Furthermore, due to being based on a binary graph, it is particularly attractive for hardware implementations. We also suggest a simplified version of the algorithm, which offers even higher gains in complexity. [ABSTRACT FROM AUTHOR]

Low density parity check (LDPC) code is a kind of linear block code with excellent performance which is proposed by Dr. Gallager. It overcomes the shortcomings of other error correcting codes and it is the nearest channel code from Shannon limit at the present discover. The decoding algorithms of LDPC code can reduce the decoding delay by the parallel decoding structure, so this has caused great concern in many fields. In recent years more and more scholars begin to study LDPC code, this makes the coding and decoding algorithms of LDPC code improved continuously. LDPC code has been widely used in many fields such as optical fiber communication, digital watermarking, underwater communication and deep space communication because of its excellent performance. Non-binary LDPC code has better performance compared with binary LDPC code. In this paper, FFT-BP decoding algorithm of non-binary LDPC code is studied, and on this basis, we proposed an improved FFT-BP algorithm. The improved decoding algorithm can effectively reduce the average iteration number. Finally, we achieved encode and decode algorithm of non-binary LDPC code with FPGA. [ABSTRACT FROM AUTHOR]

We introduce a new family of rank metric codes: Low Rank Parity Check codes (LRPC), for which we propose an efficient probabilistic decoding algorithm. This family of codes can be seen as the equivalent of classical LDPC codes for the rank metric. We then use these codes to design cryptosystems à la McEliece: more precisely we propose two schemes for key encapsulation mechanism (KEM) and public key encryption (PKE). Unlike rank metric codes used in previous encryption algorithms -notably Gabidulin codes - LRPC codes have a very weak algebraic structure. Our cryptosystems can be seen as an equivalent of the NTRU cryptosystem (and also to the more recent MDPC code-based cryptosystem) in a rank metric context, due to the similar form of the public keys. The present paper is an extended version of the article introducing LRPC codes, with important new contributions. We have improved the decoder thanks to a new approach which allows for decoding of errors of higher rank weight, namely up to $\frac {2}{3}(n-k)$ when the previous decoding algorithm only decodes up to $\frac {n-k}{2}$ errors. Our codes therefore outperform the classical Gabidulin code decoder which deals with weights up to $\frac {n-k}{2}$. This comes at the expense of probabilistic decoding, but the decoding error probability can be made arbitrarily small. The new approach can also be used to decrease the decoding error probability of previous schemes, which is especially useful for cryptography. Finally, we introduce ideal rank codes, which generalize double-circulant rank codes and allow us to avoid known structural attacks based on folding. To conclude, we propose different parameter sizes for our schemes and we obtain a public key of 3337 bits for key exchange and 5893 bits for public key encryption, both for 128 bits of security. [ABSTRACT FROM AUTHOR]

In recent times, many advanced wireless communication systems have adopted channel coding schemes to ease secure transmission and reception of wireless data over noisy perturbed channel conditions. Channel coding approaches using low-density parity-check (LDPC) codes are most interesting and fastest growing research areas in the domain of wireless communications. Due to its widespread popularity, adaptability and parallelism for cost-effective hardware implementations, LDPC codes are widely endorsed in a number of wireless communication standards. Over the years, many low complex decoding algorithms using LDPC codes were introduced to improve the data reliability of many wireless applications. This work introduces an efficient and robust offset min-sum decoding scheme for optimal decoding of LDPC codes. This improved approach introduces a new offset correction factor to suppress the error propagation during the approximation of high precision soft values within the given range boundary of signal strength-to-background noise ratio (SNR). The experimental results illustrate the competitive advantage of the proposed algorithm over several popular algorithms in terms of error rate performance, complexity reductions and convergence speed. [ABSTRACT FROM AUTHOR]

In a digital communication system, information is sent from one place to another over a noisy communication channel. It may be possible to detect and correct errors that occur during the transmission if one encodes the original information by adding redundant bits. Low–Density Parity–Check (LDPC) convolutional codes, a member of the LDPC code family, encode the original information to improve error correction capability. In practice these codes are used to decode very long information sequences, where the information arrives in subsequent packets over time, such as video streams. We consider the problem of decoding the received information with minimum error from an optimization point of view and investigate integer programming–based exact and heuristic decoding algorithms for its solution. In particular, we consider relax–and–fix heuristics that decode information in small windows. Computational results indicate that our approaches identify near–optimal solutions significantly faster than a commercial solver in high channel error rates. Our proposed algorithms can find higher quality solutions compared with the state of the art iterative decoding heuristic. [ABSTRACT FROM AUTHOR]

In this paper, the definition of threshold of elementary absorbing sets is extended to scaled Min-Sum low-density parity-check (LDPC) decoding. Based on the analysis of the behavior of scaled Min-Sum LDPC decoders in the Tanner graph of the absorbing set, it is proven that correct decoding is guaranteed with received channel messages above threshold. A fast algorithm for the threshold computation is derived. Many examples of absorbing sets taken from LDPC codes of various variable node degrees are investigated, and it is shown that all of them can be deactivated with low enough scaling factors. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely, Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code and its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity, but the optimal fraction is strictly smaller than one. We then consider techniques to further reduce the gap to capacity by means of random puncturing, and the inclusion of a certain fraction of generalized variable nodes in the graph. [ABSTRACT FROM AUTHOR]

Leafless elementary trapping sets (LETSs) are known to be the problematic structures in the error floor region of low-density parity-check (LDPC) codes over the additive white Gaussian (AWGN) channel under iterative decoding algorithms. While problems involving the general category of trapping sets, and the subcategory of elementary trapping sets (ETSs), have been shown to be NP-hard, similar results for LETSs, which are a subset of ETSs are not available. In this paper, we prove that for a general LDPC code, finding a LETS of a given size $a$ with minimum number of odd-degree check nodes $b$ is NP-hard to approximate within any approximation factor. We also prove that finding the minimum size $a$ of a LETS with a given $b$ is NP-hard to approximate within any approximation factor. Similar results are proved for elementary absorbing sets, a popular subcategory of LETSs. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new channel model termed as the $q$ -ary multi-bit channel. This channel models a memory device, where $q$ -ary symbols ($q=2^{s}$) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF($q$) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to a finite-length iterative-decoding analysis and propose an edge-labeling algorithm for the improved decoding performance. We then provide a finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves the finite-length decoding performance by orders of magnitude. [ABSTRACT FROM AUTHOR]

Compared with the non-prediction algorithm, the prediction-based symbol flipping decoding algorithm of non-binary, low-density, parity-check codes over GF($q$) can significantly improve the error performance. To escape from undesirable local maximum, this paper proposes a self-adjustment strategy to redesign the flipping metric. Combined with the prediction mechanism, our proposed algorithm provides a good tradeoff between the complexity and the error correction performance. Simulation results for the additive white Gaussian noise channel and the binary symmetric channel verify the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Low density parity check code (LDPC) is a linear block code whose performance is close to Shannon's limit. Compared with other decoding algorithms, it has low decoding complexity and flexible structure. So it becomes people's research hotspots. In recent years, LDPC code codec algorithm has been improved, and is widely used in deep space communications, optical fiber communications, mobile communications and underwater and other fields. Using chaotic sequence single-layer mapping plus pixel scrambling can complete chaotic image encryption, because some chaotic sequence is poor robust, in the process of channel transmission, it is difficult to ensure that the information can be destroyed after the attack In this paper, LDPC codes are introduced, and in the chaotic image encryption, block coding is used to form error protection to improve the stability of the image transmission process and improve the transmission quality. [ABSTRACT FROM AUTHOR]

In this paper, a new analysis method is introduced to predict the gap between Shannon limit and successful decoding signal-to-noise ratio threshold (SNR loss) of practical quasi-cyclic low-density parity check (QC-LDPC) codes. The SNR loss of a QC-LDPC code is summed up from three aspects: 1) the base matrix; 2) the number of iterations for decoding; and 3) the finite codeword length. As an example, the SNR loss of DVB-S2 and ATSC 3.0 LDPC codes is analyzed with the proposed method. Results show that a large portion of the SNR loss of DVB-S2 LDPC codes is due to their base matrices, which is significantly improved in latest ATSC 3.0 standard. Furthermore, with the aid of the proposed analysis method, newly designed QC-LDPC codes with raptor-like structure are proposed. Bit-error rate simulations show that the proposed codes are only 0.4 dB away from Shannon limit under binary-input additive white Gaussian noise channel and only 0.65 dB away from Shannon limit under binary-input Rayleigh fading channel, which are comparable with state-of-art design. [ABSTRACT FROM AUTHOR]

The joint Viterbi detector/decoder (JVDD) has been proposed as an alternative to the powerful and ubiquitous low-density parity check (LDPC) coding schemes, that are used in the majority of channel coding applications today. At shorter block lengths, the JVDD is able to outperform the state-of-the-art LDPC decoder that can be found in many of today’s systems, from deep-space communications to the digital video broadcasting standard, to the advanced television standard committee 3.0. Researchers have developed the tools to analyze the performance of the LDPC decoder, which is known as the sum-product algorithm or message passing algorithm. These tools include density evolution (DE) and extrinsic information transfer (EXIT) charts. To date, no similar such analysis exists for the JVDD. In the current paper we perform a similar preliminary analysis on the JVDD algorithm akin to what DE and EXIT charts are for the LDPC decoder. The current new analysis of the JVDD allows prediction of its probability of error based on the probability distribution functions of the metrics used within the algorithm. Our analysis plays a similar role to the abovementioned tools developed for the LDPC decoder: it allows us to predict the performance of the JVDD under various conditions (such as varying SNR’s, thresholds, and block lengths), it improves our understanding of the JVDD algorithm and the match against Monte-Carlo simulations verifies our assumptions on the main error-causing mechanisms within the JVDD. [ABSTRACT FROM AUTHOR]

The traditional degree distribution framework is a useful tool to analyze irregular low density parity check (LDPC) code. However, it cannot be used to analyze raptor-like LDPC code as it cannot describe the most important structural feature of raptor-like LDPC code, which is that any check node is connected to at most one degree-one variable node. In this paper, we propose a two-type degree distribution framework for raptor-like LDPC code. The proposed framework can describe the special structural feature of raptor-like LDPC code by dividing the check nodes into two different sets. One is the set of check nodes connected to those degree-one variable nodes, the other is the set of check nodes unconnected to those degree-one variable nodes. Furthermore, we propose a two-type discrete density evolution algorithm and a two-type Gaussian approximation density evolution algorithm based on the proposed two-type degree distribution framework to estimate the threshold of raptor-like LDPC code. During the decoding analysis, the proposed algorithms treat the check nodes belonging the two separate sets differently. Compared with the traditional multi-edge type (MET) algorithm, the proposed algorithms allow us to calculate the threshold more quickly without much sacrifice in accuracy. For some raptor-like LDPC codes adopted by the Advanced Television Systems Committee 3.0 (ATSC3.0) standard, the thresholds estimated by the proposed algorithms can be very close to the thresholds estimated by the traditional MET algorithm. The deviations are less than 0.07 dB. Meanwhile, the proposed algorithms can tremendously reduce the computation complexity because of the two-type degree distribution framework and the Gaussian approximation. [ABSTRACT FROM AUTHOR]

Double protograph low-density parity-check code has been shown to be a good solution for joint source-channel coding. Moreover, the source code and channel code (as one code pair) can be jointly optimized to achieve a good bit error rate (BER) performance. In this paper, first, an optimal design criterion of this code pair based on an extended curve-fitting algorithm is proposed. Second, a joint optimization procedure which searches for both the optimal decoding threshold and the optimal code pair is presented. In addition, for short-to-moderate-length codewords, the finite-length effect cannot be ignored and the procedure has to be modified with a new notion named length-N decoding threshold. The protograph extrinsic information transfer analysis and the BER simulation have shown that our optimized code pairs acquire lower decoding thresholds than conventional code pairs. Moreover, the optimized code pairs with short-to-moderate-length codewords can mitigate the finite-length effect and obtain low length-N decoding thresholds. [ABSTRACT FROM AUTHOR]