Given an odd prime q and an integer m ≤ q, a binary mq × q2 quasi-cyclic parity-check matrix H(m, q) can be constructed for an array low-density parity-check (LDPC) code C (m, q). In this letter, we investigate the first separating redundancy of C (m, q). We prove that H (m, q) is 1-separating for any pair of (m, q), from which we conclude that the first separating redundancy of C (m, q) is upper bounded by mq. Then we show that our upper bound on the first separating redundancy of C (m, q) is tighter than the general deterministic and constructive upper bounds in the literature. For m=2, we further prove that the first separating redundancy of C (2, q) is 2q for any odd prime q. For m ≥ 3, we conjecture that the first separating redundancy of C (m, q) is mq for any fixed m and sufficiently large q.

This paper presents a novel optimization-based decoding algorithm for LDPC codes. The proposed decoding algorithm is based on a proximal gradient method for solving an approximate maximum a posteriori (MAP) decoding problem. The key idea of the proposed algorithm is the use of a code-constraint polynomial to penalize a vector far from a codeword as a regularizer in the approximate MAP objective function. A code proximal operator is naturally derived from a code-constraint polynomial. The proposed algorithm, called proximal decoding, can be described by a simple recursive formula consisting of the gradient descent step for a negative log-likelihood function corresponding to the channel conditional probability density function and the code proximal operation regarding the code-constraint polynomial. Proximal decoding is experimentally shown to be applicable to several non-trivial channel models such as LDPC-coded massive MIMO channels, correlated Gaussian noise channels, and nonlinear vector channels. In particular, in MIMO channels, proximal decoding outperforms known massive MIMO detection algorithms, such as an MMSE detector with belief propagation decoding. The simple optimization-based formulation of proximal decoding allows a way for developing novel signal processing algorithms involving LDPC codes.

A properly designed stopping criterion for iterative decoding algorithms can save a number of iterations and lead to a considerable reduction of system latency. The symbol flipping decoding algorithms based on prediction (SFDP) have been proposed recently for efficient decoding of non-binary low-density parity-check (LDPC) codes. To detect the decoding frames with slow convergence or even non-convergence, we track the number of oscillations on the value of objective function during the iterations. Based on this tracking number, we design a simple stopping criterion for the SFDP algorithms. Simulation results show that the proposed stopping criterion can significantly reduce the number of iterations at low signal-to-noise ratio regions with slight error performance degradation.

This paper presents the bit error rate (BER) performance of frequency domain equalization (FDE) using cyclic-shifted code division multiplexing (CDM) pilot signals for single-carrier line-of-sight (LOS) - multiple-input multiple-output (MIMO) multiplexing. We propose applying different cyclic-shift resources of the same Zadoff-Chu sequence to transmission-stream-specific pilot signals that are essential for estimating the channel response for FDE and phase noise in LOS-MIMO. To validate the effectiveness of the cyclic-shifted pilot multiplexing, we use partial low-density parity-check (LDPC) coding with double Gray mapping and collaborative decoding. Simulations show that pilot signal multiplexing using a cyclic-shifted Zadoff-Chu sequence, and frequency domain averaging of the estimated channel response are effective in achieving accurate channel estimation for single-carrier LOS-MIMO. We also show that the required received signal-to-noise power ratio at the BER of 10-7 using partial LDPC coding is decreased by more than 6.6dB compared to that without LDPC coding even for the deep notch depth of -20dB regardless of the relationship between the notch frequencies in the direct and cross links for 2×2 LOS-MIMO in a Rummler fading channel. Therefore, we conclude that the CDM-based pilot signal multiplexing with different cyclic shifts is effective in accurately estimating the channel response specific to the combination sets of transmitter and receiver antennas and in achieving a low pilot-overhead loss for single-carrier LOS-MIMO.

For insertion and deletion channels, there are many coding schemes based on low-density parity-check (LDPC) codes, such as spatially coupled (SC) LDPC codes and concatenated codes of irregular LDPC codes and markers. However, most of the previous works have problems, such as poor finite-length performance and unrealistic settings for codeword lengths and decoding iterations. Moreover, when using markers, the decoder receives log-likelihood (LLR) messages with different statistics depending on code bit position. In this paper, we propose a novel concatenation scheme using protograph-based LDPC code and markers that offers excellent asymptotic/finite-length performance and a structure that controls the irregularity of LLR messages. We also present a density evolution analysis and a simple optimization procedure for the proposed concatenated coding scheme. For two decoding scenarios involving decoding complexity, both asymptotic decoding thresholds and finite-length performance demonstrate that the newly designed concatenated coding scheme outperforms the existing counterparts: the irregular LDPC code with markers, the SC-LDPC code, and the protograph LDPC code, which is optimized for an additive white Gaussian noise channel, with markers.

Low-density parity-check (LDPC) codes can be used to improve the storage reliability of multi-level cell (MLC) flash memories because of their strong error-correcting capability. In order to improve the weighted bit-flipping (WBF) decoding of LDPC codes in MLC flash memories with cell-to-cell interference (CCI), we propose two strategies of normalizing weights and adjusting log-likelihood ratio (LLR) values. Simulation results show that the WBF decoding under the proposed strategies is much advantageous in both error and convergence performances over existing WBF decoding algorithms. Based on complexity analysis, the strategies provide the WBF decoding with a good tradeoff between performance and complexity.

In racetrack memories (RM), a position error (insertion or deletion error) results from unstable data reading. For position errors in RM with multiple read-heads (RHs), we propose a protograph-based LDPC coded system specified by a protograph and a protograph-aware permutation. The protograph-aware permutation facilitates the design and analysis of the coded system. By solving a multi-objective optimization problem, the coded system attains the properties of fast convergence decoding, a good decoding threshold, and a linear minimum distance growth. In addition, the coded system can adapt to varying numbers of RHs without any modification. The asymptotic decoding thresholds with a limited number of iterations verify the good properties of the system. Furthermore, for varying numbers of RHs, the simulation results with both small and large number of iterations, exhibit excellent decoding performances, both with short and long block lengths, and without error floors.

In fifth-generation mobile communications systems (5G), grant-free non-orthogonal multiple access (NOMA) schemes have been considered as a way to accommodate the many wireless connections required for Internet of Things (IoT) devices. In NOMA schemes, both system capacity enhancement and transmission protocol simplification are achieved, and an overload test of more than one hundred percent of the transmission samples over conducted. Multi-user shared multiple access (MUSA) has been proposed as a representative scheme for NOMA. However, the performance of MUSA has not been fully analyzed nor compared to other NOMA or orthogonal multiple access schemes. Therefore, in this study, we theoretically and numerically analyze the performance of MUSA in uplink fading environments and compare it with orthogonal frequency division multiple access (OFDMA), space division multiple access-based OFDMA, low-density signature, and sparse code multiple access. The characteristics and superiority of MUSA are then clarified.

The symbol flipping decoding algorithms based on prediction (SFDP) for non-binary LDPC codes perform well in terms of error performances but converge slowly when compared to other symbol flipping decoding algorithms. In order to improve the convergence rate, we design new flipping rules with two phases for the SFDP algorithms. In the first phase, two or more symbols are flipped at each iteration to allow a quick increase of the objective function. While in the second phase, only one symbol is flipped to avoid the oscillation of the decoder when the objective function is close to its maximum. Simulation results show that the SFDP algorithms with the proposed flipping rules can reduce the average number of iterations significantly, whereas having similar performances when compared to the original SFDP algorithms.

Quasi cyclic LDPC (QC-LDPC) codes consisting of circulant permutation matrices (CPM-QC-LDPC) are one of the most attractive types of LDPC codes due to their many advantages. In this paper, we mainly do some research on CPM-QC-LDPC codes. We first propose a two-stage decoding scheme mainly based on parity check matrix transform (MT), which can efficiently improve the bit error rate performance. To optimize the tradeoff between hardware implementation complexity and decoding performance, an improved method that combines our proposed MT scheme with the existing CPM-RID decoding scheme is presented. An experiment shows that both schemes can improve the bit error rate (BER) performance. Finally, we show that the MT decoding mechanism can be applied to other types of LDPC codes. We apply the MT scheme to random LDPC codes and show that it can efficiently lower the error floor.

In this study, spatially coupled low-density parity-check (SC-LDPC) codes on the two-dimensional array erasure (2DAE) channel are devised, including a method for generating new SC-LDPC codes with a restriction on the check node constraint. A density evolution analysis confirms the improvement in the threshold of the proposed two-dimensional SC-LDPC code ensembles over the one-dimensional SC-LDPC code ensembles. We show that the BP threshold of the proposed codes can approach the corresponding maximum a posteriori (MAP) threshold of the original residual graph on the 2DAE channel. Moreover, we show that the rates of the residual graph of the two-dimensional LDPC block code ensemble are smaller than those of the one-dimensional LDPC block code ensemble. In other words, a high performance can be obtained by choosing the two-dimensional SC-LDPC codes.

Spatially “Mt. Fuji” coupled (SFC) low density parity check (LDPC) codes are constructed as a chain of block LDPC codes. A difference between the SFC-LDPC codes and the original spatially coupled (SC) LDPC codes is code lengths of the underlying block LDPC codes. The code length of the block LDPC code at the middle of the chain is larger than that at the end of the chain. It is experimentally confirmed that the bit error probability in the error floor region of the SFC-LDPC code is lower than that of the SC-LDPC code whose code length and design rate are the same as those of the SFC-LDPC code. In this letter, we calculate the weight distribution of the SFC-LDPC code and try to explain causes of the low bit error rates of the SFC-LDPC code.

Recently, IoT compatible products have been popular, and various kinds of things are IoT compliant products. In these devices, cryptosystems and authentication are not treated properly, and security measures for IoT devices are not sufficient. Requirements of authentication for IoT devices are power saving and one-to-many communication. In this paper, we propose a zero-knowledge identification scheme using LDPC codes. In the proposed scheme, the zero-knowledge identification scheme that relies on the binary syndrome decoding problem is improved and the computational cost of identification is reduced by using the sparse parity-check matrix of the LDPC codes. In addition, the security level, computational cost and safety of the proposed scheme are discussed in detail.

In this paper, we consider coded multi-input multi-output (MIMO) systems with low-density parity-check (LDPC) codes and space-time block code (STBC) in MIMO channels. The LDPC code takes the role of a channel code while the STBC provides spatial-temporal diversity. The performance of such coded MIMO system has been shown to be excellent in the past. In this paper, we present a performance analysis for an upper bound on probability of error for coded MIMO schemes. Compared to previous works, the proposed approach for the upper bound can avoid any explicit weight enumeration of codewords and provide a significant step for the upper bound by using a multinomial theorem. In addition, we propose a log domain convolution that enables us to handle huge numbers, e.g., 10500. Comparison of system simulations and numerical evaluations shows that the proposed upper bound is applicable for various coded MIMO systems.

A new type of spatially coupled low density parity check (SCLDPC) code is proposed. This code has two benefits. (1) This code requires less number of iterations to correct the erasures occurring through the binary erasure channel in the waterfall region than that of the usual SCLDPC code. (2) This code has lower error floor than that of the usual SCLDPC code. Proposed code is constructed as a coupled chain of the underlying LDPC codes whose code lengths exponentially increase as the position where the codes exist is close to the middle of the chain. We call our code spatially “Mt. Fuji” coupled LDPC (SFCLDPC) code because the shape of the graph representing the code lengths of underlying LDPC codes at each position looks like Mt. Fuji. By this structure, when the proposed SFCLDPC code and the original SCLDPC code are constructed with the same code rate and the same code length, L (the number of the underlying LDPC codes) of the proposed SFCLDPC code becomes smaller and M (the code lengths of the underlying LDPC codes) of the proposed SFCLDPC code becomes larger than those of the SCLDPC code. These properties of L and M enables the above reduction of the number of iterations and the bit error rate in the error floor region, which are confirmed by the density evolution and computer simulations.

Detecting signals in a very large multiple-input multiple-output (MIMO) system requires high complexy of implementation. Thus, belief propagation based detection has been studied recently because of its low complexity. When the transmitted signal sequence is encoded using a channel code decodable by a factor-graph-based algorithm, MIMO signal detection and channel decoding can be combined in a single factor graph. In this paper, a low density parity check (LDPC) coded MIMO system is considered, and two types of factor graphs: bipartite and tripartite graphs are compared. The former updates the log-likelihood-ratio (LLR) values at MIMO detection and parity checking simultaneously. On the other hand, the latter performs the updates alternatively. Simulation results show that the tripartite graph achieves faster convergence and slightly better bit error rate performance. In addition, it is confirmed that the LLR damping in LDPC decoding is important for a stable convergence.

It is well known that spatially coupled (SC) codes with erasure-BP decoding have powerful error correcting capability over memoryless erasure channels. However, the decoding performance of SC-codes significantly degrades when they are used over burst erasure channels. In this paper, we propose band splitting permutations (BSP) suitable for (l,r,L) SC-codes. The BSP splits a diagonal band in a base matrix into multiple bands in order to enhance the span of the stopping sets in the base matrix. As theoretical performance guarantees, lower and upper bounds on the maximal burst correctable length of the permuted (l,r,L) SC-codes are presented. Those bounds indicate that the maximal correctable burst ratio of the permuted SC-codes is given by λmax≃1/k where k=r/l. This implies the asymptotic optimality of the permuted SC-codes in terms of burst erasure correction.

Based on the codewords of the [q,2,q-1] extended Reed-Solomon (RS) code over the finite field Fq, we can construct a regular binary γq×q2 matrix H(γ,q), where q is a power of 2 and γ≤q. The matrix H(γ,q) defines a regular low-density parity-check (LDPC) code C(γ,q), called a full-length RS-LDPC code. Using some analytical methods, we completely determine the values of s(H(4,q)), s(H(5,q)), and d(C(5,q)) in this letter, where s(H(γ,q)) and d(C(γ,q)) are the stopping distance of H(γ,q) and the minimum distance of C(γ,q), respectively.

We propose a digital image watermarking method satisfying information hiding criteria (IHC) for robustness against JPEG compression, cropping, scaling, and rotation. When a stego-image is cropped, the marking positions of watermarks are unclear. To detect the position in a cropped stego-image, a marker or synchronization code is embedded with the watermarks in a lattice pattern. Attacks by JPEG compression, scaling, and rotation cause errors in extracted watermarks. Against such errors, the same watermarks are repeatedly embedded in several areas. The number of errors in the extracted watermarks can be reduced by using a weighted majority voting (WMV) algorithm. To correct residual errors in output of the WMV algorithm, we use a high-performance error-correcting code: a low-density parity-check (LDPC) code constructed by progressive edge-growth (PEG). In computer simulations using the IHC ver. 4 the proposed method could a bit error rate of 0, the average PSNR was 41.136 dB, and the computational time for synchronization recovery was less than 10 seconds. The proposed method can thus provide high image quality and fast synchronization recovery.

In this paper, we will present analysis on the fault erasure BP decoders based on the density evolution. In the fault BP decoder, the messages exchanged in a BP process are stochastically corrupted due to unreliable logic gates and flip-flops; i.e., we assume circuit components with transient faults. We derived a set of the density evolution equations for the fault erasure BP processes. Our density evolution analysis reveals the asymptotic behaviors of the estimation error probability of the fault erasure BP decoders. In contrast to the fault free cases, it is observed that the error probabilities of the fault erasure BP decoder converge to positive values, and that there exists a discontinuity in an error curve corresponding to the fault BP threshold. It is also shown that an message encoding technique provides higher fault BP thresholds than those of the original decoders at the cost of increased circuit size.