An algorithm for encoding low-density parity check (LDPC) codes is investigated. The algorithm computes parity check symbols by solving a set of sparse equations, and the triangular factorization is employed to solve the equations efficiently. It is shown analytically and experimentally that the proposed algorithm is more efficient than the Richardson's encoding algorithm if the code has a small gap.

We study a modification method for constructing low-density parity-check (LDPC) codes for solid burst erasures. Our proposed modification method is based on a column permutation technique for a parity-check matrix of the original LDPC codes. It can change the burst erasure correction capabilities without degradation in the performance over random erasure channels. We show by simulation results that the performance of codes permuted by our method are better than that of the original codes, especially with two or more solid burst erasures.

Effective and simply realizable rate compatible low-density parity-check (LDPC) codes are proposed. A parity check matrix is constructed with the progressively increased column weights (PICW) order and adopted to achieve a punctured LDPC coding scheme for a wide range of the code rates of the rate compatible systems. Using the proposed rate compatible punctured LDPC codes, low complex adaptive communication systems, such as wireless communication systems, can be achieved with the reliable transmissions.

The most powerful channel coding schemes, namely those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper presents a 2048-bit, rate-1/2 soft decision decoder for a new class of codes known as Turbo Gallager Codes. These codes are turbo codes with properly chosen component convolutional codes such that they can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief propagation algorithm working on the code Tanner graph. These coding schemes are important in practical terms for two reasons: (i) they can be encoded as classical turbo codes, giving a solution to the encoding problem of LDPC codes; (ii) they can also be decoded in a fully parallel manner, partially overcoming the routing congestion bottleneck of parallel decoder VLSI implementations thanks to the locality of the interconnections. The implemented decoder can support up to 1 Gbit/s data rate and performs up to 48 decoding iterations ensuring both high throughput and good coding gain. In order to evaluate the performance and the gate complexity of the decoder VLSI architecture, it has been synthesized in a 0.18 µm standard-cell CMOS technology.

In this paper, the performance of Tree-LDPC code [1] is presented based on the min-sum algorithm with scaling and the asymptotic performance in the water fall region is shown by density evolution. We presents that the Tree-LDPC code show a significant performance gain by scaling with the optimal scaling factor [3] which is obtained by density evolution methods. We also show that the performance of min-sum with scaling is as good as the performance of sum-product while the decoding complexity of min-sum algorithm is much lower than that of sum-product algorithm.

This paper presents a performance and thresholds for Irregular Tree-LDPC codes. We obtain optimal irregular degree distributions and threshold by the density evolution technique. It is presented that Irregular Tree-LDPC code has performance gain at low SNR.

This paper proposes a low complexity composite CDMA system based on MIMO (Multiple-Input-Multiple-Output) processing and LDPC (low-density parity-check) codec based a CDMA system, which works well even if the complicated case of multipath, multiusers and short length LDPC codes. To explore the practical application, the mobile user in the composite CDMA systems is with only two antennas and adopts short length irregular LDPC codec, each user's data has been divided two LDPC encoded substreams being sent to two transmitter antennas at base station. Since the LDPC encoded substreams of reaching the mobile user are orthogonal to each other in space and time, the CDMA system performances (BER and SINR) can be improved much, but the multipath may ruin the orthogonalilty. To solve the problems, the paper provides the algorithms of main function modules of transmitter and receivers, gives a simple method to test the girth of LDPC codes, and analyzes the performance of MIMO-LDPC CDMA systems theoretically and experimentally. The simulation results show that the hybrid CDMA systems can have better performance than the conventional CDMA systems based on single transmitted antenna at a base station.

The Reliability-Based Hybrid ARQ (RB-HARQ) scheme, which can be used with error correcting codes using soft-input soft-output (SISO) decoders such as convolutional codes and turbo codes has been proposed. In the RB-HARQ scheme, the error rate performance is improved by selecting the retransmission bits based on Log Likelihood Ratio (LLR) of each bit in the receiver. However, the receiver has to send the bit positions of retransmission bits to the transmitter. Therefore, the RB-HARQ scheme requires a great number of feedback bits. On the other hand, Low Density Parity Check (LDPC) codes are attracting a lot of interest, recently. Because LDPC codes can achieve near Shannon limit performance and be decoded easily compared to turbo code. In this paper, we evaluate the RB-HARQ scheme using LDPC code. Moreover, we propose a RB-HARQ scheme that requires a fewer feedback bits by utilizing a code structure of LDPC code. We refer to the scheme as the RB-HARQ (row base) scheme. We show that the RB-HARQ and RB-HARQ (row base) schemes using LDPC code have better error rate performance than the scheme without ARQ. We also show that the RB-HARQ (row base) scheme has a good trade-off between error rate performance and the number of feedback bits compared to the RB-HARQ scheme.

In this paper, we propose a partially-parallel LDPC decoder which achieves a high-efficiency message-passing schedule. The proposed LDPC decoder is characterized as follows: (i) The column operations follow the row operations in a pipelined architecture to ensure that the row and column operations are performed concurrently. (ii) The proposed parallel pipelined bit functional unit enables the column operation module to compute every message in each bit node which is updated by the row operations. These column operations can be performed without extending the single iterative decoding delay when the row and column operations are performed concurrently. Therefore, the proposed decoder performs the column operations more frequently in a single iterative decoding, and achieves a high-efficiency message-passing schedule within the limited decoding delay time. Hardware implementation on an FPGA and simulation results show that the proposed partially-parallel LDPC decoder improves the decoding throughput and bit error performance with a small hardware overhead.

Low-density parity-check (LDPC) codes are one of the most promising next-generation error-correcting codes. For practical use, efficient methods for encoding of LDPC codes are needed and have to be studied. However, it seems that no general encoding methods suitable for hardware implementation have been proposed so far and for randomly constructed LDPC codes there have been no other methods than the simple one using generator matrices. In this paper we show that some classes of quasi-cyclic LDPC codes based on circulant permutation matrices, specifically LDPC codes based on array codes and a special class of Sridhara-Fuja-Tanner codes and Fossorier codes can be encoded by division circuits as cyclic codes, which are very easy to implement. We also show some properties of these codes.

A fast decoding algorithm for low-density parity-check codes is presented based on graph decomposition and two-way message passing schedule. Simulations show that the convergence speed of the proposed algorithm is about twice that of the conventional belief propagation algorithm.

In this paper, we explicitly formulate the average weight and the stopping set distributions and their asymptotic exponents of two-edge type LDPC code ensembles. We also show some characteristics such as the symmetry and the conditions for zero of the weight distributions of two code ensembles. Further we investigate the relation between two code ensembles from the perspectives of the weight and stopping set distributions.

A cycle search algorithm based on a message-passing in a Tanner graph is proposed for designing good LDPC codes. By applying the message-passing algorithm with a message alphabet composed of only two messages to a cycle search, we can perform a cycle search with less computational complexity than tree-based search algorithms. Also, the proposed algorithm can be easily implemented by using an existing message-passing decoder and can easily adopt different kind of criteria for an LDPC code design with a slight modification in the node update equations.

New decoding algorithms for binary linear codes based on the concave-convex procedure are presented. Numerical experiments show that the proposed decoding algorithms surpass Belief Propagation (BP) decoding in error performance. Average computational complexity of one of the proposed decoding algorithms is only a few times greater than that of the BP decoding.

We present a simple decoding algorithm that modifies soft bit-flipping algorithm for decoding LDPC codes. In our method, a new parameter is explored to distinguish the variables (symbols) belonging to the same number of unsatisfied constraints. A token is also assigned in the method to avoid repeated flipping of the same variable, rather than using a constant taboo length. Our scheme shows a similar computational load as the taboo-based algorithm, while having a similar decoding performance as the belief propagation algorithm.

In this paper, we provide explicit representations of average weight and stopping set distributions and asymptotic expressions of their exponent for detailedly represented irregular LDPC code ensembles. Further we present numerical examples which compare a detailedly represented irregular LDPC code ensemble with a conventional one with respect to both of weight and stopping set distributions.

Density evolution has recently been used to analyze the iterative decoding of Low Density Parity Check (LDPC) codes, Turbo codes, and Serially Concatenated Convolutional Codes (SCCC). The density evolution technique makes it possible to explain many characteristics of iterative decoding including convergence of performance and preferred structures for the constituent codes. While the analytic density evolution methods were applied to LDPC codes, the simulation based density evolution methods were used for Turbo codes and SCCC due to analytic difficulties. In this paper, several density evolution ideas in the literature are used to analyze common code structures and it is shown that those ideas yield consistent results. In order to do that, we derive expressions for density evolution of SCCC with a simple 2-state constituent code. The analytic expressions are based on the sum-product and min-sum algorithms, and the thresholds are evaluated for both message passing algorithms. Particularly, for the min-sum algorithm, the density evolution with Gaussian approximation is derived and used to analyze the effect of scaling soft information. The scaling of extrinsic information slows down the convergence of soft information or avoids an overestimation effect of it and results in better performance, and its gain is maximized in particular constituent codes. Similar approaches are made for LDPC code. We show that the scaling gain is noticeable in the LDPC code as well. This scaling gain is analyzed with both density evolution and simulation performance. The expected scaling gain by density evolution matches well with the achievable scaling gain from simulation results. These results can be extended to the irregular LDPC codes based on the degree distribution for the min-sum algorithm. All density evolution algorithms used in this paper are based on the Gaussian approximation for the exchanged messages.

In this letter, we propose an iterative decoding with LDPC based unitary matrix modulated OFDM with splitting the diagonal components over the coherence bandwidth. The proposed system can obtain a frequency diversity gain by splitting the diagonal components of unitary matrix modulated symbols, and also obtain large coding gain by using LDPC code.

The codes by the u/u+v construction have the extended error capacity for additional Hamming weight. It is a simple way to construct a large size block code using smaller ones. In this letter, low density parity check (LDPC) codes on u/u+v construction are investigated. Consequently, the study can enrich the family of LDPC codes and provide some special LDPC implementations.

Recently, low-density parity-check (LDPC) codes have attracted much attention. LDPC codes can achieve the near Shannon limit performance like turbo codes. For the LDPC codes, the reduced complexity decoding algorithms referred to as uniformly most powerful (UMP) BP- and normalized BP-based algorithms were proposed for BPSK on an additive white Gaussian noise (AWGN) channel. The conventional BP and BP-based algorithms can be applied to BPSK modulation. For high bit-rate transmission, multilevel modulation is preferred. Thus, the BP algorithm for multilevel modulations is proposed in . In this paper, we propose the BP algorithm with reduced complexity for multilevel modulations, where the first likelihood of the proposed BP algorithm is modified to adjust multilevel modulations. We compare the error rate performance of the proposed algorithm with that of the conventional algorithm on AWGN and flat Rayleigh fading channels. We also propose the UMP BP- and normalized BP-based algorithms for multilevel modulations on AWGN and flat Rayleigh fading channels. We show that the error rate performance of the proposed BP algorithm is almost identical to that of the algorithm in, where the decoding complexity of the proposed BP algorithm is less than that of the algorithm in. We also show that the proposed BP-based algorithms can achieve the good trade-off between the complexity and the error rate performance.