Binary Low-density Parity-check Codes Research Articles

LDPC Design for Block Differential Modulation in Optical Communications

In coherent digital optical receivers, blind carrier phase recovery may incur errors, causing phase slips. A channel including blind phase estimation is modeled as an additive white Gaussian noise (AWGN) channel, affected by random phase slips that occur with a certain probability. For such a channel, block differential modulation (BDM) has been recently shown to achieve almost the same capacity as coherent AWGN channels. This paper proposes a concatenated coding scheme, with an inner code based on low-density parity-check (LDPC) codes, that embeds block differential demodulation, to make it robust to occasional phase slips. This additional feature comes at no practical cost and performs very close to a standard soft decoded LDPC code. The first design is a binary LDPC code, robust to phase rotations at multiples of $\pi$ , to be used with BPSK. Then, the concept is extended to quaternary LDPC codes, robust to phase rotations at multiples of $\pi /2$ , that can be used with 2-D digital constellations such as QPSK and 16-QAM.

Labeling Non-Square QAM Constellations for One-Dimensional Bit-Metric Decoding

A new modulation scheme is proposed that combines the advantages of APSK and QAM constellations, namely, the low peak power of APSK and the independent processing of the real and imaginary components of QAM. The new scheme consists of a Gray-labeled non-square QAM constellation and a bit-metric decoder. At a spectral efficiency of 3.5 bits/s/Hz, the new scheme saves 1.35 dB of peak power compared with conventional QAM. The new scheme is implemented with a binary low-density parity-check code. Simulation results are presented.

Short Turbo Codes over High Order Fields

Two classes of turbo codes constructed on high-order finite fields are introduced. The codes are derived from a particular protograph sub-ensemble of the (2,3) regular low-density parity-check (LDPC) code ensemble. The first construction results in a parallel concatenation of two non-binary, time-variant accumulators. The second construction consists of the serial concatenation of a non-binary time-variant differentiator with a non-binary time-variant accumulator, and provides a highly structured flexible encoding scheme for (2,4) LDPC codes. A cycle graph representation is also provided. The proposed codes can be decoded efficiently either as LDPC codes (via belief propagation decoding on their bipartite graphs) or as turbo codes (via the forward-backward algorithm applied to the component code trellises) by means of the fast Fourier transform. The proposed codes provide remarkable coding gains (more than 1 dB at a codeword error rate 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-4</sup> ) over binary LDPC and turbo codes in the moderate-short block length regime.

Gallager B Decoder on Noisy Hardware

Conventional communications theory assumes that the data transmission is noisy but the processing at the receiver is entirely error-free. Such assumptions may have to be revisited for advanced (silicon) technologies in which hardware failures are a major concern at the system-level. Hence, it is important to characterize the performance of a communication system with both noisy processing components and noisy data transmission. Coding systems based on low-density parity check (LDPC) codes are widely used for a variety of applications. In this paper, we focus on probabilistic analysis of the LDPC Gallager B decoder built out of faulty components. Using the density evolution technique, we find approximations for the optimal threshold of the decoder and the symbol error rate (SER) of the decoded sequence as functions of both the channel error rate and error rates of the decoder components, for both binary and non-binary regular LDPC codes. Furthermore, we study the convergence of the output SER and the decoding threshold of the decoder for different ranges of error rates. We verify our results using MATLAB simulations and hardware emulation of noisy decoders. Results presented in this paper can serve as systematic design guidelines in resource allocation for noisy decoders. Informed resource allocation is of particular relevance to emerging data storage and processing applications that need to maintain high levels of reliability despite hardware errors in advanced technologies.

Efficient detection and decoding of q-ary LDPC coded signals over partial response channels

Abstract In this study, we consider the ways for concatenating the intersymbol interference (ISI) detector with a q-ary low-density parity-check (LDPC) decoder for transmissions over partial response (PR) channels. LDPC codes allow achieving performance close to the channel capacity in additive white Gaussian noise channels, while designing receivers employing these codes for transmissions over channels affected by ISI is still an open issue. Turbo equalization schemes are considered with a novel joint message-passing-based receiver, which is derived from a recently proposed joint algorithm for binary LDPC codes. Simulation results provide performance evaluation of these systems over three different PR channels, together with an analysis of the trade-off between error-rate performance and complexity.

Iterative Linear Programming Decoding of Nonbinary LDPC Codes With Linear Complexity

The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.

Exploiting Channel Diversity in Secret Key Generation From Multipath Fading Randomness

We design and analyze a method to extract secret keys from the randomness inherent to wireless channels. We study a channel model for multipath wireless channel and exploit the channel diversity in generating secret key bits. We compare the key extraction methods based both on entire channel state information (CSI) and on single channel parameter such as the received signal strength indicators (RSSI). Due to the reduction in the degree-of-freedom when going from CSI to RSSI, the rate of key extraction based on CSI is far higher than that based on RSSI. This suggests that exploiting channel diversity and making CSI information available to higher layers would greatly benefit the secret key generation. We propose a key generation system based on low-density parity-check (LDPC) codes and describe the design and performance of two systems: one based on binary LDPC codes and the other (useful at higher signal-to-noise ratios) based on four-ary LDPC codes.

Computing the Minimum Distance of Nonbinary LDPC Codes

Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.

Non‐binary protograph low‐density parity‐check codes for space communications

SUMMARYProtograph‐based non‐binary low‐density parity‐check (LDPC) codes with ultra‐sparse parity‐check matrices are compared with binary LDPC and turbo codes (TCs) from space communication standards. It is shown that larger coding gains are achieved, outperforming the binary competitors by more than 0.3 dB on the additive white Gaussian noise channel (AWGN). In the short block length regime, the designed codes gain more than 1 dB with respect to the binary protograph LDPC codes recently proposed for the next generation up‐link standard of the Consultative Committee for Space Data Systems. Copyright © 2012 John Wiley &amp; Sons, Ltd.

A New Construction of Structured Binary Regular LDPC Codes Based on Steiner Systems with Parameter t&gt;2

This paper presents a novel method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatorial designs, known as Steiner systems. This code design approach can be considered as a generalization of the well-known method which uses the point-block incidence matrix of a Steiner 2-design for the code construction. Though the given method can be applied on any Steiner system S(t, k, v), in this paper we focus only on Steiner systems with t ≥ 3. Furthermore, we show that not only a Steiner system (X,B) itself, but also its residual design with respect to an arbitrary point x ∈ X can be employed for code construction. We also present a technique for constructing binary and non-binary QC-LDPC codes from Steiner systems. The Tanner graph of the constructed codes is free of 4-cycles and hence the codes have girth at least six. Simulation results show that the so constructed codes perform well over the AWGN channel with iterative message-passing decoding.

A Study on Nonbinary LDPC Coding and Iterative Decoding System in BPM R/W Channel

In this paper, the iterative decoding system using the nonbinary low-density parity-check (LDPC) code is studied in the magnetic recording system using a bit-patterned media (BPM) R/W channel affected by the write head field gradient, the media switching field distribution (SFD), the demagnetization field from adjacent dots, and the dot position deviation in an areal recording density of 2 Tb/in <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The performance of iterative decoding system using the nonbinary LDPC code over Galois field of GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> ) is evaluated by the computer simulation, and it is compared with the conventional iterative decoding system using the binary LDPC code. The results show that the nonbinary LDPC system provides better performance compared with the binary LDPC system.

Optimized Min-Sum Decoding Algorithm for Low Density Parity Check Codes

Low Density Parity Check (LDPC) code approaches Shannon-limit performance for binary field and long code lengths. However, performance of binary LDPC code is degraded when the code word length is small. An optimized min- sum algorithm for LDPC code is proposed in this paper. In this algorithm unlike other decoding methods, an optimization factor has been introduced in both check node and bit node of the Min- sum algorithm. The optimization factor is obtained before decoding program, and the same factor is multiplied twice in one cycle. So the increased complexity is fairly low. Simulation results show that the proposed Optimized Min-Sum decoding algorithm performs very close to the Sum-Product decoding while preserving the main features of the Min-Sum decoding, that is low complexity and independence with respect to noise variance estimation errors.

Binary SOVA and Nonbinary LDPC Codes for Turbo Equalization in Magnetic Recording Channels

Nonbinary low-density parity check (LDPC) codes can provide coding gains over binary LDPC codes at the cost of increased complexity. For magnetic recording channels, optimal detectors for the nonbinary codes are symbol-based, which are significantly more complex than binary detectors. In this paper, we investigate the feasibility of using much less complex bit-based soft output Viterbi algorithm (SOVA) as a channel detector for nonbinary LDPC codes and performing turbo iterations between the SOVA and the nonbinary LDPC decoder. We show that performance gains can still be obtained by the turbo iterations by using the bit-based SOVA, although the performance gains are suboptimal to the symbol-based detectors. This scheme can be useful in low-complexity nonbinary LDPC coding systems.

Performance and Complexity of 32 k-bit Binary LDPC Codes for Magnetic Recording Channels

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Recently, 32 k-bit sector size for hard disk drives is being investigated to take advantage of the superior performance of long error correcting codes. Meanwhile, low-density parity-check (LDPC) codes have been actively investigated for obtaining coding gains over conventional Reed-Solomon (RS) codes mainly for 4 k-bit sectors. In this paper, the coding gain of a 32 k-bit LDPC code over a 4 k-bit LDPC code, a 32 k-bit RS code, and a 4 k-bit RS code in magnetic recording channels is investigated. The decoding complexity of 32 k-bit LDPC codes and 4 k-bit LDPC codes is also discussed. It is important to evaluate whether the coding gains are enough to justify the increased complexity. Using the 32 k-bit LDPC code, 0.8-dB gain over the 32 k-bit RS code or the 4 k-bit LDPC code (the two schemes coincidentally showed similar performance) at 32 k-bit block error rate (BLER) <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$10^{-3}$</tex> </formula></emphasis>, and 1.6-dB gain over the 4 k-bit RS code were obtained. It is shown that 32 k-bit LDPC codes require a larger number of iterations than the 4 k-bit LDPC codes. It is also shown that there is much room to improve the design of 32 k-bit LDPC codes than the code used in the simulation. To illustrate the potential, quasicyclic LDPC codes with column weights up to 13 with girth 6 are investigated. </para>

High-Rate Nonbinary Regular Quasi-Cyclic LDPC Codes for Optical Communications

The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF(q) is constructed by assigning nonzero elements from GF(q) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF(q), constructed by assigning nonzero elements from GF(q) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-12</sup> . Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.

A class of quasi-cyclic LDPC codes over GF(2m)

Low Density Parity Check (LDPC) codes over GF(2m,)are an extension of binary LDPC codes. Performances of GF(2m,) LDPC codes have been shown to be higher than binary LDPC codes, but the complexity of the encoders/decoders increases. Hence there is a substantial lack of implementations for LDPC codes over GF(2m,)codes. This paper presents a class of quasi-cyclic LDPC codes over GF(2m,). These codes can alleviate the encoding/decoding complexity without excessive loss of performances. It is shown how, from a performance point of view, such codes are better than m times bigger binary codes and as good as (2m,)longer binary codes.

Next Generation FEC for High-Capacity Communication in Optical Transport Networks

Codes on graphs of interest for next generation forward error correction (FEC) in high-speed optical networks, namely turbo codes and low-density parity-check (LDPC) codes, are described in this invited paper. We describe both binary and nonbinary LDPC codes, their design, and decoding. We also discuss an FPGA implementation of decoders for binary LDPC codes. We then explain how to combine multilevel modulation and channel coding optimally by using coded modulation. Also, we describe an LDPC-coded turbo-equalizer as a candidate for dealing simultaneously with fiber nonlinearities, PMD, and residual chromatic dispersion.

New multivariate noise model and data detection using the expectation maximization algorithm

A signal sequence detector in a high areal density recording channel is required to provide robust compensation against unexpected error events. Primarily, a number of error events are caused by media noise and nonlinear distortion. The same problem of signal sequence detection remains to be solved in a future magnetic recording system that comes in predisposed to trend for recording by large-sector size instead of existing single-sector one that consists of 512 information 8-bits bytes. For the above problem, this paper shows the signal estimation method based on statistical inference for such a finite mixture model with known number of degraded noise components. Our signal detection scheme with multivariate autoregressive models for total noise and the expectation maximization algorithm is applied to maximum a posteriori estimation for multivariate mixtures of noise. Furthermore, a non-binary low-density parity-check (LDPC) code is used for an error-correcting code that satisfies the specific run-length limited condition in the proposed system. It shows that the proposed error-correcting and signal detection methods are effective in estimating signal sequences degraded by media noise and in improving the error rate performances with respect to the conventional system using the binary LDPC code and univariate autoregressive model.

Application of Nonbinary LDPC Cycle Codes to MIMO Channels

In this paper, we investigate the application of nonbinary low-density parity-check (LDPC) cycle codes over Galois field GF(q) to multiple-input multiple-output (MIMO) channels. Two types of LDPC coded systems that employ either joint or separate MIMO detection and channel decoding are considered, depending on the size of the Galois field and the modulation choice. We construct a special class of nonbinary LDPC cycle codes called the parallel sparse encodable (PSE) codes. The PSE code, consisting of a quasi-cyclic (QC) LDPC cycle code and a simple tree code, has the attractive feature that it is not only linearly encodable, but also allows parallel encoding which can reduce the encoding time significantly. We provide a systematic comparison between nonbinary coded systems and binary coded systems in both performance and complexity. Our results show that the proposed nonbinary system employing the PSE code outperforms not only the binary LDPC code specified in the 802.16e standard, but also the optimized binary LDPC code obtained using the EXIT chart methods. Through a detailed complexity analysis, we conclude that for the MIMO channel considered, the nonbinary coded systems achieve a superior performance at a receiver complexity that is comparable to that of the binary systems.

Decoding Algorithms for Nonbinary LDPC Codes Over GF$(q)$

In this letter, we address the problem of decoding nonbinary low-density parity-check (LDPC) codes over finite fields GF(q), with reasonable complexity and good performance. In the first part of the letter, we recall the original belief propagation (BP) decoding algorithm and its Fourier domain implementation. We show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding. In the second part of the letter, we introduce a simplified decoder which is inspired by the min-sum decoder for binary LDPC codes. We called this decoder extended min-sum (EMS). We show that it is possible to greatly reduce the computational complexity of the check-node processing by computing approximate reliability measures with a limited number of values in a message. By choosing appropriate correction factors or offsets, we show that the EMS decoder performance is quite good, and in some cases better than the regular BP decoder. The optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution. Our simulations on ultra-sparse codes over very-high-order fields show that nonbinary LDPC codes are promising for applications which require low frame-error rates for small or moderate codeword lengths. The EMS decoder is a good candidate for practical hardware implementations of such codes