LDPC Block Codes Research Articles

Optimizing quasi-cyclic spatially coupled LDPC codes by eliminating harmful objects

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.

Block-Based Scheduling Algorithm for Layered Decoding of Block LDPC Codes

Block-Based Scheduling Algorithm for Layered Decoding of Block LDPC Codes

Discrete MMSE Precoding for Multiuser MIMO Systems With PSK Modulation

We propose an optimal MMSE precoding technique using quantized signals with constant envelope. Unlike the existing MMSE design that relies on 1-bit resolution, the proposed approach employs uniform phase quantization and the bounding step in the branch-and-bound method is different in terms of considering the most restrictive relaxation of the nonconvex problem, which is then utilized for a suboptimal design also. We also propose robust MMSE designs where we exploit second order statistics of a channel state information mismatch. Moreover, unlike prior studies, we propose three different soft detection methods and an iterative detection and decoding scheme that allow the utilization of channel coding in conjunction with low-resolution precoding. Besides an exact approach for computing the extrinsic information, we propose two approximations with reduced computational complexity. Numerical simulations show that utilizing the MMSE criterion instead of the established maximum-minimum distance to the decision threshold yields a lower bit-error-rate in many scenarios. Furthermore, when using the MMSE criterion, a smaller number of bound evaluations in the branch-and-bound method is required for low and medium SNR. Finally, results based on an LDPC block code indicate that the proposed receive processing schemes yield a lower bit-error-rate compared to the conventional design.

Girth aware normalized min sum decoding algorithm for shorter length low density parity check codes

Recently, short block codes are in great demand due to the emergent applications requiring the transmission of a short data unit and can guarantee speedy communication, with a minimum of latency and complexity which are among the technical challenges in today’s wireless services and systems. In the context of channel coding using low density parity check (LDPC) codes, the shorter length LDPC block codes are more likely to have short cycles with lengths of 4 and 6. The effect of the cycle with the minimum size is that this one prevents the propagation of the information in the Tanner graph during the iterative process. Therefore, the message decoded by short block code is assumed to be of poor quality due to short cycles. In this work, we present a study of the evolution of the messages on check nodes during the iterative decoding process when using the LDPC decoding algorithm normalized min sum (NMS), to see the destructive effect of short cycles and justify the effectiveness of the girth aware normalized min sum (GA-NMS) decoding LDPC codes algorithm in terms of correction of the errors, particularly for the codes with short cycles 4 and 6. In addition to this, the GA-NMS algorithm is evaluated in terms of bit error rate performance and convergence behavior, using wireless regional area networks (WRAN) LDPC code, which is considered as a short block code.

Spatially Coupled LDPC Codes via Partial Superposition and Their Application to HARQ

This paper proposes a new class of spatially coupled codes, which are constructed by sending codewords of a low-density parity-check block code (LDPC-BC) in a block Markov superposition transmission (BMST) manner, resulting in the BMST-LDPC codes. Different from the conventional spatially coupled LDPC (SC-LDPC) codes, the proposed BMST-LDPC codes can have encoder/decoder implemented using the structure of the corresponding LDPC-BCs. The proposed construction is universal , which is applicable to any existing LDPC-BCs including the 5 G New Radio (5G-NR) LDPC-BCs to obtain extra coding gains. The proposed BMST-LDPC codes are also a special class of BMST codes, which have lower error floors even with an encoding memory of one, inheriting a low decoding latency. Moreover, partial superposition , where a fraction of the coded bits are superimposed onto the following codewords, is introduced to alleviate the decoding error propagation. The construction can be optimized, for random long codes, by a protograph-based extrinsic information transfer (EXIT) chart analysis, or by a one-dimensional search for structured block codes. Furthermore, the BMST-LDPC scheme is integrated with the hybrid automatic retransmission request (HARQ) over the block fading channel, improving the throughput performance. Numerical results are presented to validate our analysis and demonstrate the performance advantage of the BMST-LDPC codes over the LDPC-BCs. They also show that the proposed HARQ scheme can yield a throughput improvement of up to 10% over the conventional HARQ scheme.

INCREASING THE WIRELESS SYSTEMS INFORMATION EFFICIENCY USING LDPC CODING

Background. LDPC codes research is performed in the paper. The influence of LDPC codes on channel information efficiency is studied in scope of MCS and the benefit of using LDPC codes is specified in digits as a result of research.Objective. The purpose of the paper is LDPC correcting abilities research in modern telecom systems and their comparison to other known error-correcting codes, and research of their influence on information efficiency of telecommunication system.Methods. Simulation modeling algorithm is implemented in Matlab using Java language, what helps generating corrective matrices and allows us to find out the corrective abilities of LDPC codes with different set of parameters. Many experiments make a statistic that shows a maximum and the best corrective abilities of LDPC codes, what is the aim of applying the simulation modeling approach.Results. The programmable simulation models to generate LDPC codes and errors number determining corrected LDPC codes with different parameters are developed. It’s the first time the method is proposed of determining the ultra-long block LDPC codes to ensure the reliability characteristics specified in the channel with known parameters, characterized in that the analysis parameters LDPC code is implemented using the original statistical model and to evaluate corrective ability to code at a significant increase in the length of the block LDPC code. Also, the LDPC codes influence on channel information efficiency is researched, and shown 9...52% improvement.Conclusions. The important scientific problem is solved in the paper, – construction and use of advanced wireless transmission systems with increasing information efficiency using LDPC coding. Stages of channel resources redistribution include LDPC, OFDM, modulation, described and presented in the information efficiency coordinates and transmission system performance.

Design of Irregular SC-LDPC Codes With Non-Uniform Degree Distributions by Linear Programming

In this paper, we propose new design algorithms of irregular spatially-coupled low-density parity-check (SC-LDPC) codes with non-uniform degree distributions using linear programming (LP). In general, irregular SC-LDPC codes with non-uniform degree distributions are difficult to design with low complexity because their density evolution equations are multi-dimensional. To overcome this problem, proposed design algorithms are based on three main ideas: a local design of degree distributions, pre-computation of the input/output message relationship, and selection of a proper objective function. These ideas make it possible to design degree distributions of irregular SC-LDPC codes by solving low-complexity LP problems over the binary erasure channel (BEC). It is shown that the proposed irregular SC-LDPC codes designed by the proposed algorithms are superior to regular SC-LDPC codes in terms of both asymptotic and finite-length performances over the BEC. We also confirm that the proposed irregular SC-LDPC code achieves better performance compared with an optimized irregular block LDPC code in the same blocklength, which implies that the proposed design algorithms also provide a new way to construct capacity-approaching block LDPC codes.

Reduced Complexity Window Decoding of Spatially Coupled LDPC Codes for Magnetic Recording Systems

In channel coding theory, the performance of error correcting codes (ECCs) approaching the Shannon limit can be achieved through increasing code lengths. Unfortunately, the complexity of ECCs will be increased as the code length increases. Nowadays, the magnetic recording (MR) system takes advantage of powerful ECCs by using 4 Kbytes sector. Among the advanced ECCs, the spatially coupled LDPC (SC-LDPC) codes (also known as a LDPC convolutional code) [1]are shown to have the decoding latency and complexity lower than those of the underlying LDPC block codes (LDPC-BC). Moreover, the SC-LDPC codes with threshold decoding outperform the LDPC-BC codes [2]. Hence, the SC-LDPC codes are the strong candidate for the future MR systems, when the sector size is increased beyond 4 Kbytes. An SC-LDPC decoder can use sliding window decoding [3]whereby the received signals are decoded by sliding window along the bit sequence. The window decoder is called “uniform window decoding (U-WD),” when all variable nodes (VNs) within a window are updated. In order to reduce the complexity of window decoding, some researchers proposed the non-uniform window decoding (N-WD) [4], which do not update the VNs with no improvement in the bit error rate (BER). This approach provides about 35-50% reduction in complexity compared to U-WD. In this work, we consider the application of SC-LDPC codes in MR systems, whereby SC-LDPC decoder cooperates with BCJR detector to encounter inter-symbol interference (ISI). We propose the dynamic shifting of window decoding (DS-WD) to reduce the complexity of SC-LDPC codes. Herein, the number of shifted bits is defined according to their soft BERs which are estimated at each decoding position. In addition, we modify the N-WD [4]to reinforce our proposed algorithm called “dynamic-shifting non-uniform window decoding (DS-N-WD).” The DS-WD and DS-N-WD achieve the complexity reduction of 7% and 25% without any loss in performance compared to the N-WD algorithms.

Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length

In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.

Spatially Coupled Codes Optimized for Magnetic Recording Applications

Spatially coupled (SC) codes are a class of sparse graph-based codes known to have capacity-approaching performance. SC codes are constructed based on an underlying low-density parity-check (LDPC) code, by first partitioning the underlying block code and then putting replicas of the components together. Significant recent research efforts have been devoted to the asymptotic, ensemble-averaged study of SC codes, as these coupled variants of the existing LDPC codes offer excellent properties. While the asymptotic analysis is important, due to simplifying assumptions and averaging effects, results from the asymptotic domain are not readily translatable to the practical, finite-length setting. Despite this chasm, the finite-length analysis of SC codes is still largely unexplored. In this paper, we tackle the problem of optimized design of SC codes in the context of magnetic-recording (MR) applications. In particular, we identify combinatorial structures in the graphical representation of the code that are detrimental in the MR setting. An intriguing observation is that for the same SC code, the problematic objects for the MR channels are combinatorially different from the additive white Gaussian noise (AWGN) setting, thus necessitating a careful code design approach for the MR applications. We first demonstrate that the choice of the so-called cutting vector, which guides the code partitioning in the SC code design, directly affects the cardinality of these problematic objects. In particular, we show that the number of problematic objects is the highest - and consequently that the performance is the worst - in the case of the degenerate cutting vector, which precisely corresponds to uncoupled LDPC block codes. We, therefore, show that coupling always improves the performance and that the degree of improvement is dependent on the choice of the cutting vector. We then extend our analysis to different column weights and present SC codes that outperform block codes with unoptimized edge weights by more than 3.5 orders of magnitude and that also outperform both optimized block codes and unoptimized SC codes by more than two orders of magnitude. Through presented examples, we demonstrate high performance of the proposed code design methodology.

Design of Spatially Coupled LDPC Codes Over GF $(q)$ for Windowed Decoding

In this paper we consider the generalization of binary spatially coupled low-density parity-check (SC-LDPC) codes to finite fields GF$(q)$, $q\geq 2$, and develop design rules for $q$-ary SC-LDPC code ensembles based on their iterative belief propagation (BP) decoding thresholds, with particular emphasis on low-latency windowed decoding (WD). We consider transmission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of $(J,K)$-regular SC-LDPC code ensembles constructed over GF$(q)$ using protographs. Thresholds are calculated using protograph versions of $q$-ary density evolution (for the BEC) and $q$-ary extrinsic information transfer analysis (for the BIAWGNC). We show that WD of $q$-ary SC-LDPC codes provides significant threshold gains compared to corresponding (uncoupled) $q$-ary LDPC block code (LDPC-BC) ensembles when the window size $W$ is large enough and that these gains increase as the finite field size $q=2^m$ increases. Moreover, we demonstrate that the new design rules provide WD thresholds that are close to capacity, even when both $m$ and $W$ are relatively small (thereby reducing decoding complexity and latency). The analysis further shows that, compared to standard flooding-schedule decoding, WD of $q$-ary SC-LDPC code ensembles results in significant reductions in both decoding complexity and decoding latency, and that these reductions increase as $m$ increases. For applications with a near-threshold performance requirement and a constraint on decoding latency, we show that using $q$-ary SC-LDPC code ensembles, with moderate $q>2$, instead of their binary counterparts results in reduced decoding complexity.

Finite-Length Algebraic Spatially-Coupled Quasi-Cyclic LDPC Codes

The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.

Randomly Punctured LDPC Codes

In this paper, we present a random puncturing analysis of low-density parity-check (LDPC) code ensembles. We derive a simple analytic expression for the iterative belief propagation (BP) decoding threshold of a randomly punctured LDPC code ensemble on the binary erasure channel (BEC) and show that, with respect to the BP threshold, the strength and suitability of an LDPC code ensemble for random puncturing is completely determined by a single constant that depends only on the rate and the BP threshold of the mother code ensemble. We then provide an efficient way to accurately predict BP thresholds of randomly punctured LDPC code ensembles on the binary-input additive white Gaussian noise channel (BI-AWGNC), given only the BP threshold of the mother code ensemble on the BEC and the design rate, and we show how the prediction can be improved with knowledge of the BI-AWGNC threshold. We also perform an asymptotic minimum distance analysis of randomly punctured code ensembles and present simulation results that confirm the robust decoding performance promised by the asymptotic results. Protograph-based LDPC block code and spatially coupled LDPC code ensembles are used throughout as examples to demonstrate the results.

Searching for Binary and Nonbinary Block and Convolutional LDPC Codes

A unified approach to search for and optimize codes determined by their sparse parity-check matrices is presented. Replacing the nonzero elements of a binary parity-check matrix (the base or parent matrix) either by circulants or by companion matrices of elements from a finite field GF $(2^{m})$ , we obtain quasi-cyclic low-density parity-check (LDPC) block codes and binary images of nonbinary LDPC block codes, respectively. By substituting monomials of a formal variable $D$ , we obtain the polynomial description of an LDPC convolutional code. A set of performance measures applicable to different classes of LDPC codes is considered, and a greedy algorithm for code performance optimization is presented. The heart of the new optimization algorithm is a fast procedure for searching for LDPC codes with large girth of their Tanner graphs. For a few classes of LDPC codes, examples of codes combining good error-correcting performance with compact representation are obtained. In particular, we present optimized convolutional LDPC codes and conclude that the LDPC block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of faster than Nyquist (FTN) signaling using LDPC codes. The FTN strategy combined with a rate $R=1/2$ LDPC code of length 64 800 optimized for effective data rate $R=3/4$ gains more than 0.5 dB compared with the standard LDPC codes of the same rate and length. The obtained gain corresponds to transmission at the capacity of the binary input additive white Gaussian noise channel. In most numerical examples, we consider codes with bidiagonal structure of the parity-check matrix. This restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards.

Construction of block-LDPC codes based on quadratic permutation polynomials

A new block low-density parity-check (Block-LDPC) code based on quadratic permutation polynomials (QPPs) is proposed. The parity-check matrix of the Block-LDPC code is composed of a group of permutation submatrices that correspond to QPPs. The scheme provides a large range of implementable LDPC codes. Indeed, the most popular quasi-cyclic LDPC (QC-LDPC) codes are just a subset of this scheme. Simulation results indicate that the proposed scheme can offer similar error performance and implementation complexity as the popular QC-LDPC codes.

Performance Comparison of LDPC Block and Spatially Coupled Codes Over GF <inline-formula> <tex-math notation="TeX">$(q)$</tex-math></inline-formula>

In this paper, we compare the finite-length performance of protograph-based spatially coupled low-density parity-check (SC-LDPC) codes and LDPC block codes (LDPC-BCs) over GF(q). In order to reduce computational complexity and latency, a sliding window decoder with a stopping rule based on a soft bit-error-rate (BER) estimate is used for the q-ary SC-LDPC codes. Two regimes are considered: one when the constraint length of q-ary SC-LDPC codes is equal to the block length of q-ary LDPC-BCs and the other when the two decoding latencies are equal. Simulation results confirm that, in both regimes, (3,6)-, (3,9)-, and (3,12)-regular non-binary SC-LDPC codes can significantly outperform both binary and non-binary LDPC-BCs and binary SC-LDPC codes. Finally, we present a computational complexity comparison of q-ary SC-LDPC codes and q-ary LDPC-BCs under equal decoding latency and equal decoding performance assumptions.

Implementation of Decoders for LDPC Block Codes and LDPC Convolutional Codes Based on GPUs

In this paper, efficient LDPC block-code decoders/simulators which run on graphics processing units (GPUs) are proposed. We also implement the decoder for the LDPC convolutional code (LDPCCC). The LDPCCC is derived from a predesigned quasi-cyclic LDPC block code with good error performance. Compared to the decoder based on the randomly constructed LDPCCC code, the complexity of the proposed LDPCCC decoder is reduced due to the periodicity of the derived LDPCCC and the properties of the quasicyclic structure. In our proposed decoder architecture, Γ (Γ is a multiple of a warp) codewords are decoded together, and hence, the messages of Γ codewords are also processed together. Since all the Γ codewords share the same Tanner graph, messages of the Γ distinct codewords corresponding to the same edge can be grouped into one package and stored linearly. By optimizing the data structures of the messages used in the decoding process, both the read and write processes can be performed in a highly parallel manner by the GPUs. In addition, a thread hierarchy minimizing the divergence of the threads is deployed, and it can maximize the efficiency of the parallel execution. With the use of a large number of cores in the GPU to perform the simple computations simultaneously, our GPU-based LDPC decoder can obtain hundreds of times speedup compared with a serial CPU-based simulator and over 40 times speedup compared with an eight-thread CPU-based simulator.

Array Convolutional Low-Density Parity-Check Codes

This paper presents a design technique for obtaining regular time-invariant low-density parity-check convolutional (RTI-LDPCC) codes with low complexity and good performance. We start from previous approaches which unwrap a low-density parity-check (LDPC) block code into an RTI-LDPCC code, and we obtain a new method to design RTI-LDPCC codes with better performance and shorter constraint length. Differently from previous techniques, we start the design from an array LDPC block code. We show that, for codes with high rate, a performance gain and a reduction in the constraint length are achieved with respect to previous proposals. Additionally, an increase in the minimum distance is observed.

A 2.0 Gb/s Throughput Decoder for QC-LDPC Convolutional Codes

This paper propose a decoder architecture for low-density parity-check convolutional code (LDPCCC). Specifically, the LDPCCC is derived from a quasi-cyclic (QC) LDPC block code. By making use of the quasi-cyclic structure, the proposed LDPCCC decoder adopts a dynamic message storage in the memory and uses a simple address controller. The decoder efficiently combines the memories in the pipelining processors into a large memory block so as to take advantage of the data-width of the embedded memory in a modern field-programmable gate array (FPGA). A rate-5/6 QC-LDPCCC has been implemented on an Altera Stratix FPGA. It achieves up to 2.0 Gb/s throughput with a clock frequency of 100 MHz. Moreover, the decoder displays an excellent error performance of lower than $10^{-13}$ at a bit-energy-to-noise-power-spectral-density ratio ($E_b/N_0$) of 3.55 dB.

On the active distances of partial unit memory codes based on LDPC codes

Partial unit memory (PUM) convolutional codes constructed from block codes make it possible to obtain convolutional codes with high correcting properties. An ensemble of binary PUM convolutional codes based on low-density parity-check block codes is introduced. This ensemble is specified by semi-infinite parity-check matrices with partial unit memory. The asymptotic bounds on a free distance and active row distances are derived for codes of the specified ensemble. It is proved that the ensemble codes have the positive slope of active row distances.