Random Low-density Parity-check Codes Research Articles

Asymptotic Average Multiplicity of Structures Within Different Categories of Trapping Sets, Absorbing Sets, and Stopping Sets in Random Regular and Irregular LDPC Code Ensembles

The performance of low-density parity-check (LDPC) codes in the error floor region is closely related to some substructures of the code's Tanner graph, collectively referred to as trapping sets (TSs). In this paper, we study the asymptotic average number of different types of trapping sets such as elementary TSs (ETS), leafless ETSs (LETS), absorbing sets (ABS), elementary ABSs (EABS), and stopping sets (SS), in random variable-regular and irregular LDPC code ensembles. We demonstrate that, regardless of the type of the TS, as the code's length tends to infinity, the average number of a given structure tends to infinity, to a positive constant, or to zero, if the structure contains no cycle, only one cycle, or more than one cycle, respectively. For the case where the structure contains a single cycle, we derive the asymptotic expected multiplicity of the structure by counting the average number of its constituent cycles and all the possible ways that the structure can be constructed from the cycle. This, in general, involves computing the expected number of cycles of a certain length with a certain given combination of node degrees, or computing the expected number of cycles of a certain length expanded to the desired structure by the connection of trees to its nodes. The asymptotic results obtained in this work, which are independent of the block length and only depend on the code's degree distributions, are shown to be accurate even for finite-length codes.

BP-LED Decoding Algorithm for LDPC Codes Over AWGN Channels

A new method is presented for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel. The proposed method termed belief-propagation–list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder fails to find a unique solution. The asymptotic exponent of the average list size for random regular LDPC codes from the Gallager ensemble is analyzed. Furthermore, a few examples of irregular quasi-cyclic LDPC as well as randomly constructed regular LDPC codes of short and moderate lengths are studied by simulations and their performance is compared to the tightened upper bound on the LDPC ensemble-average performance and the upper bound on the average performance of random linear codes under ML decoding. A comparison of the BP decoding and BP-LED performance of the WiMAX standard codes and performance of the near-ML BEAST decoding are presented. The new algorithm is applied to decoding a short nonbinary (NB) LDPC code over extensions of the binary Galois field. The obtained simulation results are compared to the tightened upper bound on the ensemble-average performance of the binary image of regular NB LDPC codes.

LP/SDP Hierarchy Lower Bounds for Decoding Random LDPC Codes

Random $(d_{v},d_{c})$ - regular low-density parity-check (LDPC) codes, where each variable is involved in $d_{v}$ parity checks and each parity check involves $d_{c}$ variables are well-known to achieve the Shannon capacity of the binary symmetric channel, for sufficiently large $d_{v}$ and $d_{c}$ , under exponential time decoding. However, polynomial time algorithms are only known to correct a much smaller fraction of errors. One of the most powerful polynomial-time algorithms with a formal analysis is the linear programming (LP) decoding algorithm of Feldman et al., which is known to correct an $\Omega (1/d_{c})$ fraction of errors. In this paper, we show that fairly powerful extensions of LP decoding, based on the Sherali–Adams and Lasserre hierarchies, fail to correct much more errors than the basic LP-decoder. In particular, we show that: 1) for any values of $d_{v}$ and $d_{c}$ , a linear number of rounds of the Sherali–Adams LP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code; and 2) for any value of $d_{v}$ and infinitely many values of $d_{c}$ , a linear number of rounds of the Lasserre SDP hierarchy cannot correct more than an $O(1/d_{c})$ fraction of errors on a random $(d_{v},d_{c})$ -regular LDPC code. Our proofs use a new stretching and collapsing technique that allows us to leverage recent progress in the study of the limitations of LP/SDP hierarchies for Maximum Constraint Satisfaction Problems (Max-CSPs). The problem then reduces to the construction of special balanced pairwise independent distributions for Sherali–Adams and special cosets of balanced pairwise independent subgroups for Lasserre. Our (algebraic) construction for the Lasserre hierarchy is based on designing sets of points in ${\mathbb F}_{q}^{d}$ (for $q$ any power of 2 and $d = 2,3$ ) with special hyperplane-incidence properties—constructions that may be of independent interest. An intriguing consequence of our work is that expansion seems to be both the strength and the weakness of random regular LDPC codes. Some of our techniques are more generally applicable to a large class of Boolean CSPs called Min-Ones. In particular, for $k$ -Hypergraph Vertex Cover, we obtain an improved integrality gap of $k-1-\epsilon $ that holds after a linear number of rounds of the Lasserre hierarchy, for any $k = q+1$ with $q$ an arbitrary prime power. The best previous gap for a linear number of rounds was equal to $2-\epsilon $ and due to Schoenebeck.

On the Tanner Graph Cycle Distribution of Random LDPC, Random Protograph-Based LDPC, and Random Quasi-Cyclic LDPC Code Ensembles

In this paper, we study the cycle distribution of random low-density parity-check (LDPC) codes, randomly constructed protograph-based LDPC codes, and random quasi-cyclic (QC) LDPC codes. We prove that for a random bipartite graph, with a given (irregular) degree distribution, the distributions of cycles of different length tend to independent Poisson distributions, as the size of the graph tends to infinity. We derive asymptotic upper and lower bounds on the expected values of the Poisson distributions that are independent of the size of the graph, and only depend on the degree distribution and the cycle length. For a random lift of a bi-regular protograph, we prove that the asymptotic cycle distributions are essentially the same as those of random bipartite graphs as long as the degree distributions are identical. For random QC-LDPC codes, however, we show that the cycle distribution can be quite different from the other two categories. In particular, depending on the protograph and the value of $c$ , the expected number of cycles of length $c$ , in this case, can be either $\Theta (N)$ or $\Theta (1)$ , where $N$ is the lifting degree (code length). We also provide numerical results that match our theoretical derivations. Our results provide a theoretical foundation for emperical results that were reported in the literature but were not well-justified. They can also be used for the analysis and design of LDPC codes and associated algorithms that are based on cycles.

Protograph-Based Quasi-Cyclic LDPC Coding for Ultrahigh Density Magnetic Recording Channels

In this paper, protograph-based quasi-cyclic (QC) low-density parity-check (LDPC) codes are proposed for the efficient structure coding of LDPC codes in 2-D intersymbol interference (ISI) ultrahigh density magnetic recording channels, such as bit-patterned magnetic recording and 2-D magnetic recording, where a reduced-complexity 2-D detector based on the iterative row-column soft detection feedback with Gaussian approximation is employed instead of the full 2-D Bahl-Cocke-Jelinek-Raviv detector. The proposed protograph-based codes, whose base matrices are constructed according to the degree sequences for 2-D-ISI channels, have a low-complexity QC encoder structure with a readily parallelizable protograph decoder structure. Simulation results show that the proposed protograph-based QC-LDPC codes outperform the random LDPC codes optimized for additive white Gaussian noise channels. Furthermore, the performance of the QC codes is similar to that exhibited by ISI-optimized random LDPC codes designed for 2-D-ISI channels, while retaining the implementational benefits.

Design of efficiently encodable nonbinary LDPC codes for adaptive coded modulation

This paper is concerned with constructions of nonbinary low-density parity-check (LDPC) codes for adaptive coded modulations (ACM). A new class of efficiently encodable structured nonbinary LDPC codes are proposed. The defining parity-check matrices are composed of scalar circulant sub-matrices which greatly reduce the storage requirement when compared with random LDPC codes. With this special structure of parity-check matrix, an efficient encoding algorithm is presented. Based on the proposed codes, a family of variablerate/variable-field nonbinary LDPC codes is designed for the ACM system. When combined with matched-size signal constellations, the family of constructed codes can achieve a wide range of spectral efficiency. Furthermore, the resultant ACM system can be implemented via a set of encoder and decoder. Simulation results show that the proposed nonbinary LDPC codes for the ACM system perform well.

Non-volatile memory reduction based on 1-D memory space mapping of a specific set of QC-LDPC codes

Abstract Supporting a great diversity of multi-rate H-matrices for multiple communication protocols requires a large amount of non-volatile memory, which may consume a large silicon area or logic elements and constrain the implementation of an overall decoder. Therefore, schemes for memory reduction are necessary to make the parity-check storage more compact. This study proposes a specific set of quasi-cyclic low-density parity-check (LDPC) (QC-LDPC) codes which can transfer a traditional two-dimensional (2-D) parity-check matrix (H-matrix) into a one-dimensional (1-D) memory space. Compared to the existing schemes, the proposed codes and memory reduction scheme do achieve significant reduction rates. Within a fixed memory space, many more H-matrices for diverse communication protocols can be saved via the proposed QC-LDPC codes, which are well constructed from modified Welch-Costas sequences. Furthermore, relatively good error performances, which outperform computer-generated random LDPC codes and Sridhara-Fuja-Tanner codes, are also shown in our simulation results. Consequently, we conclude that the proposed QC-LDPC codes can enlarge the capacity for saving much more low-BER (bit error rate) H-matrices within a fixed memory space.

Modeling of 2-D Magnetic Recording and a Comparison of Data Detection Schemes

Two-dimensional magnetic recording (TDMR) together with shingled magnetic recording (SMR) are technologies proposed to extend the life of conventional granular magnetic recording. The grain flipping probability (GFP) model has been proposed to mimic the performance of micromagnetic ( μ-mag) simulations for the purpose of signal reproduction. Other work in TDMR includes the proposal of a Gaussian mixture model (GMM) that produces improved likelihood information at the output of the detector, combined with low density parity check (LDPC) codes. The contribution of this paper is threefold. First, we aim to simulate a TDMR/SMR recording system with the GFP model, both with and without the GMM detector, and with various random and structured LDPC codes, of both 4 k and 16 k block lengths, to determine areal densities that might be achieved. Second, we perform a comparison of the various model order reduced (MOR) GFP implementations to compare the effect of writing with various factors taken out of the picture. Third, we perform a validation of the GFP model and the setup as a whole, by running the system with a parameter set close to that of conventional recording. The results of these experiments give an assurance of the validity of our model, give an indication of the expected density that might be achieved in a TDMR/SMR system, and give a direction for which parameter(s) in magnetic recording systems might be optimized to yield the most gain.

A Novel Error-Correcting System Based on Product Codes for Future Magnetic Recording Channels

We propose a novel construction of product codes for high-density magnetic recording based on binary low-density parity check (LDPC) codes and binary image of Reed-Solomon (RS) codes. Moreover, two novel algorithms are proposed to decode the codes in the presence of AWGN errors and scattered hard errors (SHEs). Simulation results show that at a bit-error rate (BER) of approximately 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-8</sup> , our method allows improving the error performance by approximately 1.9 dB compared with that of a hard decision decoder of RS codes of the same length and code rate. For the mixed error channel, including random noise and SHEs, the signal-to-noise ratio is set at 5 dB, and 150 to 400 SHEs are randomly generated. The bit-error performance of the proposed product code shows a significant improvement over that of equivalent random LDPC codes or serial concatenation of LDPC and RS codes.

A differential binary message-passing LDPC decoder

In this paper, we propose a binary message-passing algorithm for decoding low-density parity-check (LDPC) codes. The algorithm substantially improves the performance of purely hard-decision iterative algorithms with a small increase in the memory requirements and the computational complexity. We associate a reliability value to each nonzero element of the code's parity-check matrix, and differentially modify this value in each iteration based on the sum of the extrinsic binary messages from the check nodes. For the tested random and finitegeometry LDPC codes, the proposed algorithm can perform as close as about 1 dB and 0.5 dB to belief propagation (BP) at the error rates of interest, respectively. This is while, unlike BP, the algorithm does not require the estimation of channel signal to noise ratio. Low memory and computational requirements and binary message-passing make the proposed algorithm attractive for high-speed low-power applications.

A deterministic construction of irregular LDPC codes

In order to solve high encoding complexities of irregular low-density parity-check (LDPC) codes, a deterministic construction of irregular LDPC codes with low encoding complexities was proposed. And the encoding algorithms were designed, whose complexities are linear equations of code length. The construction and encoding algorithms were derived from the effectively encoding characteristics of repeat-accumulate (RA) codes and masking technique. Firstly, the new construction modified parity-check matrices of RA codes to eliminate error floors of RA codes. Secondly, the new constructed parity-check matrices were based on Vandermonde matrices, this deterministic algebraic structure was easy for hardware implementation. Theoretic analysis and experimental results show that, at a bit-error rate of 10×10−4, the new codes with lower encoding complexities outperform Mackay’s random LDPC codes by 0.4–0.6 dB over an additive white Gauss noise (AWGN) channel.

Probabilistic Analysis of Linear Programming Decoding

We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous non-asymptotic results for LDPC codes, and in particular exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a generalized matching. An interesting by-product of our analysis is to establish the existence of ``probabilistic expansion'' in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst-case setting.

Joint linear interleaver design for concatenated zigzag codes

The design of a class of well-structured low-density parity-check (LDPC) codes, namely linear interleaver based concatenated zigzag (LICZ) codes, is investigated. With summary distances as the design metric, short LICZ codes with large minimum distances can be constructed. Moreover, an efficient cycle-based method is proposed to compute the minimum distances of LICZ codes. Simulation results show that LICZ codes outperform both CZ codes with random interleavers and LDPC codes by the progressive edge growth algorithm.

Low density parity check codes for the relay channel

We propose Low Density Parity Check (LDPC) code designs for the half-duplex relay channel. Our designs are based on the information theoretic random coding scheme for decode-and-forward relaying. The source transmission is decoded with the help of side information in the form of additional parity bits from the relay. We derive the exact relationships that the component LDPC code profiles in the relay coding scheme must satisfy. These relationships act as constraints for the density evolution algorithm which is used to search for good relay code profiles. To speed up optimization, we outline a Gaussian approximation of density evolution for the relay channel. The asymptotic noise thresholds of the discovered relay code profiles are a fraction of a decibel away from the achievable lower bound for decode-and-forward relaying. With random component LDPC codes, the overall relay coding scheme performs within 1.2 dB of the theoretical limit.

Structured LDPC codes for high-density recording: large girth and low error floor

High-rate low-density parity-check (LDPC) codes are the focus of intense research in magnetic recording because, when decoded by the iterative sum-product algorithm, they show decoding performance close to the Shannon capacity. However, cycles, especially short cycles, are harmful to LDPC codes. The paper describes the partition-and-shift LDPC (PS-LDPC) codes, a new class of regular, structured LDPC codes that can be designed with large girth and arbitrary large minimum distance. Large girth leads to more efficient iterative decoding and codes with better error-floor properties than random LDPC codes. PS-LDPC codes can be designed for any desired column weight and with flexible code rates. The paper details the girth and distance properties of the codes and their systematic construction and presents analytical and simulation performance results that show that, in the high signal-to-noise ratio region, PS-LDPC codes outperform random codes, alleviating the error floor phenomenon.

Construction of well-structured quasi-cyclic low-density parity check codes

Two classes of well-structured binary low-density parity check codes (LDPC) are described. The first class is based on a branch of combinatorial mathematics, known as the balanced incomplete block design (BIBD). Construction of three types of codes derived from BIBD designs is illustrated in addition to a family of LDPC codes constructed by decomposition of incidence matrices of the proposed BIBD designs. The decomposition technique reduces the density of the parity check matrix and hence reduces the number of short cycles, which generally lead to better performing LDPC codes. The second class of well-structured LDPC codes, Vandermonde or array LDPC codes, are defined by a small number of parameters and cover a large set of code lengths and rates with different column weights. The presented LDPC codes are quasi-cyclic with no cycles of length four; hence simple encoding while maintaining good performance is achieved. Furthermore, the codes are compared with known random LDPC codes, in order to assess their relative achievable performance/complexity trade-offs. It is shown that well-structured LDPC codes perform very similar to the known random LDPC codes.

Multilevel LDPC Codes Design for Multimedia Communication CDMA System

We design multilevel coding (MLC) with a semi-bit interleaved coded modulation (BICM) scheme based on low density parity check (LDPC) codes. Different from the traditional designs, we joined the MLC and BICM together by using the Gray mapping, which is suitable to transmit the data over several equivalent channels with different code rates. To perform well at signal-to-noise ratio (SNR) to be very close to the capacity of the additive white Gaussian noise (AWGN) channel, random regular LDPC code and a simple semialgebra LDPC (SA-LDPC) code are discussed in MLC with parallel independent decoding (PID). The numerical results demonstrate that the proposed scheme could achieve both power and bandwidth efficiency.

Irregular low-density parity-check codes: construction and performance on perpendicular magnetic recording channels

In this paper, we propose an algorithm to design an irregular low-density parity-check code (LDPC), with a given degree distribution pair, from a random regular LDPC code. The proposed algorithm splits columns and rows of the regular code in order to achieve the desired degree distribution pair. Kou, Lin, and Fossorier have used column splitting to increase code rate. However, we use the splitting technique to achieve a given degree distribution pair in the bipartite graph of the resultant irregular code. The bit-error rate (BER) performance of high-rate irregular LDPC codes, constructed using this algorithm, were simulated on equalized perpendicular magnetic recording channels. In addition, this paper compares the performance of these irregular codes with that of random regular LDPC codes.