Variable Node Degree Research Articles

Improved ADMM Penalized Decoder for Irregular Low-Density Parity-Check Codes

Based on linear programming (LP) decoding with alternating direction method of multipliers (ADMM), Liu et al. proposed an ADMM penalized decoder for low-density parity-check (LDPC) codes that can improve the error rate performance at low signal-to-noise ratios. In this letter, we propose a new ADMM penalized decoder to improve the error rate performance further for irregular LDPC codes. The proposed decoder modifies the penalty term of the objective function by assigning different penalty parameters for different variable node degrees. Simulations over three irregular LDPC codes show that the proposed decoder performs considerably better than the ADMM penalized decoder.

Erasure Floor Analysis of Distributed LT Codes

We investigate the erasure floor performance of distributed Luby transform (DLT) codes for transmission within a multi-source, single-relay, and single-destination erasure-link network. In general, Luby transform (LT) codes exhibit a high erasure floor due to poor minimum-distance properties, which can be improved by maximizing the minimum variable-node degree. The same behavior is observed for DLT codes, and therefore a new combining scheme at the relay is proposed to maximize the minimum variable-node degree in the decoding graph. Furthermore, the encoding process at the sources and the combining scheme at the relay are coordinated to improve the transmission overhead. To characterize the asymptotic performance of the proposed DLT codes, we derive closed-form density-evolution expressions, considering both lossless and lossy source-relay channels, respectively. To support the asymptotic analysis, we evaluate the performance of the proposed DLT codes by numerical examples and demonstrate that the numerical results correspond closely to the analysis. Significant improvements in both the erasure floor and transmission overhead are obtained for the proposed DLT codes, as compared to conventional DLT codes.

Towards optimal edge weight distribution and construction of field‐compatible low‐density parity‐check codes over GF( q )

Non-binary low-density parity-check (NB-LDPC) codes can be directly constructed by using algebraic methods, or indirectly constructed by mapping well-designed binary parity-check matrices to non-binary parity-check matrices. Given the Tanner graph (TG) of a NB-LDPC code, the selection of edge weights in the TG significantly affects the performance of the NB-LDPC code. The authors introduce an edge weight distribution (EWD) parameter for the TG of NB-LDPC codes. By utilising particle swarm optimisation (PSO), the EWD is optimised and it has been demonstrated that the optimal EWD approaches a two-element distribution for large field size and high average variable-node degree. With the optimised EWD, the authors construct a class of field-compatible LDPC (FC-LDPC) codes over GF(q) whose parity-check matrices only include elements 0, 1 and 2, and can be encoded and decoded over different field sizes. The simulations demonstrate that the performance of the proposed FC-LDPC codes improves monotonically with increasing field size, and significantly outperforms that of the corresponding algebraic NB-LDPC codes or NB-LDPC codes generated with uniform distribution of non-zero elements over GF(q).

Reliability-Based Iterative Decoding Algorithm for LDPC Codes With Low Variable-Node Degree

In this letter, a new reliability-based iterative majority-logic decoding (RBI-MLGD) algorithm that computes the extrinsic information of previous iterations of the variable node is proposed. This decoding algorithm is called historical-extrinsic reliability-based iterative decoder (HE-RBID) and improves the bit-error-rate performance of the previous RBI-MLGD. HE-RBID is particularly interesting for codes with a low degree of variable node, where traditional RBI-MLGD algorithms do not provide a good behavior. HE-RBID ensures a good performance without searching for the minimum at the check node and only by exchanging 1-bit messages between variable and check nodes.

On Characterization of Elementary Trapping Sets of Variable-Regular LDPC Codes

In this paper, we study the graphical structure of elementary trapping sets (ETSs) of variable-regular low-density parity-check (LDPC) codes. ETSs are known to be the main cause of error floor in LDPC coding schemes. For the set of LDPC codes with a given variable node degree dl and girth g, we identify all the nonisomorphic structures of an arbitrary class of (a, b) ETSs, where a is the number of variable nodes and b is the number of odd-degree check nodes in the induced subgraph of the ETS. This paper leads to a simple characterization of dominant classes of ETSs (those with relatively small values of a and b) based on short cycles in the Tanner graph of the code. For such classes of ETSs, we prove that any set S in the class is a layered superset (LSS) of a short cycle, where the term layered is used to indicate that there is a nested sequence of ETSs that starts from the cycle and grows, one variable node at a time, to generate S. This characterization corresponds to a simple search algorithm that starts from the short cycles of the graph and finds all the ETSs with LSS property in a guaranteed fashion. Specific results on the structure of ETSs are presented for d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sub> = 3, 4, 5, 6, g = 6, 8, and a, b ≤ 10 in this paper. The results of this paper can be used for the error floor analysis and for the design of LDPC codes with low error floors.

Jointly optimized rate-compatible UEP-LDPC codes for half-duplex co-operative relay networks

This paper addresses the design of low-density parity-check (LDPC) codes for half-duplex co-operative relay networks. Structured rate-compatible codes with unequal error protection (UEP) are designed through joint optimization of the codes for the channels between source and relay and source and destination. The proposed codes clearly outperform simpler LDPC codes which are not optimized for relay channels and puncturing-based rate-compatible LDPC codes, and they show a significant performance improvement over the direct link communication depending on the position of relay. The optimization algorithm for the proposed codes is based on density evolution using the Gaussian approximation and optimal variable node degree distributions are found through iterative linear programming. Interestingly, they anyhow show performance which is almost comparable to the performance of codes optimized through a more complex non-linear optimization algorithm. We analyze the performance of our proposed codes with short, medium and long block lengths, and with low and high rates under realistic assumptions, i.e., imperfect decoding of the codeword at relay and variant signal-to-noise ratio within a single codeword.

이중 편파 MIMO를 쓰는 DVB-T2 시스템의 비트 매퍼 성능 분석

초고해상도의 비디오와 다채널 오디오를 통한 극사실적인 방송 서비스를 지향하는 UHDTV (Ultra-High Definition TeleVision)가 차세대 방송 표준으로 논의되고 있으나 기존의 지상파 방송 시스템의 데이터 전송률이 UHDTV 방송 서비스의 요구 전송률에 크게 미치지 못하기 때문에 새로운 지상파 방송 시스템의 개발 또는 현재 시스템의 전송률 증가에 대한 연구가 필요한 상황이다. 이를 위해 기존의 DVB-T2 (Digital Video Broadcasting-2nd generation Terrestrial) 시스템에 고차 성상 변조 및 이중 편파 안테나를 이용한 다중 입출력 시스템을 적용하여 데이터 전송량을 증가시키는 연구가 진행되고 있다. 비정규 LDPC (Low-Density Parity Check) 부호를 사용하는 DVB-T2에 기반한 다중 입출력 시스템의 성능을 최적화하기 위해, 다중 입출력 시스템을 위한 최적의 비트 매퍼 설계에 대한 연구가 필요하다. 그러나 현재까지 진행되어 온 비트 매퍼 설계는 단일 입출력 시스템에 대해 국한되어 있다. 따라서 본 논문에서는 다중 입출력 시스템의 위한 최적의 비트 매퍼를 설계하기 위한 사전 연구로, 이중 편파 다중 입출력 시스템의 VND (Variable Node Degree) 분포를 나타내는 파라미터를 새롭게 정의하고 정의된 파라미터에 따른 시스템의 수신 성능을 분석하였다. The UHDTV system, which provides realistic service with ultra-high definite video and multi-channel audio, has been studied as a next generation broadcasting service. Since the conventional digital terrestrial transmission system is not capable to cover the increased transmission data rate of the UHDTV service, there are great necessity of researches about increase of data rate. Accordingly, the researches has been studied to increase the transmission data rate of the DVB-T2 system using dual-polarized MIMO technique and high order modulation. In order to optimize the MIMO DVB-T2 system where irregular LDPC codes are used, it is necessary to study the design of the bit mapper that matches the LDPC code and QAM symbols in MIMO channel. However, the research related to the design of the bit mapper has been limited to the SISO system. Therefore, this paper defines a new parameter that indicates the VND distribution of MIMO DVB-T2 system and performs the performance analysis according to the parameter which will be helpful for designing a MIMO bit mapper.

EXIT-Chart-Based LDPC Code Design for 2D ISI Channels

In this paper, optimization of low-density parity-check (LDPC) codes to approach the symmetric information rate (SIR) of two-dimensional (2-D) intersymbol interference (ISI) channels is proposed for high-density magnetic recording, such as bit-patterned magnetic recording (BPMR) and 2-D magnetic recording (TDMR). The code design makes use of the modified Extrinsic Information Transfer (EXIT) chart, where the optimal variable node degree is searched by selecting the best check node degree to fit the check node decoder (CND) EXIT curve to the EXIT curve of the variable node decoder (VND) curve combined with 2-D detector. Simulation results show that LDPC codes with code length 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> bits optimized for a 2-D ISI channel corresponding to 4 Tb/in <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> recording density can achieve bit error rate 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> at signal-to-noise ratio 0.33 dB away from the SIR. To our knowledge, this is the first capacity-approaching LDPC code successfully optimized for a 2-D ISI channel.

Interactive Encoding and Decoding Based on Binary LDPC Codes With Syndrome Accumulation

Interactive encoding and decoding based on binary low-density parity-check codes with syndrome accumulation (SA-LDPC-IED) is proposed and investigated. Assume that the source alphabet is <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">GF</b> (2), and the side information alphabet is finite. It is first demonstrated how to convert any classical universal lossless code <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Cn</i> (with block length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> and side information available to both the encoder and decoder) into a universal SA-LDPC-IED scheme. It is then shown that with the word error probability approaching 0 subexponentially with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , the compression rate (including both the forward and backward rates) of the resulting SA-LDPC-IED scheme is upper bounded by a functional of that of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Cn</i> , which in turn approaches the compression rate of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Cn</i> for each and every individual sequence pair ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xn</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">yn</i> ) and the conditional entropy rate H ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> | <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</i> ) for any stationary, ergodic source and side information ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</i> ) as the average variable node degree <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l̅</i> of the underlying LDPC code increases without bound. When applied to the class of binary source and side information ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X</i> , <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Y</i> ) correlated through a binary symmetrical channel with crossover probability unknown to both the encoder and decoder, the resulting SA-LDPC-IED scheme can be further simplified, yielding even improved rate performance versus the bit error probability when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l̅</i> is not large. Simulation results (coupled with linear time belief propagation decoding) on binary source-side information pairs confirm the theoretic analysis and further show that the SA-LDPC-IED scheme consistently outperforms the Slepian-Wolf coding scheme based on the same underlying LDPC code. As a by-product, probability bounds involving LDPC established in the course are also interesting on their own and expected to have implications on the performance of LDPC for channel coding as well.

Windowed Decoding of Spatially Coupled Codes

Spatially coupled codes have been of interest recently owing to their superior performance over memoryless binary-input channels. The performance is good both asymptotically, since the belief propagation thresholds approach the Shannon limit, as well as for finite lengths, since degree-2 variable nodes that result in high error floors can be completely avoided. However, to realize the promised good performance, one needs large blocklengths. This in turn implies a large latency and decoding complexity. For the memoryless binary erasure channel, we consider the decoding of spatially coupled codes through a windowed decoder that aims to retain many of the attractive features of belief propagation, while trying to reduce complexity further. We characterize the performance of this scheme by defining thresholds on channel erasure rates that guarantee a target erasure rate. We give analytical lower bounds on these thresholds and show that the performance approaches that of belief propagation exponentially fast in the window size. We give numerical results including the thresholds computed using density evolution and the erasure rate curves for finite-length spatially coupled codes.

The Cycle Consistency Matrix Approach to Absorbing Sets in Separable Circulant-Based LDPC Codes

For low-density parity-check (LDPC) codes operating over additive white Gaussian noise channels and decoded using message-passing decoders with limited precision, absorbing sets have been shown to be a key factor in error floor behavior. Focusing on this scenario, this paper introduces the cycle consistency matrix (CCM) as a powerful analytical tool for characterizing and avoiding absorbing sets in separable circulant-based (SCB) LDPC codes. SCB codes include a wide variety of regular LDPC codes such as array-based LDPC codes as well as many common quasi-cyclic codes. As a consequence of its cycle structure, each potential absorbing set in an SCB LDPC code has a CCM, and an absorbing set can be present in an SCB LDPC code only if the associated CCM has a nontrivial null space. CCM-based analysis can determine the multiplicity of an absorbing set in an SCB code, and CCM-based constructions avoid certain small absorbing sets completely. While these techniques can be applied to an SCB code of any rate, lower rate SCB codes can usually avoid small absorbing sets because of their higher variable-node degree. This paper focuses attention on the high-rate scenario in which the CCM constructions provide the most benefit. Simulation results demonstrate that under limited-precision decoding the new codes have steeper error-floor slopes and can provide one order of magnitude of improvement in the low-frame-error-rate region.

Stochastic Decoding of LDPC Codes over GF(q)

Despite the outstanding performance of non-binary low-density parity-check (LDPC) codes over many communication channels, they are not in widespread use yet. This is due to the high implementation complexity of their decoding algorithms, even those that compromise performance for the sake of simplicity. In this paper, we present three algorithms based on stochastic computation to reduce the decoding complexity. The first is a purely stochastic algorithm with error-correcting performance matching that of the sum-product algorithm (SPA) for LDPC codes over Galois fields with low order and a small variable node degree. We also present a modified version which reduces the number of decoding iterations required while remaining purely stochastic and having a low per-iteration complexity. The second algorithm, relaxed half-stochastic (RHS) decoding, combines elements of the SPA and the stochastic decoder and uses successive relaxation to match the error-correcting performance of the SPA. Furthermore, it uses fewer iterations than the purely stochastic algorithm and does not have limitations on the field order and variable node degree of the codes it can decode. The third algorithm, NoX, is a fully stochastic specialization of RHS for codes with a variable node degree 2 that offers similar performance, but at a significantly lower computational complexity. We study the performance and complexity of the algorithms; noting that all have lower per-iteration complexity than SPA and that RHS can have comparable average per-codeword computational complexity, and NoX a lower one.

Design of LT Codes with Equal and Unequal Erasure Protection over Binary Erasure Channels

The erasure floor performance of Luby Transform (LT) codes is mainly determined by the minimum variable-node degree. Thus we propose a modified encoding scheme that maximizes the minimum variable-node degree for transmission over binary erasure channels. The proposed scheme leads to an almost-regular variable-node degree distribution. The encoding process is generalized to accommodate arbitrary variable-node degree distributions for additional improved performance. The asymptotic performance is investigated using density evolution and compared with a conventional LT code. The scheme is further extended to enable a higher level of unequal erasure protection.

A PEG Construction of Finite-Length LDPC Codes with Low Error Floor

The progressive-edge-growth (PEG) algorithm of Hu et al. is modified to improve the error floor performance of the constructed low-density parity-check (LDPC) codes. To improve the error floor, the original PEG algorithm is equipped with an efficient algorithm to find the dominant elementary trapping sets (ETS's) that are added to the Tanner graph of the under-construction code by the addition of each variable node and its adjacent edges. The aim is to select the edges, among the candidates available at each step of the original PEG algorithm, that prevent the creation of dominant ETS's. The proposed method is applicable to both regular and irregular variable node degree distributions. Simulation results are presented to demonstrate the superior ETS distribution and error floor performance of the constructed codes compared to similar codes constructed by the original and other modifications of the PEG algorithm.

Efficient Stochastic Decoding of Non-Binary LDPC Codes with Degree-Two Variable Nodes

In this letter, we present an optimized version of the relaxed half-stochastic (RHS) algorithm targeted at non-binary LDPC codes with a variable node degree equal to two. The new algorithm has significantly fewer multiplications while maintaining an error-correction performance and a decoding iteration count identical to those of the original.

Rate-Compatible LDPC Convolutional Codes Achieving the Capacity of the BEC

In this paper, we propose a new family of rate-compatible regular low-density parity-check (LDPC) convolutional codes. The construction is based on graph extension, i.e., the codes of lower rates are generated by successively extending the graph of the base code with the highest rate. Theoretically, the proposed rate-compatible family can cover all the rational rates from 0 to 1. In addition, the regularity of degree distributions simplifies the code optimization. We prove analytically that all the LDPC convolutional codes of different rates in the family are capable of achieving the capacity of the binary erasure channel (BEC). The analysis is extended to the general binary memoryless symmetric channel, for which a capacity-approaching performance can be achieved. Analytical thresholds and simulation results for finite check and variable node degrees are provided for both BECs and binary-input additive white Gaussian noise channels. The results confirm that the decoding thresholds of the rate-compatible codes approach the corresponding Shannon limits over both channels.

Accumulate-Repeat-Delay-Partial-Delay Codes

Inspired by Accumulate-Repeat-Accumulate (ARA) codes, in this letter we propose a new channel coding scheme called Accumulate-Repeat-Delay-Partial-Delay (ARDPD) codes. Contrary to ARA codes, ARDPD codes simultaneously achieve good threshold and linear minimum distance. Based on EXIT for LDPC codes, we show that for maximum variable node degree 5, the threshold of ARDPD codes outperforms the best known unstructured irregular LDPC codes. Iterative decoding simulation results show that the proposed codes perform as predicted.

Successive Maximization for Systematic Design of Universally Capacity Approaching Rate-Compatible Sequences of LDPC Code Ensembles over Binary-Input Output-Symmetric Memoryless Channels

A systematic construction of capacity achieving low-density parity-check (LDPC) code ensemble sequences over the Binary Erasure Channel (BEC) has been proposed by Saeedi et al. based on a method, here referred to as Successive Maximization (SM). In SM, the fraction of degree-i nodes are successively maximized starting from i = 2 with the constraint that the ensemble remains convergent over the channel. In this paper, we propose SM to design universally capacity approaching rate-compatible LDPC code ensemble sequences over the general class of Binary-Input Output-Symmetric Memoryless (BIOSM) channels. This is achieved by first generalizing the SM method to other BIOSM channels to design a sequence of capacity approaching ensembles called the parent sequence. The SM principle is then applied to each ensemble within the parent sequence, this time to design rate-compatible puncturing schemes. As part of our results, we extend the stability condition which was previously derived for degree-2 variable nodes to other variable node degrees as well as to the case of rate-compatible codes. Consequently, we prove that using the SM principle, one is able to design universally capacity achieving rate-compatible LDPC code ensemble sequences over the BEC. Unlike the previous results in the literature, the proposed SM approach is naturally extendable to other BIOSM channels. The performance of the rate-compatible schemes designed based on our method is comparable to those designed by optimization.

New Sequences of Capacity Achieving LDPC Code Ensembles Over the Binary Erasure Channel

In this paper, new sequences <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(\lambda ^{n},\rho ^{n})$</tex> </formula> of capacity achieving low-density parity-check (LDPC) code ensembles over the binary erasure channel (BEC) is introduced. These sequences include the existing sequences by Shokrollahi <etal xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> as a special case. For a fixed code rate <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$R$</tex></formula> , in the set of proposed sequences, Shokrollahi's sequences are superior to the rest of the set in that for any given value of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$n$</tex></formula> , their threshold is closer to the capacity upper bound <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$1- R$</tex></formula> . For any given <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\delta $</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$0 &lt; \delta &lt; 1-R$</tex></formula> , however, there are infinitely many sequences in the set that are superior to Shokrollahi's sequences in that for each of them, there exists an integer number <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n_{0}$</tex> </formula> , such that for any <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n &gt; n_{0}$</tex></formula> , the sequence <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(\lambda ^{n},\rho ^{n})$</tex></formula> requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\delta $</tex></formula> -neighborhood of the capacity upper bound <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$1-R$</tex></formula> . Moreover, it is proven that the check-regular subset of the proposed sequences are asymptotically quasi-optimal, i.e., their decoding complexity increases only logarithmically with the relative increase of the threshold. A stronger result on asymptotic optimality of some of the proposed sequences is also established.

Multilevel LDPC-Coded High-Speed Optical Systems: Efficient Hard Decoding and Code Optimization

We consider a multilevel coding scheme employing low-density parity-check (LDPC) codes and high-order modulations for high-speed optical transmissions, where the coherent receiver performs either parallel independent decoding (PID) or multistage decoding (MSD). To meet the severe complexity constraint imposed by the ultrahigh data rate of the emerging optical transmission systems, we focus on hard-decision decoding of LDPC codes. A new LDPC hard decoding method is developed, which is equivalent to the Gallager decoding algorithm B, but is more efficient in terms of circuit implementation, since no variable node degree information is needed. Two variants of this decoder is also proposed, which offers significant performance gain for finite-length codes. We optimize the system by allocating rates and designing profiles for component codes. Both numerical evaluations and simulation results show that the optimized multilevel coding systems with either PID or MSD substantially outperform the optimized single-level LDPC-coded system.