Spatially-Coupled Code Design for Partial-Response Channels: Optimal Object-Minimization Approach

Spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern storage devices. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for AWGN and Flash channels. The OO partitioning stage operates on the protograph of the SC code, while the CPO optimizes the circulant powers, in order to minimize the number of detrimental objects. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by around 3 orders of magnitude in FER and 1.1 dB in SNR, and more intriguingly, outperform structured block codes of the same length by around 1.6 orders of magnitude in FER and 0.4 dB in SNR.

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title: Coding and Signal Processing for Magnetic Recording Systems
abstract: RECORDING SYSTEMS A BriefHistory of Magnetic Storage, Dean Palmer Physics of Longitudinal and Perpendicular Recording, Hong Zhou, Tom Roscamp, Roy Gustafson, Eric Boernern, and Roy Chantrell The Physics of Optical Recording, William A. Challener and Terry W. McDaniel Head Design Techniques for Recording Devices, Robert E. Rottmayer COMMUNICATION AND INFORMATION THEORY OF MAGNETIC RECORDING CHANNELS Modeling the Recording Channel, Jaekyun Moon Signal and Noise Generation for Magnetic Recording Channel Simulations, Xueshi Yang and Erozan M. Kurtas Statistical Analysis of Digital Signals and Systems, Dragana Bajic and Dusan Drajic Partial Response Equalization with Application to High Density Magnetic Recording Channels, John G. Proakis An Introduction to Error-Correcting Codes, Mario Blaum. Message-Passing Algorithm, Sundararajan Sankaranarayanan and Bane Vasic Modulation Codes for Storage Systems, Brian Marcus and Emina Soljanin Information Theory of Magnetic Recording Channels, Zheng Zhang, Tolga M. Duman, and Erozan M. Kurtas Capacity of Partial Response Channels, Shaohua Yang and Aleksandar Kavcic  Binary Codes with Error Correction Capability, Hendrik C. Ferreira and Willem A. Clarke Constrained Coding and Error-Control Coding, John L. Fan Convolutional Codes for Partial-Response Channels, Bartolomeu F. Uchoa-Filho, Mark A. Herro, Miroslav Despotovic and Vojin Senk Capacity-Approaching Codes for Partial Response Channels, Nedeljko Varnica, Xiao Ma, and Aleksandar Kavcic Coding and Detection for Multitrack Systems, Bane Vasic and Olgica Milenkovic Two-Dimensional Data Detection and Error Control, Brian M. King and Mark A. Neifeld SIGNAL PROCESSING FOR READ CHANNELS Adaptive Timing Recovery for Partial Response Channels, Pervez M. Aziz and Viswanath Annampedu Interpolated Timing Recovery, Piya Kovintavewat, John R. Barry, M. Fatih Erden, and Erozan M. Kurtas Adaptive Equalization Architectures for Partial Response Channels, Pervez M. Aziz Head Position Estimation, Ara Patapoutian Servo Signal Processing, Pervez M. Aziz Data Detection, Miroslav Despotovic and Vojin Senk Detection Methods for Data-dependent Noise in Storage Channels, Erozan M. Kurtas, Jongseung Park, Xueshi Yang, William Radich, and Aleksandar Kavcic Read/Write Channel Implementation, Borivoje Nikolic, Michael Leung, Engling Yeo, and Kiyoshi Fukahori ITERATIVE DECODING Turbo Codes, Mustafa N. Kaynak, Tolga M. Duman, and Erozan M. Kurtas An Introduction to LDPC Codes, William E. Ryani Concatenated Single-Parity Check Codes for High-Density Digital Recording Systems, Jing Li, Krishna R. Narayanan, Erozan Kurtas, and Travis R. Oenning Structured Low-Density Parity-Check Codes, Bane Vasic, Erozan M. Kurtas, Alexander Kuznetsov, and Olgica Milenkovic Turbo Coding for Multitrack Recording Channels, Zheng Zhang, Tolga M. Duman, and Erozan M. Kurtas

title: Non-Binary LDPC Codes for Magnetic Recording Channels: Error Floor Analysis and Optimized Code Design
abstract: In this paper, we provide a comprehensive analysis of the error floor along with code optimization guidelines for structured and regular non-binary low-density parity-check (NB-LDPC) codes in magnetic recording (MR) applications. While the topic of the error floor performance of binary LDPC codes over additive white Gaussian noise (AWGN) channels has recently received considerable attention, very little is known about the error floor performance of NB-LDPC codes over other types of channels, despite the early results demonstrating superior characteristics of NB-LDPC codes relative to their binary counterparts. We first show that, due to the outer looping between the detector and the decoder in the receiver, the error profile of NB-LDPC codes over partial-response (PR) channels is qualitatively different from the error profile over AWGN channels-this observation motivates us to introduce new combinatorial objects aimed at capturing decoding errors that dominate the PR channel error floor region. We call these objects balanced absorbing sets (BASs), which are viewed as a special subclass of previously introduced absorbing sets (ASs). Aided by these new objects (BASs), we develop a method that combines analytical equations and biased simulations to predict the error floor performance of NB-LDPC codes over PR channels without the need to execute extensive Monte Carlo (MC) simulations. We show that explicitly incorporating the inter-symbol interference of MR channels into our prediction method makes the accuracy of the error floor estimate within 0.2 of an order of magnitude from the traditional MC simulation. In addition, we prove that, due to the more restrictive definition of BASs (relative to the more general class of ASs), an additional degree of freedom can be exploited in the code design for PR channels. We then demonstrate that the proposed code optimization aimed at removing dominant BASs offers performance improvements in the frame error rate in the error floor region by up to 2.5 orders of magnitude over the unoptimized designs. Our code optimization technique carefully, yet provably, removes BASs from the code while preserving its overall structure (node degree, quasi-cyclic property, regularity, and so forth). The resulting codes outperform the existing binary and NB-LDPC solutions for PR channels by about 2.5 and 1.25 orders of magnitude, respectively.

title: On the application of factor graphs and the sum-product algorithm to ISI channels
abstract: In this paper, based on the application of the sum-product (SP) algorithm to factor graphs (FGs) representing the joint a posteriori probability (APP) of the transmitted symbols, we propose new iterative soft-input soft-output (SISO) detection schemes for intersymbol interference (ISI) channels. We have verified by computer simulations that the SP algorithm converges to a good approximation of the exact marginal APPs of the transmitted symbols if the FG has girth at least 6. For ISI channels whose corresponding FG has girth 4, the application of a stretching technique allows us to obtain an equivalent girth-6 graph. For sparse ISI channels, the proposed algorithms have advantages in terms of complexity over optimal detection schemes based on the Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm. They also allow a parallel implementation of the receiver and the possibility of a more efficient complexity reduction. The application to joint detection and decoding of low-density parity-check (LDPC) codes is also considered and results are shown for some partial-response magnetic channels. Also in these cases, we show that the proposed algorithms have a limited performance loss with respect to that can be obtained when the optimal "serial" BCJR algorithm is used for detection. Therefore, for their parallel implementation, they represent a favorable alternative to the modified "parallel" BCJR algorithm proposed in the literature for the application to magnetic channels.

title: Non-Binary Protograph-Based LDPC Codes for 2-D-ISI Magnetic Recording Channels
abstract: 2-D inter-symbol-interference (2-D-ISI), consisting of ISI in the down track direction and inter-track interference along the cross-track direction, is a major factor that severely degrades the performance of ultra-high density magnetic recording systems. Due to its superior performance and the protograph structure which facilitates high-speed encoding and decoding, the protograph codes have shown a high potential to tackle 2-D-ISI. However, so far, no work has been reported on the design of non-binary protograph-based low-density parity-check (NP-LDPC) codes for 2-D-ISI channels. In this paper, we first propose a modified protograph extrinsic information transfer analysis. In conjunction with the asymptotic ensemble weight distribution analysis, they both serve as theoretical tools to analyze the performance of NP-LDPC codes designed for the 2-D-ISI channels. By further applying a fast search approach, we construct two types of NP-LDPC codes for 2-D-ISI channels. Both theoretical analyses and simulated results show that the proposed codes outperform the existing binary LDPC codes that optimized for 2-D-ISI channels as well as the non-binary quasi-cyclic LDPC code, at both low and high SNR regions.

title: A General Non-Binary LDPC Code Optimization Framework Suitable for Dense Flash Memory and Magnetic Storage
abstract: Transmission channels underlying modern dense storage systems, e.g., Flash memory and magnetic recording (MR) systems, significantly differ from canonical channels, like additive white Gaussian noise (AWGN) channels. While existing low-density parity-check (LDPC) codes optimized for symmetric, AWGN-like channels are being actively considered for Flash applications, we demonstrate that, due to channel asymmetry, such approaches are inadequate. We introduce a refined definition of absorbing sets, which we call general absorbing sets of type two (GASTs), and study the combinatorial properties of GASTs. We then present the weight consistency matrix (WCM), which succinctly captures key properties in a GAST. Furthermore, we show how to customize the WCM definition such that it suits other special subclasses of GASTs. Based on these new concepts, we then develop a new, general combinatorial code optimization framework, which we call the WCM framework, and demonstrate its effectiveness on the realistic highly-asymmetric normal-Laplace mixture (NLM) Flash channel. Moreover, we show that our framework can be customized to optimize non-binary LDPC (NB-LDPC) codes for other asymmetric channels, channels with memory (incorporated in MR systems), and canonical symmetric channels. For all the channels we have simulated NB-LDPC codes over, the codes optimized using the WCM framework enjoy at least 1 order, and up to nearly 2 orders of magnitude performance gain in the uncorrectable bit error rate (UBER) or the frame error rate (FER) relative to the unoptimized codes. Our simulations also show that codes optimized for symmetric channels are not the best choice for asymmetric channels.

title: A Combinatorial Methodology for Optimizing Non-Binary Graph-Based Codes: Theoretical Analysis and Applications in Data Storage
abstract: Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbate the error floor problem by widening the spectrum of problematic objects that contribute to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. In addition, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with additional soft information, and spatially coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels.

title: Time-varying periodic convolutional codes with low-density parity-check matrix
abstract: We present a class of convolutional codes defined by a low-density parity-check matrix and an iterative algorithm for decoding these codes. The performance of this decoding is close to the performance of turbo decoding. Our simulation shows that for the rate R=1/2 binary codes, the performance is substantially better than for ordinary convolutional codes with the same decoding complexity per information bit. As an example, we constructed convolutional codes with memory M=1025, 2049, and 4097 showing that we are about 1 dB from the capacity limit at a bit-error rate (BER) of 10/sup -5/ and a decoding complexity of the same magnitude as a Viterbi decoder for codes having memory M=10.

title: Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation
abstract: We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felstrom and Zigangirov, perform so well over this channel. We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and a gap to capacity, we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding. In fact, most codes in this ensemble have this property. The quantifier universal refers to the single ensemble/code that is good for all channels but we assume that the channel is known at the receiver. The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble. We conclude by discussing some interesting open problems.

title: Spatially Coupled Codes Optimized for Magnetic Recording Applications
abstract: 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.

title: Windowed Decoding of Protograph-Based LDPC Convolutional Codes Over Erasure Channels
abstract: We consider a windowed decoding scheme for LDPC convolutional codes that is based on the belief-propagation (BP) algorithm. We discuss the advantages of this decoding scheme and identify certain characteristics of LDPC convolutional code ensembles that exhibit good performance with the windowed decoder. We will consider the performance of these ensembles and codes over erasure channels with and without memory. We show that the structure of LDPC convolutional code ensembles is suitable to obtain performance close to the theoretical limits over the memoryless erasure channel, both for the BP decoder and windowed decoding. However, the same structure imposes limitations on the performance over erasure channels with memory.

title: A novel combinatorial framework to construct spatially-coupled codes: Minimum overlap partitioning
abstract: Spatially-coupled (SC) codes are a family of graph-based codes that have attracted significant attention thanks to their capacity approaching performance. An SC code is constructed by partitioning an underlying block code into a number of components, and coupling their copies together. The number of components is determined by the memory parameter. In this paper, we study a finite length construction for the circulant-based SC codes. We introduce a new partitioning scheme, which we call minimum overlap partitioning, that outperforms previous methods. We also present a general approach for the enumeration of problematic objects in the error-floor regime that can be applied to any circulant-based SC code and to a variety of partitioning schemes. Compared to the uncoupled block codes, an SC code constructed by the new approach has more than 1.5 and 3 orders of magnitude performance improvement for the memory 1 and 2, respectively. Additionally, it outperforms the existing method of partitioning via cutting vectors by at least half an order of magnitude; this performance advantage becomes more pronounced for SC codes with higher memories.

title: Edge spreading design of high rate array-based SC-LDPC codes
abstract: Absorbing sets (ASs) are combinatorially defined objects existing in the Tanner graph of a low-density parity-check (LDPC) code that have been shown to cause failures in the iterative message-passing decoder when transmission occurs over the additive white Gaussian noise channel. In this paper, we propose an edge spreading approach to construct high rate array-based spatially-coupled LDPC codes by jointly optimizing the AS spectrum and the minimum distance. By considering general edge spreadings and by considering a larger memory, we show that strictly better codes can be constructed, both in terms of achievable minimum distance for small-to-moderate block lengths and in terms of the number of small ASs.

title: Finite-Length Construction of High Performance Spatially-Coupled Codes via Optimized Partitioning and Lifting
abstract: Spatially-coupled (SC) codes are a family of graph-based codes that have attracted significant attention, thanks to their capacity approaching performance and low decoding latency. An SC code is constructed by partitioning an underlying block code into a number of components and coupling their copies together. In this paper, we first introduce a general approach for the enumeration of detrimental combinatorial objects in the graph of finite-length SC codes. Our approach is general in the sense that it effectively works for SC codes with various partitioning schemes, column weights, and memories. Next, we present a two-stage framework for the construction of high performance binary SC codes optimized for the additive white Gaussian noise channels; we aim at minimizing the number of detrimental combinatorial objects in the error floor region. In the first stage, we deploy a novel partitioning scheme, called the optimal overlap partitioning, to produce the optimal partitioning corresponding to the smallest number of detrimental objects. In the second stage, we apply a new circulant power optimizer to further reduce the number of detrimental objects in the lifted graph. SC codes constructed by our new framework have up to two orders of magnitude error floor performance improvement and up to 0.6 dB SNR gain compared to prior state-of-the-art SC codes.

title: High performance non-binary spatially-coupled codes for flash memories
abstract: Modern dense Flash memory devices operate at very low error rates, which require powerful error correcting coding (ECC) techniques. An emerging class of graph-based ECC techniques that has broad applications is the class of spatially-coupled (SC) codes, where a block code is partitioned into components that are then rewired multiple times to construct an SC code. Here, our focus is on SC codes with the underlying circulant-based structure. In this paper, we present a three-stage approach for the design of high performance non-binary SC (NB-SC) codes optimized for practical Flash channels; we aim at minimizing the number of detrimental general absorbing sets of type two (GASTs) in the graph of the designed NB-SC code. In the first stage, we deploy a novel partitioning mechanism, called the optimal overlap partitioning, which acts on the protograph of the SC code to produce optimal partitioning corresponding to the smallest number of detrimental objects. In the second stage, we apply a new circulant power optimizer to further reduce the number of detrimental GASTs. In the third stage, we use the weight consistency matrix framework to manipulate edge weights to eliminate as many as possible of the GASTs that remain in the NB-SC code after the first two stages (that operate on the unlabeled graph of the code). Simulation results reveal that NB-SC codes designed using our approach outperform state-of-the-art NB-SC codes when used over Flash channels.

title: Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices
abstract: In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than 12, and a relatively mild necessary and sufficient condition for the code to have a girth of 6, 8,10, or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.

title: Spatially-Coupled Code Design for Partial-Response Channels: Optimal Object-Minimization Approach
abstract: Spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern storage devices. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for AWGN and Flash channels. The OO partitioning stage operates on the protograph of the SC code, while the CPO optimizes the circulant powers, in order to minimize the number of detrimental objects. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by around 3 orders of magnitude in FER and 1.1 dB in SNR, and more intriguingly, outperform structured block codes of the same length by around 1.6 orders of magnitude in FER and 0.4 dB in SNR.

title: EXIT Chart Analysis of Nonbinary Protograph LDPC Codes for Partial Response Channels
abstract: Low-density parity-check (LDPC) codes over finite fields GF(q) provide an error correction in the noisy partial response (PR) channels. The extrinsic information transfer (EXIT) chart can be used to predict the threshold decoding of protograph LDPC codes, however, previous works only consider the binary protograph LDPC codes in the PR channels. In this paper, we propose to perform the EXIT chart analysis on the nonbinary protograph LDPC codes for the PR channels. Unlike prior works, the actual extrinsic information of the channel detector is measured, then the extrinsic information to the variable nodes is generated with the measured statistics. Moreover, since the mutual information of the variable nodes depends on GF(q), we use the Monte Carlo method to approximate the mutual information. The analysis on the regular (2,4) code, regular (3,6) code, RA code, and AR3A code on the PR channels reveal that, for the PR! channel, the RA code outperforms the others for q = 2, 4, and 8, but the regular (2,4) code is the best for q = 16 and 32. For the PR2 channel, the RA code is the best code for q = 2 and 4, but the regular (2,4) code is the best code for q > 4. The simulation of the codes for q = 4 and 16 are then used to confirm the theoretical results.

title: A Communication-Theoretic Framework for 2-DMR Channel Modeling: Performance Evaluation of Coding and Signal Processing Methods
abstract: We develop a communication theoretic framework for modeling 2-D magnetic recording channels. Using the model, we define the signal-to-noise ratio (SNR) for the channel considering several physical parameters, such as the channel bit density, code rate, bit aspect ratio, and noise parameters. We analyze the problem of optimizing the bit aspect ratio for maximizing SNR. The read channel architecture comprises a novel 2-D joint self-iterating equalizer and detection system with noise prediction capability. We evaluate the system performance based on our channel model through simulations. The coded performance with the 2-D equalizer detector indicates ~ 5.5 dB of SNR gain over uncoded data.

title: Turbo decoding for partial response channels
abstract: The partial response channel can be viewed as a rate-1 encoder in which the output alphabet differs from the input alphabet. In serially concatenated coding schemes, the partial response channel can serve as the inner encoder. Previous work on the application of turbo decoding techniques to partial response channels has focused on using a parallel concatenation of convolutional encoders as the outer code and the partial response channel as the inner code. This system requires three a posteriori probability (APP) detectors-one matched to the channel and two matched to the constituent encoders. A simplified system is presented that uses as its outer code a single convolutional code and as its inner code the partial response channel. The simplified system requires only two APP detectors, offering significant savings in complexity and computation time. This single convolutional code system is shown to perform as well as the more complicated system, offering substantial gains over uncoded systems. Simulation results for three magnetic recording channel models are presented: a partial response channel with additive white Gaussian noise, an equalized Lorentzian channel model, and a media noise model called the microtrack model. Since the use of an outer Reed-Solomon code is anticipated in an actual system, the burst-error statistics are investigated. System performance with various interleaver designs and precoders is also investigated.

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