International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / Low-Density Parity-Check (LDPC) codes are powerful codes capable of nearly achieving the Shannon channel capacity. This paper presents a tutorial introduction to LDPC codes, with a detailed description of the decoding algorithm. The algorithm propagates information about bit and check probabilities through a tree obtained from the Tanner graph for the code. This paper may be useful as a supplement in a course on error-control coding or digital communication.

We consider transmission over a binary erasure wiretap channel using the code construction method introduced by Rathi et al. based on two edge type Low-Density Parity-Check (LDPC) codes and the coset encoding scheme. By generalizing the method of computing conditional entropy for standard LDPC ensembles introduced by Méasson, Montanari, and Urbanke to two edge type LDPC ensembles, we show how the equivocation for the wiretapper can be computed. We find that relatively simple constructions give very good secrecy performance and are close to the secrecy capacity. / <p>Copyright 2010 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. QC 20120110</p>

This thesis focuses on designing Low-Density Parity-Check (LDPC) codes for forward-error-correction. The target application is real-time multimedia communications over packet networks. We investigate two code design issues, which are important in the target application scenarios, designing LDPC codes with low decoding latency, and constructing capacity-approaching LDPC codes with very low error probabilities. On designing LDPC codes with low decoding latency, we present a framework for optimizing the code parameters so that the decoding can be fulfilled after only a small number of iterative decoding iterations. The brute force approach for such optimization is numerical intractable, because it involves a difficult discrete optimization programming. In this thesis, we show an asymptotic approximation to the number of decoding iterations. Based on this asymptotic approximation, we propose an approximate optimization framework for finding near-optimal code parameters, so that the number of decoding iterations is minimized. The approximate optimization approach is numerically tractable. Numerical results confirm that the proposed optimization approach has excellent numerical properties, and codes with excellent performance in terms of number of decoding iterations can be obtained. Our results show that the numbers of decoding iterations of the codes by the proposed design approach can be as small as one-fifth of the numbers of decoding iterations of some previously well-known codes. The numerical results also show that the proposed asymptotic approximation is generally tight for even non-extremely limiting cases. On constructing capacity-approaching LDPC codes with very low error probabilities, we propose a new LDPC code construction scheme based on $2$-lifts. Based on stopping set distribution analysis, we propose design criteria for the resulting codes to have very low error floors. High error floors are the main problems of previously constructed capacity-approaching codes, which prevent them from achieving very low error probabilities. Numerical results confirm that codes with very low error floors can be obtained by the proposed code construction scheme and the design criteria. Compared with the codes by the previous standard construction schemes, which have error floors at the levels of $10^{-3}$ to $10^{-4}$, the codes by the proposed approach do not have observable error floors at the levels higher than $10^{-7}$. The error floors of the codes by the proposed approach are also significantly lower compared with the codes by the previous approaches to constructing codes with low error floors.

This thesis focuses on designing Low-Density Parity-Check (LDPC) codes for forward-error-correction. The target application is real-time multimedia communications over packet networks. We investigate two code design issues, which are important in the target application scenarios, designing LDPC codes with low decoding latency, and constructing capacity-approaching LDPC codes with very low error probabilities. On designing LDPC codes with low decoding latency, we present a framework for optimizing the code parameters so that the decoding can be fulfilled after only a small number of iterative decoding iterations. The brute force approach for such optimization is numerical intractable, because it involves a difficult discrete optimization programming. In this thesis, we show an asymptotic approximation to the number of decoding iterations. Based on this asymptotic approximation, we propose an approximate optimization framework for finding near-optimal code parameters, so that the number of decoding iterations is minimized. The approximate optimization approach is numerically tractable. Numerical results confirm that the proposed optimization approach has excellent numerical properties, and codes with excellent performance in terms of number of decoding iterations can be obtained. Our results show that the numbers of decoding iterations of the codes by the proposed design approach can be as small as one-fifth of the numbers of decoding iterations of some previously well-known codes. The numerical results also show that the proposed asymptotic approximation is generally tight for even non-extremely limiting cases. On constructing capacity-approaching LDPC codes with very low error probabilities, we propose a new LDPC code construction scheme based on $2$-lifts. Based on stopping set distribution analysis, we propose design criteria for the resulting codes to have very low error floors. High error floors are the main problems of previously constructed capacity-approaching codes, which prevent them from achieving very low error probabilities. Numerical results confirm that codes with very low error floors can be obtained by the proposed code construction scheme and the design criteria. Compared with the codes by the previous standard construction schemes, which have error floors at the levels of $10^{-3}$ to $10^{-4}$, the codes by the proposed approach do not have observable error floors at the levels higher than $10^{-7}$. The error floors of the codes by the proposed approach are also significantly lower compared with the codes by the previous approaches to constructing codes with low error floors.

A family of low-density lattice codes (LDLC) is studied based on Construction-A for lattices. The family of Construction-A codes is already known to contain a large capacity-achieving subset. Parallels are drawn between coset non-binary low-density parity-check (LDPC) codes and nested low-density Construction-A lattices codes. Most of the related research in LDPC domain assumes optimal power allocation to encoded codeword. The source coding problem of mapping message to power optimal codeword for any LDPC code is in general, NP-hard. In this thesis, we present a novel method for encoding and decoding lattice based on non-binary LDPC codes using message-passing algorithms. / text

ITC/USA 2007 Conference Proceedings / The Forty-Third Annual International Telemetering Conference and Technical Exhibition / October 22-25, 2007 / Riviera Hotel & Convention Center, Las Vegas, Nevada / Next generation satellite communication systems require efficient coding schemes that enable high data rates, require low overhead, and have excellent bit error rate performance. A newly rediscovered class of block codes called Low Density Parity Check (LDPC) codes has the potential to revolutionize forward error correction (FEC) because of the very high coding rates. This paper presents a brief overview of LDPC coding and decoding. An LDPC algorithm developed by Goddard Space Flight Center is discussed, and an overview of an accompanying VHDL development by L-3 Communications Cincinnati Electronics is presented.

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / With the discovery of Turbo Codes in 1993, interest in developing error control coding schemes that approach channel capacity has intensified. Some of this interest has been focused on lowdensity parity-check (LDPC) codes due to their high performance characteristics and reasonable decoding complexity. A great deal of literature has focused on performance of regular and irregular LDPC codes of various rates and on a variety of channels. This paper presents the simulated performance results of a serial concatenated LDPC coding system on an AWGN channel. Performance and complexity comparisons between this serial LDPC system and typical LDPC systems are made.

International Telemetering Conference Proceedings / October 20-23, 2003 / Riviera Hotel and Convention Center, Las Vegas, Nevada / A joint source-channel coding scheme (JSCCS) used in applications, like sending images, voice, music etc. over internet/ wireless networks, involves source coding to compress the information and channel coding to detect/ correct errors, introduced by the channel. In this paper, we investigate the unequal error protection (UEP) capability of a class of low-density parity-check (LDPC) codes in a JSCCS. This class of irregular LDPC codes is constructed from cyclic difference families (CDFs).

Free-Space Optical (FSO) communication through the terrestrial atmospheric channel offers many benefits in the wireless communications arena, like power efficiency; suitability for secure communications; absence of electromagnetic interference; and potentially very high bandwidth. An optical beam propagating through the atmosphere is subject to optical turbulence. Optical turbulence is a random process that distorts the intensity and phase structure of a propagating optical beam and induces a varying signal at the receiver of an FSO communication link. This phenomenon (usually referred to as scintillation) degrades the performance of the FSO link by increasing the probability of error. In this dissertation we seek to characterize the effects of the scintillation-induced power fluctuations by determining the channel capacity of the optical link using numerical methods. We find that capacity decreases monotonically with increasing turbulence strength in weak turbulence conditions, but it is non-monotonic in strong turbulence conditions. We show that low-density parity-check (LDPC) codes provide strong error control capabilities in this channel if a perfect interleaver is used. Multiple transmit optical beams can be used to reduce scintillation. We characterize the spatial correlation of the atmospheric optical channel and determine a scintillation model for the multiple-beam scheme. With this model we can predict the effective reduction in scintillation as a function of the system design parameters. A Multi-channel FSO communications system based on orbital angular momentum (OAM)-carrying beams is studied. We analyze the effects of turbulence on the system and find that turbulence induces attenuation and crosstalk among OAM channels. Based on a model in which the constituent channels are binary symmetric and crosstalk is a Gaussian noise source, we find optimal sets of OAM states at each turbulence condition studied, and determine the aggregate capacity of the multi-channel system at those conditions. At very high data rates the FSO channel shows inter-symbol interference (ISI). We address the problem of joint sequence detection in ISI channels and decoding of LDPC codes. We derive the belief propagation equations that allow the simultaneous detection and decoding of a LDPC codeword in a ISI channel.

L’objectif de la thèse est l’étude des codes correcteurs d’erreurs au niveau applicatif (Application Layer – Forward Error Correction, ou AL-FEC) pour les communications par satellite. Dans ce contexte, pendant les deux première années de thèse, nous avons proposé de nouvelles méthodes d’analyse, de construction et d’optimisation des codes à effacements définis par des matrices de parité à faible densité (code LDPC, pour « Low Density Parity Check » en anglais). La troisième année de la thèse a été consacrée à : (1) La suite des études portant sur de nouvelles méthodes de construction des codes LDPC non-binaires. D’une part, nous avons développé un nouvel algorithme (Scheduled-PEG) qui permet d’optimiser la construction des codes LDPC non-binaires pas rapport aux métriques de performance spécifiques à la couche application, notamment dans le cadre des systèmes de diffusion de contenu (broadcasting). D’autre part, nous avons proposé une nouvelle méthode de construction de codes à faible rendement, qui utilise l’image binaire étendue d’un code LDPC non-binaire. Ces études ont fait l’objet de deux publications dans deux conférences internationales : (a) “Scheduled-PEG construction of LDPC codes for Upper-Layer FEC”, International Workshop on Coding and Cryptography, April 2011, Paris, France. (b) “Extended Non-Binary Low-Density Parity-Check Codes over Erase Channels”, IEEE International Symposium on Wireless Communication Systems, November 2011, Aachen, Germany. (2) Une étude portant sur l’analyse asymptotique de codes cluster-LDPC non-binaires. Cette nouvelle classe de codes – introduite récemment (ISIT’2011) – se distingue par ses excellentes propriétés en termes de distance minimale. Notre étude a permis de déterminer de manière analytique la capacité de correction des codes cluster-LDPC non-binaires, aussi bien pour le décodage itératif par propagation de croyances (BP, pour « Belief Propagation ») que pour le décodage par maximum de vraisemblance (ML, pour « Maximum Likelihood »). Ces résultats seront intégrés à une publication scientifique sur les codes cluster-LDPC, en cours de rédaction, qui sera soumise à « IEEE Transactions on Information Theory », avant la fin de l’année 2011. (3) Une étude portant sur une méthode de construction des codes LDPC qui permet de réduire de manière significative le plancher d’erreur (« error floor ») du code, sans dégrader ses performances dans la région de « waterfall ». Ainsi, nous avons proposé la structuration de la matrice de parité du code, de manière à intégrer une partie irrégulière, optimisée pour la partie « waterfall », et une partie régulière, qui permet de réduire le plancher d’erreur du code. Cette étude fera l’objet d’une publication dans une conférence internationale (à déterminer), à soumettre début 2012. / The advent of content distribution, IPTV, video-on-demand and other similar services accelerate the demand for reliable data transmission over highly heterogeneous networks and toward terminals potentially heterogeneous too. In this context, Forward Error Correction (FEC) codes that operate at the transport or the Application Layer (AL-FEC) are used in conjunction with the FEC codes implemented at the physical layer, in order to improve the overall performance of the communication system. AL-FEC codes are aimed at recovering erased data packets and they are essential in many multicast/broadcast environments, no matter the way the information is transported, for instance using a wired or wireless link, and a terrestrial, satellite-based or hybrid infrastructure.This thesis addresses the design of Low Density Parity Check (LDPC) codes for AL-FEC applications. One the one hand, we provide an asymptotical analysis of non-binary LDPC codes over erasure channels, as well as waterfall and error-floor optimization techniques for finite-length codes. On the other hand, new concepts and coding techniques are developed in order to fully exploit the potential of non-binary LDPC codes.The first contribution of this thesis consists of the analysis and optimization of two new ensembles of LDPC codes. First, we have derived the density evolution equations for a very general ensemble of non-binary LDPC codes with rank-deficient coefficients. This allows improving the code performance, as well as designing ensembles of LDPC codes that can be punctured in an effective manner. The second approach allows the asymptotical optimization of a particular ensemble of LDPC codes, while ensuring low error-floors at finite lengths.The second contribution is the construction of finite length LDPC codes with good waterfall and error floor performance. Two approaches were investigated, according to the metric used to evaluate the code. The “Scheduled” Progressive Edge Growth (SPEG) algorithm is proposed, in order to optimize the waterfall performance of the code. Another method is proposed which consists in optimizing a specific structure of the parity check matrix. This approach gives low error-floors.The third contribution investigates a new technique of rate adaptability for non-binary LDPC codes. We propose a new method to generate “on-the-fly” incremental redundancy, which allows designing codes with flexible coding rates, in order to cope with severe channel conditions or to enable Fountain-like distribution applications.The fourth contribution focuses on a new class of LDPC codes, called non-binary cluster-LDPC codes. We derive exact equations of the density evolution for the iterative decoding and an upper bound for the maximum-likelihood decoding.Finally, we propose a practical solution to the problem of reliable communication via satellite to high-speed trains. Here, the challenge is that obstacles present along the track regularly interrupt the communication. Our solution offers optimal performance with a minimum amount of redundancy.