In this thesis we study sparse classes of graphs and their properties usable for design of algorithms and data structures. Our specific focus is on the con- cepts of bounded expansion and tree-depth, developed in recent years mainly by J. Nešetřil and P. Ossona de Mendez. We first give a brief introduction to the theory as whole and survey tools and results from related areas of parametrised complexity and algorithmic model theory. The main part of the thesis, application of the theory, presents two new dynamic data structures. The first is for keeping a tree-depth decomposition of a graph, the second counts appearances of fixed subgraphs in a given graph. The time and space complexity of operations of both structures is guaranteed to be low when used for sparse graphs. 1

L'utilisation de la modulation à phase continue (CPM) est particulièrement intéressante lorsque le canal de communication comporte une forte non-linéarité et un support spectral limité, en particulier pour la voie aller, lorsque l'on dispose d'un amplificateur par porteuse à bord du satellite, et pour la voie retour où le terminal d'émission travaille à saturation. De nombreuses études ont été effectuées sur le sujet mais les solutions proposées reposent sur la démodulation/décodage itératif des CPM couplées à un code correcteur d'erreur de type convolutif ou bloc. L'utilisation de codes LDPC n'a pas été à ce jour abordée de façon précise. En particulier, il n'existe pas à notre connaissance de travaux sur l'optimisation des codes basés sur des graphes creux adaptés à ce type de schémas. Dans cette étude, nous proposons d'effectuer l'analyse asymptotique et le design d'un schéma Turbo-CPM basé sur des graphes creux. Une étude du récepteur associé comportant les fonctions de démodulation sera également effectuée. / The use of the continuous phase modulation (CPM) is interesting when the channel represents a strong non-linearity and in the case of limited spectral support; particularly for the uplink, where the satellite holds an amplifier per carrier, and for downlinks where the terminal equipment works very close to the saturation region. Numerous studies have been conducted on this issue but the proposed solutions use iterative CPM demodulation/decoding concatenated with convolutional or block error correcting codes. The use of LDPC codes has not yet been introduced. Particularly, no works, to our knowledge, have been done on the optimization of sparse graph-based codes adapted for the context described here. In this study, we propose to perform the asymptotic analysis and the design of turbo-CPM systems based on the optimization of sparse graph-based codes. Moreover, an analysis on the corresponding receiver will be done.

Depuis deux décennies, la révolution technologique est avant tout numérique entrainant une forte croissance de la quantité de données à stocker. Le rythme de cette croissance est trop importante pour les solutions de stockage matérielles, provoquant une augmentation du coût de l'octet. Il est donc nécessaire d'apporter une amélioration des solutions de stockage ce qui passera par une augmentation de la taille des réseaux et par la diminution des copies de sauvegarde dans les centres de stockage de données. L'objet de cette thèse est d'étudier l'utilisation des codes en graphe dans les réseaux de stockage de donnée. Nous proposons un nouvel algorithme combinant construction de codes en graphe et allocation des noeuds de ce code sur le réseau. Cet algorithme permet d'atteindre les hautes performances des codes MDS en termes de rapport entre le nombre de disques de parité et le nombre de défaillances simultanées pouvant être corrigées sans pertes (noté R). Il bénéficie également des propriétés de faible complexité des codes en graphe pour l'encodage et la reconstruction des données. De plus, nous présentons une étude des codes LDPC Spatiallement-Couplés permettant d'anticiper le comportement de leur décodage pour les applications de stockage de données.Il est généralement nécessaire de faire des compromis entre différents paramètres lors du choix du code correcteur d'effacement. Afin que ce choix se fasse avec un maximum de connaissances, nous avons réalisé deux études théoriques comparatives pour compléter l'état de l'art. La première étude s'intéresse à la complexité de la mise à jour des données dans un réseau dynamique établi et déterminons si les codes linéaires utilisés ont une complexité de mise à jour optimale. Dans notre seconde étude, nous nous sommes intéressés à l'impact sur la charge du réseau de la modification des paramètres du code correcteur utilisé. Cette opération peut être réalisée lors d'un changement du statut du fichier (passage d'un caractère hot à cold par exemple) ou lors de la modification de la taille du réseau. L'ensemble de ces études, associé au nouvel algorithme de construction et d'allocation des codes en graphe, pourrait mener à la construction de réseaux de stockage dynamiques, flexibles avec des algorithmes d'encodage et de décodage peu complexes. / For two decades, the numerical revolution has been amplified. The spread of digital solutions associated with the improvement of the quality of these products tends to create a growth of the amount of data stored. The cost per Byte reveals that the evolution of hardware storage solutions cannot follow this expansion. Therefore, data storage solutions need deep improvement. This is feasible by increasing the storage network size and by reducing data duplication in the data center. In this thesis, we introduce a new algorithm that combines sparse graph code construction and node allocation. This algorithm may achieve the highest performance of MDS codes in terms of the ratio R between the number of parity disks and the number of failures that can be simultaneously reconstructed. In addition, encoding and decoding with sparse graph codes helps lower the complexity. By this algorithm, we allow to generalize coding in the data center, in order to reduce the amount of copies of original data. We also study Spatially-Coupled LDPC (SC-LDPC) codes which are known to have optimal asymptotic performance over the binary erasure channel, to anticipate the behavior of these codes decoding for distributed storage applications. It is usually necessary to compromise between different parameters for a distributed storage system. To complete the state of the art, we include two theoretical studies. The first study deals with the computation complexity of data update and we determine whether linear code used for data storage are update efficient or not. In the second study, we examine the impact on the network load when the code parameters are changed. This can be done when the file status changes (from a hot status to a cold status for example) or when the size of the network is modified by adding disks. All these studies, combined with the new algorithm for sparse graph codes, could lead to the construction of new flexible and dynamical networks with low encoding and decoding complexities.

Traditional solutions to information security in communication systems act in the application layer and are oblivious to the effects in the physical layer. Physical-layer security methods, of which information-theoretic security is a special case, try to extract security from the random effects in the physical layer. In information-theoretic security, there are two asymptotic notions of secrecy---weak and strong secrecy This dissertation investigates the problem of information-theoretic strong secrecy on the binary erasure wiretap channel (BEWC) with a specific focus on designing practical codes. The codes designed in this work are based on analysis and techniques from error-correcting codes. In particular, the dual codes of certain low-density parity-check (LDPC) codes are shown to achieve strong secrecy in a coset coding scheme. First, we analyze the asymptotic block-error rate of short-cycle-free LDPC codes when they are transmitted over a binary erasure channel (BEC) and decoded using the belief propagation (BP) decoder. Under certain conditions, we show that the asymptotic block-error rate falls according to an inverse square law in block length, which is shown to be a sufficient condition for the dual codes to achieve strong secrecy. Next, we construct large-girth LDPC codes using algorithms from graph theory and show that the asymptotic bit-error rate of these codes follow a sub-exponential decay as the block length increases, which is a sufficient condition for strong secrecy. The secrecy rates achieved by the duals of large-girth LDPC codes are shown to be an improvement over that of the duals of short-cycle-free LDPC codes.

Digital technology has made complex signal processing possible in communication systems and greatly improved the performance and quality of most modern telecommunication systems. The telecommunication industry and specifically mobile wireless telephone and computer networks have shown phenomenal growth in both the number of subscribers and emerging services, resulting in rapid consumption of common resources of which the electromagnetic spectrum is the most important. Technological advances and research in digital communication are necessary to satisfy the growing demand, to fuel the demand and to exploit all the possibilities and business opportunities. Efficient management and distribution of resources facilitated by state-of-the-art algorithms are indispensable in modern communication networks. The challenge in communication system design is to construct a system that can accurately reproduce the transmitted source message at the receiver. The channel connecting the transmitter and receiver introduces detrimental effects and limits the reliability and speed of information transfer between the source and destination. Typical channel effects encountered in mobile wireless communication systems include path loss between the transmitter and receiver, noise caused by the environment and electronics in the system, and fading caused by multiple paths and movement in the communication channel. In multiple access systems, different users cause interference in each other’s signals and adversely affect the system performance. To ensure reliable communication, methods to overcome channel effects must be devised and implemented in the system. Techniques used to improve system performance and capacity include temporal, frequency, polarisation and spatial diversity. This dissertation is concerned mainly with temporal or time diversity. Channel coding is a temporal diversity scheme and aims to improve the system error performance by adding structured redundancy to the transmitted message. The receiver exploits the redundancy to infer with greater accuracy which message was transmitted, compared with uncoded systems. Sparse graph codes are channel codes represented as sparse probabilistic graphical models which originated in artificial intelligence theory. These channel codes are described as factor graph structures with bit nodes, representing the transmitted codeword bits, and bit-constrained or check nodes. Each constraint involves only a small number of code bits, resulting in a sparse factor graph with far fewer connections between bit and check nodes than the maximum number of possible connections. Sparse graph codes are iteratively decoded using message passing or belief propagation algorithms. Three classes of iteratively decodable channel codes are considered in this study, including low-density parity-check (LDPC), Turbo and repeat-accumulate (RA) codes. The modulation platform presented in this dissertation is a spectrally efficient wideband system employing orthogonal complex spreading sequences (CSSs) to spread information sequences over a wider frequency band in multiple modulation dimensions. Special features of these spreading sequences include their constant envelopes and power output, providing communication range or device battery life advantages. This study shows that multiple layer modulation (MLM) can be used to transmit parallel data streams with improved spectral efficiency compared with single-layer modulation, providing data throughput rates proportional to the number of modulation layers at performances equivalent to single-layer modulation. Alternatively, multiple modulation layers can be used to transmit coded information to achieve improved error performance at throughput rates equivalent to a single layer system / Dissertation (MEng (Electronic Engineering))--University of Pretoria, 2007. / Electrical, Electronic and Computer Engineering / unrestricted

One of the most important technologies used in modern communication systems is channel coding. Channel coding dates back to a paper published by Shannon in 1948 [1] entitled “A Mathematical Theory of Communication”. The basic idea behind channel coding is to send redundant information (parity) together with a message to make the transmission more error resistant. There are different types of codes that can be used to generate the parity required, including block, convolutional and concatenated codes. A special subclass of codes consisting of the codes mentioned in the previous paragraph, is sparse graph codes. The structure of sparse graph codes can be depicted via a graphical representation: the factor graph which has sparse connections between its elements. Codes belonging to this subclass include Low-Density-Parity-Check (LDPC) codes, Repeat Accumulate (RA), Turbo and fountain codes. These codes can be decoded by using the belief propagation algorithm, an iterative algorithm where probabilistic information is passed to the nodes of the graph. This dissertation focuses on noisy decoding of fountain codes using belief propagation decoding. Fountain codes were originally developed for erasure channels, but since any factor graph can be decoded using belief propagation, noisy decoding of fountain codes can easily be accomplished. Three fountain codes namely Tornado, Luby Transform (LT) and Raptor codes were investigated during this dissertation. The following results were obtained: <ol> <li>The Tornado graph structure is unsuitable for noisy decoding since the code structure protects the first layer of parity instead of the original message bits (a Tornado graph consists of more than one layer).</li> <li> The successful decoding of systematic LT codes were verified.</li> <li>A systematic Raptor code was introduced and successfully decoded. The simulation results show that the Raptor graph structure can improve on its constituent codes (a Raptor code consists of more than one code).</li></ol> Lastly an LT code was used to replace the convolutional incremental redundancy scheme used by the 2G mobile standard Enhanced Data Rates for GSM Evolution (EDGE). The results show that a fountain incremental redundancy scheme outperforms a convolutional approach if the frame lengths are long enough. For the EDGE platform the results also showed that the fountain incremental redundancy scheme outperforms the convolutional approach after the second transmission is received. Although EDGE is an older technology, it still remains a good platform for testing different incremental redundancy schemes, since it was one of the first platforms to use incremental redundancy. / Dissertation (MEng)--University of Pretoria, 2008. / Electrical, Electronic and Computer Engineering / MEng / unrestricted