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Fountain codes have demonstrated great results for the binary erasure channel and have already been incorporated into several international standards to recover lost packets at the application layer. These include multimedia broadcast/multicast sessions and digital video broadcasting on global internet-protocol. The rateless property of Fountain codes holds great promise for noisy channels. These are more sophisticated mathematical models representing errors on communications links rather than only erasures. The practical implementation of Fountain codes for these channels, however, is hampered by high decoding cost and delay. In this work we study trapping sets in Fountain codes over noisy channels and their effect on the decoding process. While trapping sets have received much attention for low-density parity-check (LDPC) codes, to our knowledge they have never been fully explored for Fountain codes. Our study takes into account the different code structure and the dynamic nature of Fountain codes. We show that 'error-free' trapping sets exist for Fountain codes. When the decoder is caught in an error-free trapping set it actually has the correct message estimate, but is unable to detect this is the case. Thus, the decoding process continues, increasing the decoding cost and delay for naught. The decoding process for rateless codes consists of one or more decoding attempts. We show that trapping sets may reappear as part of other trapping sets on subsequent decoding attempts or be defeated by the reception of more symbols. Based on our observations we propose early termination methods that use trapping set detection to obtain improvements in realized rate, latency, and decoding cost for Fountain codes. / Thesis (Master, Electrical & Computer Engineering) -- Queen's University, 2009-10-29 14:33:06.548

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.