The availability of feedback in communication channels can significantly increase the reliability of transmission while decreasing the encoding and decoding complexity. Most of the applications like cellular telephony, satellite communications and internet involve two-way transmission. Hence, it is important to devise coding schemes which utilize the advantages of feedback. Most of the results in code designs, which make use of feedback, concentrate on noiseless and instantaneous feedback. But in real-time systems, the feedback is usually noisy, and is available at the transmitter after some delay. Hence, it is important that we characterize the gains obtained in this case over that of one-way channels. We consider binary erasure channels to keep the problem tractable. For the erasure channels with noisy feedback, we have designed and analyzed a concatenated coding scheme, which achieves lower probability of error than any forward error correcting code of the same rate. Hence, it is shown that even noisy feedback can be useful in increasing the reliability of the channel. We have designed and analyzed a coding scheme using Low Density Parity Check (LDPC) codes along with selective retransmission strategy, which utilizes the limited (but noiseless), delayed feedback to achieve low frame error rates even with small blocklengths, at rates close to capacity. Furthermore, our scheme provides a way to trade off feedback bandwidth for reliability. The complexity of this scheme is lower than that of a forward error correcting code (FEC) of same blocklength and comparable performance. We have shown that our scheme performs better than the Automatic Repeat Request (ARQ) protocol which makes use of 1 bit feedback to signal retransmissions. For fair comparisons, we have also incorporated the rate loss due to the bits which are fed back in addition to the retransmitted bits. Thus, we have shown that for two-way communications with complexity and delay constraints, it is better to utilize the availability of feedback than to use just FEC.

The availability of feedback in communication channels can significantly increase the reliability of transmission while decreasing the encoding and decoding complexity. Most of the applications like cellular telephony, satellite communications and internet involve two-way transmission. Hence, it is important to devise coding schemes which utilize the advantages of feedback. Most of the results in code designs, which make use of feedback, concentrate on noiseless and instantaneous feedback. But in real-time systems, the feedback is usually noisy, and is available at the transmitter after some delay. Hence, it is important that we characterize the gains obtained in this case over that of one-way channels. We consider binary erasure channels to keep the problem tractable. For the erasure channels with noisy feedback, we have designed and analyzed a concatenated coding scheme, which achieves lower probability of error than any forward error correcting code of the same rate. Hence, it is shown that even noisy feedback can be useful in increasing the reliability of the channel. We have designed and analyzed a coding scheme using Low Density Parity Check (LDPC) codes along with selective retransmission strategy, which utilizes the limited (but noiseless), delayed feedback to achieve low frame error rates even with small blocklengths, at rates close to capacity. Furthermore, our scheme provides a way to trade off feedback bandwidth for reliability. The complexity of this scheme is lower than that of a forward error correcting code (FEC) of same blocklength and comparable performance. We have shown that our scheme performs better than the Automatic Repeat Request (ARQ) protocol which makes use of 1 bit feedback to signal retransmissions. For fair comparisons, we have also incorporated the rate loss due to the bits which are fed back in addition to the retransmitted bits. Thus, we have shown that for two-way communications with complexity and delay constraints, it is better to utilize the availability of feedback than to use just FEC.

Low-density parity-check (LDPC) codes are linear error correcting codes which are capable of performing near channel capacity. Furthermore, they admit efficient decoding algorithms that provide near optimum performance. Their main disadvantage is that most LDPC codes have relatively complex encoders. In this thesis, we begin by giving a detailed discussion of the sum-product decoding algorithm, we then study the performance of LDPC codes on the binary erasure channel under sum-product decoding to obtain criteria for the design of codes that allow reliable transmission at rates arbitrarily close to channel capacity. Using these criteria we show how such codes are designed. We then present experimental results and compare them with theoretical predictions. Finally, we provide an overview of several approaches to solving the complex encoder problem.

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This thesis explores the queueing performance of a wireless communication system that transmits packets over a correlated erasure channel using the IEEE 802.11 protocol suit. The channel states and the queue length together form a Markov chain. Exploiting this mathematical structure, the probability of the queue exceeding a certain threshold can be obtained. Most previous contributions in this area treat code-rate selection, channel erasure probability and network congestion separately. In this thesis, a simple integrated approach, which jointly considers these factors, is introduced. This approach becomes especially valuable for capturing the performance of delay-sensitive communication systems over time-varying channels. This thesis starts with a review of related work about correlated bit-erasure wireless channel models. A numerical study is then conducted to demonstrate the importance of optimizing overall system performance, and how this process impacts error-control coding at the physical layer. Following this exercise, a packet-erasure channel model with a Poisson arrival process is analyzed. The Baum-Welch algorithm is subsequently presented as a means to estimate the parameters of wireless communication systems. Furthermore, a matrix geometric method for obtaining the stationary distribution of the ensuing Markov chain is discussed. This offers a new perspective on wireless communication in the context of delay-sensitive applications. To complement the analysis platform put forth in this work, illustrative numerical results are contained in the last section of the thesis. From these results, design guidelines for improving the performance of delay-sensitive wireless communication systems are established. Although these results are obtained under simplifying assumptions, the overall methodology applies to more general situations, especially for wide-band delay-sensitive wireless communication applications.

Ce travail est consacré à la conception, l'analyse et l'optimisation des codes correcteurs d'effacements de niveau applicatif (AL-FEC). Nous nous intéressons à une famille des codes LDPC généralisés (GLDPC), nommés les codes GLDPC-Staircase, qui sont com- posés d'un code LDPC-Staircase (code de base) ainsi que des codes Reed-Solomon (RS) (codes externes). Dans la première partie de cette thèse, nous commençons par montrer que les codes RS ayant une construction basée sur la matrice "quasi" Hankel sont les codes MDS les plus appropriés pour obtenir la structure des codes GLDPC-Staircase. Ensuite, nous proposons un nouveau type de décodage à ces codes, baptisé le décodage hybride (IT/RS/ML), pour atteindre les caspacités de correction du décodage par maximum de vraisemblance (ML) avec de faible complexité. Afin d'étudier l'impact de la structure des codes GLDPC- Staircase sur le décodage, nous proposons une autre construction : ils se diffèrent sur la nature des symboles de redondance LDPC générés. Puis, pour prédire le seuil de décodage et les écarts par rapport à la limite de Shannon des codes GLDPC-Staircase, nous élaborons une analyse asymptotique en fonction de la technique d'évolution de densité (DE), la technique EXIT (Extrinsic Information Transfer) et le théorème d'air. Finalement, en se basant sur l'analyse à taille finie et l'analyse asymptotique, nous ajustons les différentes paramètres internes de ces codes pour obtenir la configuration optimale sous le décodage hybride (IT/RS/ML). La deuxième partie de la thèse est consacrée à l'étude des codes GLDPC-Staircase dans diverses situations. Tout d'abord, nous montrons que ces codes ont des performances asymptotiquement très proches de la limite de Shannon. En plus, à taille finie, ils permet- tent d'atteindre d'excellentes capacités de correction d'effacements (i.e., très proches de celles des codes MDS) peu importe la taille des objets : très faible overhead de décodage, faible plancher d'erreur, et une zone "waterfall" raide. Nous montrons aussi que les performances de ces codes sont trés proches des codes RaptorQ et surpassent celles des codes Raptor, les codes LDPC-Staircase, et un autre code GLDPC avec une construction différente. Finalement, nous proposons une méthodologie générale pour étudier l'impact de l'ordonnancement des paquets envoyés sur les performances des codes GLDPC-Staircase sur un grand nombre des canaux à effacements (avec des pertes en rafale ou pas). Cette étude montre le meilleur ordonnancement de paquets. Tous les résultats mentionnés ci-dessus montrent que les codes GLDPC-Staircase peuvent considérés comme des codes AL-FEC universels.

L'objet de cette thèse est l'étude des codes LDPC quantiques. Dans un premier temps, nous travaillons sur des constructions topologiques de codes LDPC quantiques. Nous proposons de construire une famille de codes couleur basée sur des pavages hyperboliques. Nous étudions ensuite les paramètres d'une famille de codes basée sur des graphes de Cayley.Dans une seconde partie, nous examinons les performances de ces codes. Nous obtenons une borne supérieure sur les performances des codes LDPC quantiques réguliers sur le canal à effacement quantique. Ceci prouve que ces codes n'atteignent pas la capacité du canal à effacement quantique. Dans le cas du canal de dépolarisation, nous proposons un nouvel algorithme de décodage des codes couleur basé sur trois décodages de codes de surface. Nos simulations numériques montrent de bonnes performances dans le cas des codes couleur toriques.Pour finir, nous nous intéressons au phénomène de percolation. La question centrale de la théorie de la percolation est la détermination du seuil critique. Le calcul exacte de ce seuil est généralement difficile. Nous relions la probabilité de percolation dans certains pavages réguliers du plan hyperbolique à la probabilité d'erreur de décodage pour une famille de codes hyperboliques. Nous en déduisons une borne sur le seuil critique de ces pavages hyperboliques basée sur des résultats de théorie de l'information quantique. Il s'agit d'une application de la théorie de l'information quantique à un problème purement combinatoire. / This thesis is devoted to the study of quantum LDPC codes. The first part presents some topological constructions of quantum LDPC codes. We introduce a family of color codes based on tilings of the hyperbolic plane. We study the parameters of a family of codes based on Cayley graphs.In a second part, we analyze the performance of these codes. We obtain an upper bound on the performance of regular quantum LDPC codes over the quantum erasure channel. This implies that these codes don't achieve the capacity of the quantum erasure channel. In the case of the depolarizing channel, we propose a new decoding algorithm of color codes based on three surface codes decoding. Our numerical results show good performance for toric color codes.Finally, we focus on percolation theory. The central question in percolation theory is the determination of the critical probability. Computing the critical probability exactly is usually quite difficult. We relate the probability of percolation in some regular tilings of the hyperbolic plane to the probability of a decoding error for hyperbolic codes on the quantum erasure channel. This leads to an upper bound on the critical probability of these hyperbolic tilings based on quantum information. It is an application of quantum information to a purely combinatorial problem.

This research is concerned with the reliable delivery of scalable compressed data over lossy communication channels. Recent works proposed several strategies for assigning optimal code redundancies to elements of scalable data, which form a linear structure of dependency, under the assumption that all source elements are encoded onto a common group of network packets. Given large data and small network packets, such schemes require very long channel codes with high computational complexity. In networks with high loss, small packets are more desirable than long packets. The first contribution of this thesis is to propose a strategy for optimally assigning elements of the scalable data to clusters of packets, subject to constraints on packet size and code complexity. Given a packet cluster arrangement, the scheme then assigns optimal code redundancies to the source elements, subject to a constraint on transmission length. Experimental results show that the proposed strategy can outperform the previous code assignment schemes subject to the above-mentioned constraints, particularly at high channel loss rates. Secondly, we modify these schemes to accommodate complex structures of dependency. Source elements are allocated to clusters of packets according to their dependency structure, subject to constraints on packet size and channel codeword length. Given a packet cluster arrangement, the proposed schemes assign optimal code redundancies to the source elements, subject to a constraint on transmission length. Experimental results demonstrate the superiority of the proposed strategies for correctly modelling the dependency structure. The last contribution of this thesis is to propose a scheme for optimizing protection of scalable data where limited retransmission is possible. Previous work assumed that retransmission is not possible. For most real-time or interactive applications, however, retransmission of lost data may be possible up to some limit. In the present work we restrict our attention to streaming sources (e.g., video) where each source element can be transmitted in one or both of two time slots. An optimization algorithm determines the transmission and level of protection for each source element, using information about the success of earlier transmissions. Experimental results confirm the benefit of limited retransmission.

Orientador: Marcelo Firer / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-23T13:47:51Z (GMT). No. of bitstreams: 1 Lemes_LeandroCruvinel_D.pdf: 1250342 bytes, checksum: e1a6a639b2a96ae11f0c70d7507040aa (MD5) Previous issue date: 2013 / Resumo: Considerando canais discretos, sem memória e com apagamento, obtemos limitantes superiores e inferiores para as probabilidades de erro de decodificação e de ocorrências de ambiguidade de códigos corretores de erro lineares. Os limitantes dependem da hierarquia de pesos e dos espectros generalizados e melhoram os limitantes conhecidos. Encontramos expressões exatas para essas probabilidades nos casos em que o código é AMDS ou MDS / Abstract: Considering an erasure channel, we improve upper and lower bounds for error decoding and ambiguity probabilities of linear error-correcting codes. The given bounds depend on the generalized weight hierarchy and spectrum of a code. We find explicit formulae in the case of AMDS and MDS codes / Doutorado / Matematica / Doutor em Matemática

Ce travail est dédié à la conception, l’analyse et l’évaluation des performances de nouveaux schémas de codage appropriés aux systèmes de stockage distribué. La première partie de ce travail est consacrée à l’étude des performances des codes spatialement couplés pour les canaux à effacements. Une nouvelle méthode de couplage spatial des ensembles classiques de contrôle de parité à faible densité (LDPC) est proposée. La méthode est inspirée du codage en couches. Les arêtes des ensembles locaux et celles définissant le couplage spatial sont construites séparément. Nous proposons également de saturer le seuil d’un ensemble Root-LDPC par couplage spatial de ses bits de parité dans le but de faire face aux évanouissements quasi-statiques. Le couplage spatial est dans un deuxième temps appliqué à un ensemble Root-LDPC, ayant une double diversité, conçu pour un canal à effacements par blocs à 4 états. Dans la deuxième partie de ce travail, nous considérons les codes produits non-binaires avec des composantes MDS et leur décodage algébrique itératif ligne-colonne sur un canal à effacements. Les effacements indépendants et par blocs sont considérés. Une représentation graphique compacte du code est introduite avec laquelle nous définissions la notion de coloriage à double diversité. Les ensembles d’arrêt sont définis et une caractérisation complète est donnée. La performance des codes produits à composantes MDS, avec et sans coloration, à double diversité est analysée en présence d’effacements indépendants et par blocs. Les résultats numériques montrent aussi une excellente performance en présence d’effacements à probabilité inégale due au coloriage ayant une double diversité. / This work is dedicated to the design, analysis, and the performance evaluation of new coding schemes suitable for distributed storage systems. The first part is devoted to spatially coupled codes for erasure channels. A new method of spatial coupling for low-density parity-check ensembles is proposed. The method is inspired from overlapped layered coding. Edges of local ensembles and those defining the spatial coupling are separately built. We also propose to saturate the whole Root-LDPC boundary via spatial coupling of its parity bits to cope with quasi-static fading. Then, spatial coupling is applied on a Root-LDPC ensemble with double diversity designed for a channel with 4 block-erasure states. In the second part of this work, we consider non-binary product codes with MDS components and their iterative row-column algebraic decoding on the erasure channel. Both independent and block erasures are considered. A compact graph representation is introduced on which we define double-diversity edge colorings via the rootcheck concept. Stopping sets are defined and a full characterization is given in the context of MDS components. A differential evolution edge coloring algorithm that produces colorings with a large population of minimal rootcheck order symbols is presented. The performance of MDS-based product codes with and without double-diversity coloring is analyzed in presence of both block and independent erasures. Furthermore, numerical results show excellent performance in presence of unequal erasure probability due to double-diversity colorings.