Secure symmetrical multilevel diversity coding (S-SMDC) is a source coding problem, where a total of L - N discrete memoryless sources (S1,...,S_L-N) are to be encoded by a total of L encoders. This thesis considers a natural generalization of SMDC to the secure communication setting with an additional eavesdropper. In a general S-SMDC system, a legitimate receiver and an eavesdropper have access to a subset U and A of the encoder outputs, respectively. Which subsets U and A will materialize are unknown a priori at the encoders. No matter which subsets U and A actually occur, the sources (S1,...,Sk) need to be perfectly reconstructable at the legitimate receiver whenever |U| = N +k, and all sources (S1,...,S_L-N) need to be kept perfectly secure from the eavesdropper as long as |A| <= N. A precise characterization of the entire admissible rate region is established via a connection to the problem of secure coding over a three-layer wiretap network and utilizing some properties of basic polyhedral structure of the admissible rate region. Building on this result, it is then shown that superposition coding remains optimal in terms of achieving the minimum sum rate for the general secure SMDC problem.

Symmetrical multilevel diversity coding (SMDC) is a classical model for coding over distributed storage. In this setting, a simple separate encoding strategy known as superposition coding was shown to be optimal in terms of achieving the minimum sum rate and the entire admissible rate region of the problem in the literature. The proofs utilized carefully constructed induction arguments, for which the classical subset entropy inequality of Han played a key role. This thesis includes two parts. In the first part the existing optimality proofs for classical SMDC are revisited, with a focus on their connections to subset entropy inequalities. First, a new sliding-window subset entropy inequality is introduced and then used to establish the optimality of superposition coding for achieving the minimum sum rate under a weaker source-reconstruction requirement. Second, a subset entropy inequality recently proved by Madiman and Tetali is used to develop a new structural understanding to the proof of Yeung and Zhang on the optimality of superposition coding for achieving the entire admissible rate region. Building on the connections between classical SMDC and the subset entropy inequalities developed in the first part, in the second part the optimality of superposition coding is further extended to the cases where there is an additional all-access encoder, an additional secrecy constraint or an encoder hierarchy.

Symmetrical Multilevel Diversity Coding (SMDC) is a network compression problem for which a simple separate coding strategy known as superposition coding is optimal in terms of achieving the entire admissible rate region. Carefully constructed induction argument along with the classical subset entropy inequality of Han played a key role in proving the optimality. This thesis considers a generalization of SMDC for which, in addition to the randomly accessible encoders, there is also an all-access encoder. It is shown that superposition coding remains optimal in terms of achieving the entire admissible rate region of the problem. Key to our proof is to identify the supporting hyperplanes that define the boundary of the admissible rate region and then build on a generalization of Han's subset inequality. As a special case, the (R0,Rs) admissible rate region, which captures all possible tradeoffs between the encoding rate, R0, of the all-access encoder and the sum encoding rate, Rs, of the randomly accessible encoders, is explicitly characterized. To provide explicit proof of the optimality of superposition coding in this case, a new sliding-window subset entropy inequality is introduced and is shown to directly imply the classical subset entropy inequality of Han.

This dissertation will present several novel techniques that use cooperation and diversity to improve the performance of multihop Wireless Sensor Networks, as measured by throughput, delay, and reliability, beyond what is achievable with conventional error control technology. We will investigate the applicability of these new technologies to Wireless Body Area Networks (WBANs) an important emerging class of wireless sensor networks. WBANs, which promise significant improvement in the reliability of monitoring and treating people's health, comprise a number of sensors and actuators that may either be implanted in vivo or mounted on the surface of the human body, and which are capable of wireless communication to one or more external nodes that are in close proximity to the human body. Our focus in this research is on enhancing the performance of WBANs, especially for emerging real-time in vivo traffic such as streaming real-time video during surgery. Because of the nature of this time-sensitive application, retransmissions may not be possible. Furthermore, achieving minimal energy consumption, with the required level of reliability is critical for the proper functioning of many wireless sensor and body area networks. Additionally, regardless of the traffic characteristics, the techniques we introduce strive to realize reliable wireless sensor networks using (occasionally) unreliable components (wireless sensor nodes). To improve the performance of wireless sensor networks, we introduce a novel technology Cooperative Network Coding, a technology that synergistically integrates the prior art of Network Coding with Cooperative Communications. With the additional goal of further minimizing the energy consumed by the network, another novel technology Cooperative Diversity Coding was introduced and is used to create protection packets at the source node. For representative applications, optimized Cooperative Diversity Coding or Cooperative Network Coding achieves ≥ 25% energy savings compared to the baseline Cooperative Network Coding scheme. Cooperative Diversity Coding requires lees computational complexity at the source node compared to Cooperative Network Coding. To improve the performance and increase the robustness and reliability of WBANs, two efficient feedforward error-control technologies, Cooperative Network Coding (CDC) and Temporal Diversity Coding (TDC), are proposed. Temporal Diversity Coding applies Diversity Coding in time to improve the WBAN's performance. By implementing this novel technique, it is possible to achieve significant improvement (50%) in throughput compared to extant WBANs. An example of an implementation of in vivo real-time application, where TDC can improve the communications performance, is the MARVEL (Miniature Anchored Robotic Videoscope for Expedited Laparoscopy) research platform developed at USF. The MARVEL research platform requires high bit rates (100 Mbps) for high-definition transmission. Orthogonal Frequency Division Multiplexing (OFDM), a widely used technology in fourth generation wireless networks (4G) that achieves high transmission rates over dispersive channels by transmitting serial information through multiple parallel carriers. Combining Diversity Coding with OFDM (DC-OFDM) promises high reliability communications while preserving high transmission rates. Most of the carriers transport original information while the remaining (few) carriers transport diversity coded (protection) information. The impact of DC-OFDM can extend far beyond in vivo video medical devices and other special purpose wireless systems and may find significant application in a broad range of ex vivo wireless systems, such as LTE, 802.11, 802.16.

This dissertation focuses on the use of subcarrier multiplexing (SCM) in multimode fibers, utilizing carrier frequencies above what is generally utilized for multimode fiber transmission, to achieve high bit rates. In the high frequency region (i.e., frequencies larger than the intermodal bandwidth), the magnitude response of multimode fiber does not decrease monotonically as a function of the frequency but is shown to become relatively flat (but with several deep nulls) with an amplitude below that at DC. The statistical properties of this frequency response at high frequencies are analyzed. The probability density function of the magnitude response at high frequencies is found to be a Rayleigh density function. The average amplitude in this high frequency region does not depend on the frequency but depends on the number of modes supported by the fiber. To transmit a high bit rate signal over the multimode fiber, subcarrier multiplexing is adopted. The performance of the SCM multimode fiber system is presented. The performance of the SCM system is significantly degraded if there are some subcarriers located at the deep nulls of the fiber. Equalization and spread spectrum techniques are investigated but are shown to be not effective in combating the effects of these nulls. To cancel the effects of these deep nulls, training process and diversity coding are considered. The basic theory of diversity coding is given. It is found that the performances of the system with training process and the system with diversity coding are almost identical. However, diversity coding is more appropriate since it requires less system complexity. Finally, the practical limits and capacity of the SCM multimode fiber system are investigated. It is shown that a signal with a bit rate of 1.45 Gbps can be transmitted over a distance up to 5 km. / Ph. D.

This dissertation studies fundamental limits of modern digital communication systems in presence/absence of delay and secrecy constraints. In the first part of this dissertation, we consider a typical time-division wireless communication system wherein the channel strengths of the wireless users vary with time with a power constraint at the base station and which is not subject to any delay constraint. The objective is to allocate resources to the wireless users in an equitable manner so as to achieve a specific throughput. This problem has been looked at in different ways by previous researchers. We address this problem by developing a systematic way of designing scheduling schemes that can achieve any point on the boundary of the rate region. This allows us to map a desired throughput to a specific scheduling scheme which can then be used to service the wireless users. We then propose a simple scheme by which users can cooperate and then show that a cooperative scheduling scheme enlarges the achievable rate region. A simple iterative algorithm is proposed to find the resource allocation parameters and the scheduling scheme for the cooperative system. In the second part of the dissertation, a downlink time-division wireless sys- tem that is subject to a delay constraint is studied, and the rate region and optimal scheduling schemes are derived. The result of this study concludes that the achievable throughput of users decrease as the delay constraint is increased. Next, we consider a problem motivated by cognitive radio applications which has been proposed as a means to implement efficient reuse of the licensed spectrum. Previous research on this topic has focussed largely on obtaining fundamental limits on achievable throughput from a physical layer perspective. In this dissertation, we study the impact of im- posing Quality of Service constraints (QoS) on the achievable throughput of users. The result of this study gives insights on how the cognitive radio system needs to be operated in the low and high QoS constraint regime. Finally, the third part of this dissertation is motivated by the need for commu- nicating information not only reliably, but also in a secure manner. To this end, we study a source coding problem, wherein multiple sources needs to be communicated to a receiver with the stipulation that there is no direct channel from the transmitter to the receiver. However, there are many \agents" that can help carry the information from the transmitter to the receiver. Depending on the reliability that the transmit- ter has on each of the agents, information is securely encoded by the transmitter and given to the agents, which will be subsequently given to the receiver. We study the overhead that the transmitter has to incur for transmitting the information to the receiver with the desired level of secrecy. The rate region for this problem is found and simple achievable schemes are proposed. The main result is that, separate secure coding of sources is optimal for achieving the sum-rate point for the general case of the problem and the rate region for simple case of this problem.

Secure communication under channel uncertainty is an important and challenging problem in physical-layer security and cryptography. In this dissertation, we take a fundamental information-theoretic view at three concrete settings and use them to shed insight into efficient secure communication techniques for different scenarios under channel uncertainty. First, a multi-input multi-output (MIMO) Gaussian broadcast channel with two receivers and two messages: a common message intended for both receivers (i.e., channel uncertainty for decoding the common message at the receivers) and a confidential message intended for one of the receivers but needing to be kept asymptotically perfectly secret from the other is considered. A matrix characterization of the secrecy capacity region is established via a channel-enhancement argument and an extremal entropy inequality previously established for characterizing the capacity region of a degraded compound MIMO Gaussian broadcast channel. Second, a multilevel security wiretap channel where there is one possible realization for the legitimate receiver channel but multiple possible realizations for the eavesdropper channel (i.e., channel uncertainty at the eavesdropper) is considered. A coding scheme is designed such that the number of secure bits delivered to the legitimate receiver depends on the actual realization of the eavesdropper channel. More specifically, when the eavesdropper channel realization is weak, all bits delivered to the legitimate receiver need to be secure. In addition, when the eavesdropper channel realization is strong, a prescribed part of the bits needs to remain secure. We call such codes security embedding codes, referring to the fact that high-security bits are now embedded into the low-security ones. We show that the key to achieving efficient security embedding is to jointly encode the low-security and high-security bits. In particular, the low-security bits can be used as (part of) the transmitter randomness to protect the high-security ones. Finally, motivated by the recent interest in building secure, robust and efficient distributed information storage systems, the problem of secure symmetrical multilevel diversity coding (S-SMDC) is considered. This is a setting where there are channel uncertainties at both the legitimate receiver and the eavesdropper. The problem of encoding individual sources is first studied. A precise characterization of the entire admissible rate region is established via a connection to the problem of secure coding over a three-layer wiretap network and utilizing some basic polyhedral structure of the admissible rate region. Building on this result, it is then shown that the simple coding strategy of separately encoding individual sources at the encoders can achieve the minimum sum rate for the general S-SMDC problem.

Au cours des 15 dernières années, des progrès spectaculaires dans l'analyse et la conception des codes définis par des graphes bipartites et décodables par des algorithmes itératifs ont permis le développement de systèmes de correction d'erreurs, avec des performances de plus en plus proches la limite théorique de Shannon. Dans ce contexte, un rôle déterminant a été joué par la famille des codes à matrice de parité creuse, appelés codes LDPC (pour " Low-Density Parity-Check ", en anglais), introduit par Gallager au début des années 60 et décrits plus tard en termes de graphes bipartites. Négligés pendant de longues années, ces codes ont été redécouverts à la fin des années 90, après que la puissance du décodage itératif a été mise en évidence grâce à l'invention des Turbo-codes. Ce n'est qu'au début des années 2000 que les techniques nécessaires à l'analyse et l'optimisation des codes LDPC ont été développées, techniques qui ont permis ensuite la construction des codes avec des performances asymptotiques proches de la limite de Shannon. Cette remarquable avancée a motivé l'intérêt croissant de la communauté scientifique et soutenu le transfert rapide de cette technologie vers le secteur industriel. Plus récemment, un intérêt tout particulier a été porté aux codes LDPC définis sur des alphabets non-binaires, grâce notamment à leur meilleure capacité de correction en " longueur finie ". Bien que Gallager ait déjà proposé l'utilisation des alphabets non-binaires, en utilisant l'arithmétique modulaire, les codes LDPC non-binaires définis sur les corps finis n'ont étés étudiés qu'à partir de la fin des années 90. Il a été montré que ces codes offrent de meilleures performances que leurs équivalents binaires lorsque le bloc codé est de longueur faible à modérée, ou lorsque les symboles transmis sur le canal sont eux-mêmes des symboles non- binaires, comme par exemple dans le cas des modulations d'ordre supérieur ou des canaux à antennes multiples. Cependant, ce gain en performance implique un coût non négligeable en termes de complexité de décodage, qui peut entraver l'utilisation des codes LDPC non binaires dans des systèmes réels, surtout lorsque le prix à payer en complexité est plus important que le gain en performance. Cette thèse traite de l'analyse et de la conception des codes LDPC non binaires pour des canaux à évanouissements. L'objectif principal de la thèse est de démontrer que, outre le gain en performance en termes de capacité de correction, l'emploi des codes LDPC non binaires peut apporter des bénéfices supplémentaires, qui peuvent compenser l'augmentation de la complexité du décodeur. La " flexibilité " et la " diversité " représentent les deux bénéfices qui seront démontrées dans cette thèse. La " flexibilité " est la capacité d'un système de codage de pouvoir s'adapter à des débits (rendements) variables tout en utilisant le même encodeur et le même décodeur. La " diversité " se rapporte à sa capacité d'exploiter pleinement l'hétérogénéité du canal de communication. La première contribution de cette thèse consiste à développer une méthode d'approximation de l'évolution de densité des codes LDPC non-binaires, basée sur la simulation Monte-Carlo d'un code " infini ". Nous montrons que la méthode proposée fournit des estimations très fines des performances asymptotiques des codes LDPC non-binaires et rend possible l'optimisation de ces codes pour une large gamme d'applications et de modèles de canaux. La deuxième contribution de la thèse porte sur l'analyse et la conception de système de codage flexible, utilisant des techniques de poinçonnage. Nous montrons que les codes LDPC non binaires sont plus robustes au poinçonnage que les codes binaires, grâce au fait que les symboles non-binaires peuvent être partialement poinçonnés. Pour les codes réguliers, nous montrons que le poinçonnage des codes non-binaires obéit à des règles différentes, selon que l'on poinçonne des symboles de degré 2 ou des symboles de degré plus élevé. Pour les codes irréguliers, nous proposons une procédure d'optimisation de la " distribution de poinçonnage ", qui spécifie la fraction de bits poinçonnés par symbole non-binaire, en fonction du degré du symbole. Nous présentons ensuite des distributions de poinçonnage optimisées pour les codes LDPC non binaires, avec des performances à seulement 0,2 - 0,5 dB de la capacité, pour des rendements poinçonnés variant de 0,5 à 0,9. La troisième contribution de la thèse concerne les codes LDPC non binaires transmis sur un canal de Rayleigh à évanouissements rapides, pour lequel chaque symbole modulé est affecté par un coefficient d'évanouissement différent. Dans le cas d'une correspondance biunivoque entre les symboles codés et les symboles modulés (c.-à-d. lorsque le code est définit sur un corps fini de même cardinalité que la constellation utilisée), certains symboles codés peuvent être complètement noyés dans le bruit, dû aux évanouissements profonds du canal. Afin d'éviter ce phénomène, nous utilisons un module d'entrelacement au niveau bit, placé entre l'encodeur et le modulateur. Au récepteur, le module de désentrelacement apporte de la diversité binaire en entrée du décodeur, en atténuant les effets des différents coefficients de fading. Nous proposons un algorithme d'entrelacement optimisé, inspirée de l'algorithme " Progressive Edge-Growth " (PEG). Ainsi, le graphe bipartite du code est élargi par un nouvel ensemble de nœuds représentant les symboles modulés, et l'algorithme proposé établit des connections entre les nœuds représentant les symboles modulés et ceux représentant les symboles codés, de manière à obtenir un graphe élargi de maille maximale. Nous montrons que l'entrelaceur optimisé permet d'obtenir un gain de performance par rapport à un entrelaceur aléatoire, aussi bien en termes de capacité de correction que de détection d'erreurs. Enfin, la quatrième contribution de la thèse consiste en un schéma de codage flexible, permettant d'atteindre la diversité maximale d'un canal à évanouissements par blocs. La particularité de notre approche est d'utiliser des codes Root-LDPC non binaires couplés avec des codes multiplicatifs non binaires, de manière à ce que le rendement de codage puisse facilement s'adapter au nombre de blocs d'évanouissement. Au niveau du récepteur, une simple technique de combinaison de diversité est utilisée en entrée du décodeur. Comme conséquence, la complexité du décodage reste inchangée quel que soit le nombre de blocs d'évanouissement et le rendement du code utilisé, tandis que la technique proposée apporte un réel bénéfice en termes de capacité de correction.

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