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In this thesis, we consider the problem of designing space-time block codes that have low maximum-likelihood (ML) decoding complexity. We present a unified framework for determining the worst-case ML decoding complexity of space-time block codes. We use this framework to not only determine the worst-case ML decoding complexity of our own constructions, but also to show that some popular constructions of space-time block codes have lower ML decoding complexity than was previously known. Recognizing the practical importance of the two transmit and two receive antenna system, we propose the asymmetric golden code, which is designed specifically for low ML decoding complexity. The asymmetric golden code has the lowest decoding complexity compared to previous constructions of space-time codes, regardless of whether the channel varies with time. We also propose the embedded orthogonal space-time codes, which is a family of codes for an arbitrary number of antennas, and for any rate up to half the number of antennas. The family of embedded orthogonal space-time codes is the first general framework for the construction of space-time codes with low-complexity decoding, not only for rate one, but for any rate up to half the number of transmit antennas. Simulation results for up to six transmit antennas show that the embedded orthogonal space-time codes are simultaneously lower in complexity and lower in error probability when compared to some of the most important constructions of space-time block codes with the same number of antennas and the same rate larger than one. Having considered the design of space-time block codes with low ML decoding complexity on the transmitter side, we also develop efficient algorithms for ML decoding for the golden code, the asymmetric golden code and the embedded orthogonal space-time block codes on the receiver side. Simulations of the bit-error rate performance and decoding complexity of the asymmetric golden code and embedded orthogonal codes are used to demonstrate their attractive performance-complexity tradeoff.

Parmi les différents critères qu'une fonction booléenne doit satisfaire en cryptographie, on s'intéresse à la non-linéarité. Pour une fonction booléenne donnée, cette notion mesure la distance de Hamming qui la sépare des fonctions de degré au plus 1. C'est un critère naturel pour évaluer la complexité d'une fonction cryptographique, celle-ci ne devant pas admettreune approximation qui soit simple, comme par une fonction de degré 1, ou plus généralement une fonction de bas degré. Ainsi, il est important de considérer plus généralement, la non-linéarité d'ordre supérieur, qui pour un ordre donné r, mesure la distance d'une fonction donnée à l'ensemble des fonctions de degré au plus r. Cette notion est également importante pour les fonctions vectorielles, i.e., celles à plusieurs sorties. Quand le nombre de variables est grand, presque toutes les fonctions ont une non-linéarité (d'ordre 1) voisine d'une certaine valeur, assez élevée. Dans un premier travail, on étend ce résultat à l'ordre 2. Cette méthode qui consiste à observer comment les boules de Hamming recouvrent l'hypercube des fonctions booléennes, nous conduit naturellement vers une borne de décodage théorique des codes de Reed-Muller d'ordre 1, coïncidant au même endroit où se concentre la non-linéarité de presque toutes les fonctions ; une approche nouvelle pour un résultat pas entièrement nouveau. On étudie aussi la non-linéarité des fonctions vectorielles. On montre avec une approche différente, que le comportement asymptotique est le même que celui des fonctions booléennes: une concentration de la non-linéarité autour d'une valeur assez élevée. / Among the different criteria that a Boolean function must satisfy in symmetric cryptography, we focus on the nonlinearity of these. This notion measures the Hamming distance between a given function and the set of functions with degree at most 1. It is a natural criterion to evaluate the complexity of a cryptographic function that must not have a simple approximation as by a function of degree 1, or more generally, a function of low degree. Hence, it is important to consider the higher order nonlinearity, which for a given order r, measures the distance between a given function and the set of all functions of degree at most r. This notion is equally important for multi-output Boolean functions. When the number of variables is large enough, almost all Boolean functions have nonlinearities lying in a small neighbourhood of a certain high value. We prove that this fact holds when considering the second-order nonlinearity. Our method which consists in observing how the Hamming balls pack the hypercube of Boolean functions led quite naturally to a theoretical decoding bound for the first-order Reed-Muller code, coinciding with the concentration point of the nonlinearity of almost all functions. This was a new approach for a result which is not entirely new. We also studied the nonlinearity of multi-output functions. We proved with a different approach, that the asymptotic behaviour of multi-output functions is the same as the single-output ones: a concentration of the nonlinearity around a certain large value.

This thesis investigates a graph and information theoretic approach to design and analysis of low-density parity-check (LDPC) codes and wireless networks. In this work, both LDPC codes and wireless networks are considered as random graphs. This work proposes solutions to important theoretic and practical open problems in LDPC coding, and for the first time introduces a framework for analysis of finite wireless networks. LDPC codes are considered to be one of the best classes of error-correcting codes. In this thesis, several problems in this area are studied. First, an improved decoding algorithm for LDPC codes is introduced. Compared to the standard iterative decoding, the proposed decoding algorithm can result in several orders of magnitude lower bit error rates, while having almost the same complexity. Second, this work presents a variety of bounds on the achievable performance of different LDPC coding scenarios. Third, it studies rate-compatible LDPC codes and provides fundamental properties of these codes. It also shows guidelines for optimal design of rate-compatible codes. Finally, it studies non-uniform and unequal error protection using LDPC codes and explores their applications to data storage systems and communication networks. It presents a new error-control scheme for volume holographic memory (VHM) systems and shows that the new method can increase the storage capacity by more than fifty percent compared to previous schemes. This work also investigates the application of random graphs to the design and analysis of wireless ad hoc and sensor networks. It introduces a framework for analysis of finite wireless networks. Such framework was lacking from the literature. Using the framework, different network properties such as capacity, connectivity, coverage, and routing and security algorithms are studied. Finally, connectivity properties of large-scale sensor networks are investigated. It is shown how unreliability of sensors, link failures, and non-uniform distribution of nodes affect the connectivity of sensor networks.

This thesis investigates a graph and information theoretic approach to design and analysis of low-density parity-check (LDPC) codes and wireless networks. In this work, both LDPC codes and wireless networks are considered as random graphs. This work proposes solutions to important theoretic and practical open problems in LDPC coding, and for the first time introduces a framework for analysis of finite wireless networks. LDPC codes are considered to be one of the best classes of error-correcting codes. In this thesis, several problems in this area are studied. First, an improved decoding algorithm for LDPC codes is introduced. Compared to the standard iterative decoding, the proposed decoding algorithm can result in several orders of magnitude lower bit error rates, while having almost the same complexity. Second, this work presents a variety of bounds on the achievable performance of different LDPC coding scenarios. Third, it studies rate-compatible LDPC codes and provides fundamental properties of these codes. It also shows guidelines for optimal design of rate-compatible codes. Finally, it studies non-uniform and unequal error protection using LDPC codes and explores their applications to data storage systems and communication networks. It presents a new error-control scheme for volume holographic memory (VHM) systems and shows that the new method can increase the storage capacity by more than fifty percent compared to previous schemes. This work also investigates the application of random graphs to the design and analysis of wireless ad hoc and sensor networks. It introduces a framework for analysis of finite wireless networks. Such framework was lacking from the literature. Using the framework, different network properties such as capacity, connectivity, coverage, and routing and security algorithms are studied. Finally, connectivity properties of large-scale sensor networks are investigated. It is shown how unreliability of sensors, link failures, and non-uniform distribution of nodes affect the connectivity of sensor networks.

Space-Time Block Coding is an efficient communication technique used in multiple-input multiple-output wireless systems. The complexity with which a Space-Time Block Code (STBC) can be decoded is important from an implementation point of view since it directly affects the receiver complexity and speed. In this thesis, we address the problem of designing low complexity decoding techniques for STBCs, and constructing STBCs that achieve high rate and full-diversity with these decoders. This thesis is divided into two parts; the first is concerned with the optimal decoder, viz. the maximum-likelihood (ML) decoder, and the second with non-ML decoders. An STBC is said to be multigroup ML decodable if the information symbols encoded by it can be partitioned into several groups such that each symbol group can be ML decoded independently of the others, and thereby admitting low complexity ML decoding. In this thesis, we first give a new framework for constructing low ML decoding complexity STBCs using codes over the Klein group, and show that almost all known low ML decoding complexity STBCs can be obtained by this method. Using this framework we then construct new full-diversity STBCs that have the least known ML decoding complexity for a large set of choices of number of transmit antennas and rate. We then introduce the notion of Asymptotically-Good (AG) multigroup ML decodable codes, which are families of multigroup ML decodable codes whose rate increases linearly with the number of transmit antennas. We give constructions for full-diversity AG multigroup ML decodable codes for each number of groups g > 1. For g > 2, these are the first instances of g-group ML decodable codes that are AG or have rate more than 1. For g = 2 and identical delay, the new codes match the known families of AG codes in terms of rate. In the final section of the first part we show that the upper triangular matrix R encountered during the sphere-decoding of STBCs can be rank-deficient, thus leading to higher sphere-decoding complexity, even when the rate is less than the minimum of the number of transmit antennas and the number receive antennas. We show that all known AG multigroup ML decodable codes suffer from such rank-deficiency, and we explicitly derive the sphere-decoding complexities of most known AG multigroup ML decodable codes. In the second part of this thesis we first study a low complexity non-ML decoder introduced by Guo and Xia called Partial Interference Cancellation (PIC) decoder. We give a new full-diversity criterion for PIC decoding of STBCs which is equivalent to the criterion of Guo and Xia, and is easier to check. We then show that Distributed STBCs (DSTBCs) used in wireless relay networks can be full-diversity PIC decoded, and we give a full-diversity criterion for the same. We then construct full-diversity PIC decodable STBCs and DSTBCs which give higher rate and better error performance than known multigroup ML decodable codes for similar decoding complexity, and which include other known full-diversity PIC decodable codes as special cases. Finally, inspired by a low complexity essentially-ML decoder given by Sirianunpiboon et al. for the two and three antenna Perfect codes, we introduce a new non-ML decoder called Adaptive Conditional Zero-Forcing (ACZF) decoder which includes the technique of Sirianunpiboon et al. as a special case. We give a full-diversity criterion for ACZF decoding, and show that the Perfect codes for two, three and four antennas, the Threaded Algebraic Space-Time code, and the 4 antenna rate 2 code of Srinath and Rajan satisfy this criterion. Simulation results show that the proposed decoder performs identical to ML decoding for these five codes. These STBCs along with ACZF decoding have the best error performance with least complexity among all known STBCs for four or less transmit antennas.

It is well known that multiple antennas at the transmitter and receiver are imperative for reliable and high data-rate communication over wireless channels. However, these systems essentially need multiple radio frequency (RF) chains owing to multiple antennas, and hence pose challenges for applications with limited form-factor. Antenna Selection (AS) techniques alleviate this problem by using only a subset of the total available antennas and hence require only a few RF chains compared to the number of antennas. These systems operate in a closed-loop scenario, where the information fed back from the receiver is used for the transmit antenna subset selection. In contrast to this, a novel open-loop technique known as spatial modulation (SM) was recently proposed that uses a single RF-chain at the transmitter and achieves a higher spectral efficiency compared to single-input and AS based systems. The work in the thesis mainly focuses on the following aspects of SM system: Study of Mutual Information in SM systems operating in open-loop and closed-loop scenarios: We study the achievable mutual information in the SM system operating with finite and Gaussian input alphabet, and compare the results with that of the SIMO and AS based systems. Reduced-complexity maximum-likelihood (ML) decoding algorithms for SM systems: We propose ML-optimal sphere decoders for SM systems with arbitrary number of transmit antennas. Furthermore, a reduced-complexity ML detector is also proposed whose computational complexity is lowest among the known existing detectors in the literature. Transmit diversity techniques for SM systems: The conventional SM system achieves a transmit diversity order of one. We propose a complex interleaved orthogonal design baaed SM scheme that achieves a transit diversity order of two, while offering symbol-by- symbol ML decodability. Transmit antenna subset selection algorithms for SM systems: The SM system is considered in the closed-loop scenario, where only a subset of the total number of transmit antennas is chosen based on the information fed back by the receiver. Specifically, the Euclidean distance and capacity optimized antenna selection algorithms are studied in comparison with the conventional AS based systems. SM system operating in dispersive channels: The SM system operating in a dispersive channel with the aid of zero-padding is studied. It is shown that the SM system achieves full receive-diversity and multipath-diversity with ML decoding, but offers a decoding complexity that is exponential in the number of multipaths. Furthermore, a reduced complexity linear receiver is proposed that achieves achieves full multipath as well as receive-diversity, while offering a decoding complexity order same as that of the SM system operating in a frequency-flat channel.

One of the most popular ways to exploit the advantages of a multiple-input multiple-output (MIMO) system is using space time block coding. A space time block code (STBC) is a finite set of complex matrices whose entries consist of the information symbols to be transmitted. A linear STBC is one in which the information symbols are linearly combined to form a two-dimensional code matrix. A well known method of maximum-likelihood (ML) decoding of such STBCs is using the sphere decoder (SD). In this thesis, new constructions of STBCs with low sphere decoding complexity are presented and various ways of characterizing and reducing the sphere decoding complexity of an STBC are addressed. The construction of low sphere decoding complexity STBCs is tackled using irreducible matrix representations of Clifford algebras, cyclic division algebras and crossed-product algebras. The complexity reduction algorithms for the STBCs constructed are explored using tree based search algorithms. Considering an STBC as a vector space over the set of weight matrices, the problem of characterizing the sphere decoding complexity is addressed using quadratic form representations. The main results are as follows. A sub-class of fast decodable STBCs known as Block Orthogonal STBCs (BOSTBCs) are explored. A set of sufficient conditions to obtain BOSTBCs are explained. How the block orthogonal structure of these codes can be exploited to reduce the SD complexity of the STBC is then explained using a depth first tree search algorithm. Bounds on the SD complexity reduction and its relationship with the block orthogonal structure are then addressed. A set of constructions to obtain BOSTBCs are presented next using Clifford unitary weight designs (CUWDs), Coordinate-interleaved orthogonal designs (CIODs), cyclic division algebras and crossed product algebras which show that a lot of codes existing in literature exhibit the block orthogonal property. Next, the dependency of the ordering of information symbols on the SD complexity is discussed following which a quadratic form representation known as the Hurwitz-Radon quadratic form (HRQF) of an STBC is presented which is solely dependent on the weight matrices of the STBC and their ordering. It is then shown that the SD complexity is only a function of the weight matrices defining the code and their ordering, and not of the channel realization (even though the equivalent channel when SD is used depends on the channel realization). It is also shown that the SD complexity is completely captured into a single matrix obtained from the HRQF. Also, for a given set of weight matrices, an algorithm to obtain a best ordering of them leading to the least SD complexity is presented using the HRQF matrix.