Applications of Random Graphs to Design and Analysis of LDPC Codes and Sensor Networks
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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.