The explosive demand for capacity in wireless and wireline communication systems has promoted tremendous research in error control technologies. The discovery and re-discovery of capacity-achieving error correction codes (ECC) in the last decade has not only revolutionized the coding theory, but also opened up a very pervasive scope of well-proven and emerging practical applications, including, for example, wireless communications, multi-user detection, turbo equalization, digital data storage systems, distributed compression, and quantum error correction. In this dissertation, we study emerging topics on the theory and practice of error correction coding technologies and explore their capabilities beyond their conventional applications.The first topic involves the Slepian-Wolf coding problem on binary memoryless sources. The Slepian-Wolf coding problem considers multiple physically-separated non-communicating sources sending statistically-correlated data to a common destination. We first propose a simple and powerful framework for symmetric and asymmetric Slepian-Wolf coding on binary memoryless sources. The new scheme, termed as symmetric syndrome-former inverse-syndrome-former framework (SSIF), can be efficiently applied to any linear channel code, incurs no rate loss when converting the channel code to the Slepian-Wolf code, and can achieve an arbitrary point in the Slepian-Wolf rate region. The feasibility and optimality of the proposed framework is rigorously proven using properties of linear codes and cosets. Hamming codes, turbo product codes, convolutional codes, turbo codes and low-density parity-check (LDPC) codes are provided as examples to demonstrate the generality of the framework. We also analyze the two major categories of SW coding, the syndrome approach and the parity approach. Our study covers both theoretical and practical aspects, including the practical implementation of the source encoder and decoder for both approaches, the criterion for designing the base channel codes, and their respective performances and robustness in both the noiseless and the noisy transmission environments. We show that the syndrome approach is optimal for the noiseless (syndrome) channel case, yet the parity approach is more robust and less error-sensitive. The second problem we study is quantum stabilizer codes, a fundamental error control technology for quantum information and computing. Elucidated in [3], developed in depth in [4] and [5], and exemplified in many research papers [6] [7], stabilizer codes have perhaps the only near-mature theory in quantum coding. The majority of the up-to-date research on quantum ECC codes focuses on one subclass of stabilizer codes, known as CSS codes. We first performed a detailed classification from the perspective of constructing stabilizer codes. From this classification, we notice that all the existing constructions only tackle a small subclass (type-I codes) of potential constructions. Especially, systematic constructions for the type-II codes are almost completely lacking. We discuss properties of the type-II codes and define their core codes. We further develop systematic ways to construct two rich classes of non-CSS stabilizer codes: quantum LDPC codes based on classical quaci-cyclic (QC) LDPC codes, and quantum convolutional codes based on classical LDPC-convolutional codes. Both of these codes belong to the type-I non-CSS codes. To the best of the authors' knowledge, they are also the first non-CSS quantum LDPC codes and non-CSS convolutional quantum codes in the literature. Because they root to powerful classical codes, they are capable of correcting a large number of quantum errors, including bursty errors, in a block. Additionally, both classes of codes enjoy a wide range of lengths and rates, including very high rates that approach 1. Besides constructions, we also investigate decoding strategies, especially for quantum convolutional codes, where no performance curves were reported in literature before our work. Finally, we analyze properties of general stabilizer codes, first on a popular group of CSS-LDPC codes, then on the degeneracy feature of stabilizer codes, and finally on the distance spectrum property of stabilizer codes.
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