Iterative Decoding Techniques and their Applications

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Over the past decade, iterative decoding methods have received a great deal of interest due to the astonishing error performances achieved first by turbo codes, and more recently by low-density parity check (LDPC) codes. However, since the vast majority of these results have been either theoretically derived for, or achieved by computer simulations of very long codes, severalissues remain problematic in the implementation of LDPC codes.This research addresses some of these problems and can be divided into three major areas:(1) Algebraic construction of LDPC codes:This research investigates the construction of LDPC codes based on algebra and finite geometries. These constructions are motivated by the possibility of describing families of codes in a very specific and unambiguous way, which is impossible for codes generated with the aid of a computer. Furthermore, tight bounds if not exact values of important parameters such as the code minimum Hamming distance can be determined. This feature remains interesting to evaluate the error performance of applications requiring very small bit error rates such as storage systems. In addition, the algebraic structure of the codes constructed can be exploited to perform simple encoding or decoding, such as encoding of cyclic or quasi-cyclic codes based on linear feedback shift registers.(2) Near-optimum reduced-latency iterative decoding of short to medium length LDPC codes:Although very impressive results have been derived and achieved with long LDPC codes,codes of shorter lengths are more likely to be used in practical applications such as wireless communication systems. For codes of medium length, iterative decoding alone requires a lot of iterations and falls significantly short of the optimum error performance. The investigators devise new decoding methods which, combined with iterative decoding, both reduce the number ofrequired iterations and improve the error performance of current approaches.(3) Simplification of iterative decoding methods:The best iterative decoding methods for LDPC codes depend on the operating signal-to-noise ratio and are subject to numerical problems inherent to the evaluation of probabilities or log-likelihood ratios. The investigators study new decoding methods which become universal and aremuch less sensitive to numerical problems while maintaining near-optimum error performance. Such properties are important for implementing these promising new decoding approaches in commercial products.