Improved random redundant iterative HDPC decoding
IEEE Transactions on Communications
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A decade ago, the introduction of turbo codes and iterative message passing algorithms revolutionized the theory and practice of coding. In the ensuing years, the coding theory community has become adept at designing codes from good graphical models---that is, models which imply low-complexity, near-optimal iterative message passing algorithms. Specifically, modern codes are constructed by connecting a large number of simple local codes together via a rich, random-like, cyclic interconnection network. A key observation from this work is that the introduction of cycles to graphical models can enable massive complexity reductions in model, and thus decoding, complexity. Whereas constructive graphical modeling problems (e.g. code design) have been widely addressed by the coding theory community, less is understood about the inverse problem of model extraction. Specifically, can good graphical models be obtained for existing algebraic codes, or more generally, for arbitrary systems? What tradeoffs exist between model complexity and cyclic topology for a given code? If good models can exist, how can they be obtained, or extracted? This dissertation presents a theoretical framework for the study of extractive graphical modeling problems. The limits of extraction are first considered and a number of new results are presented on the space of graphical models for a given (fixed) code. Most significantly, a new characterization of the tradeoff between cyclic topology and complexity in graphical models for linear codes is provided. Inasmuch as the cyclic topology of a graphical model is related to the performance of the decoding algorithms it implies, this tree-inducing cut-set bound provides insight into the fundamental limits of graphical model extraction. Extraction is then treated formally using the language of combinatorial optimization and a number of novel heuristics for both defining and solving this optimization problem are presented.The results of a number of related problems that arose in the aforementioned study of graphical model extraction are also reported. Novel optimal soft-in soft-out (SISO) decoding algorithms are described for Reed-Solomon codes and for first-order Reed-Muller codes. A practically realizable---yet remarkably successful---suboptimal SISO decoding algorithm for arbitrary linear block codes based on massively redundant Tanner graphs is also developed. Finally, an efficient algorithm for counting short cycles in bipartite graphs is described.