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It has long been recognized that graphical representations of parallel concatenated codes contain loops. These loops make it extremely difficult to use the standard probabilistic methods for graphs to obtain any insight as to the functionality of turbo decoding. Graph-based techniques do, however, lead to useful extensions and generalizations of the turbo decoding concept. In this paper we use a graph-theoretic approach to develop a class of parallel turbo decoding algorithms that are faster and provide better limiting performance than the classical turbo decoding algorithm,.