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This book introduces an advanced list decoding system for both Reed-Solomon codes and algebraic- geometric codes. This list decoding system can offer better error-correction capability than the currently used decoding scheme, which provides a solution for future generation communication systems and HDTV/DVD systems. It introduces the decoding system from an engineering point of view with a practical insight. Addressing its implementation issues, several complexity reducing methods are introduced. This book also provides some future solution to replace Reed-Solomon codes the algebraic-geometric codes. It shows compared with the Reed-Solomon codes, the algebraic-geometric codes have a larger size and with better error- correction capability. Its hard-decision and soft- decision list decoding systems are also introduced.