The results obtained during the 25-year evolution of the error-correction coding optimization theory (OT) and multithreshold decoding (MTD) methods, which have been created on its basis, are presented. These iterative algorithms, with each symbol correction iteration, always find decisions of strictly increasing likelihood, and can achieve optimum results that would normally require exhaustive search of all possible code words. Research results on MTDs and other error-correction methods for binary and non-binary codes used to send messages over channels with binary, symbolic errors and erasures are presented. It is shown MTDs simply decode very long codes, which are the only ones capable of supporting the effective implementation of error correction at high channel noise levels. Assessments of software implementation complexity show the advantage of MTD over other methods in terms of the number of operations per bit with comparable efficiency. It reviews the capabilities of symbolic codes, discovered by the authors, and the corresponding, simple to implement special symbolic MTD decoders, which are easier and more efficient than all other known methods of decoding non-binary codes. The methodological basis of the OT and the new paradigms for successful research into the theory and applied issues of error-correction coding are discussed. General conclusions are formulated on the study, and directions for further development of work on MTD are suggested.Keywords: error-correction coding, multithreshold decoding, symbolic codes, self-orthogonal codes, Viterbi algorithm, flash memory, Earth Remote Sensing, highly reliable data storage, optical communications channels, codes with a directly controlled metric, divergent coding Accepted: 08.02.201708.02. DOI: 10.21046/207008.02. -7401-2017
History of the subjectIn 2015, 25 years were passed from the date of defense of the dissertation (Zolotarev, 1990), where many basic results were proved for the codes that are simple from the contemporary viewpoint and were later systemized and represented within the optimization coding theory . Based on the development of the ideas of majority decoding , the optimization theory enabled us to look in an absolutely different way at the iterative error correction problem, the initial multithreshold decoding (MTD) methods of which were patented as early as 1972 (Zolotarev, 1972).Nowadays, all main stages of the design and research of MTD algorithms are performed using special powerful optimization procedures whose effectiveness and complexity grow quickly.At the same time, the complexity of the MTD technique is a minimum and increases only linearly with a code length. Owing to an increase in the number of iterations of error correction, the MTD potential grows at a rather low complexity of the algorithm itself. At present, as compared to the
