Abstract
The influence of the theory of information on development of the error correcting coding theory has been studied. Main differences between the probabilistic approach and the deterministic approach in the analysis of error-correcting capabilities of different classes of linear codes have been demonstrated. The automaton hierarchical models for analysis of permutation decoding of cyclic codes have been developed and a cyclic permutation generator based on two Moore automata has been proposed. A study has been carried out into the regular and irregular states of linear finite-state machines (LFSM) based on the automaton representation of cyclic codes. A possibility of significant simplification of decoding of cyclic codes based on conversion of irregular LFSM syndromes into regular ones using permutations has been shown. The formalized methods for determination of error-correcting capabilities of iteratively decoded cyclic codes (IDCC) have been devised. They imply the replacementof traditional complete checking of all possible options for comparison of code words to directional search for the solution of the assigned problem, which leads to a significant time saving for calculations. The algorithm for determination of error-correcting capabilities of IDCC with respect to double errors is given. It has been shown that all iterative codes increase their error-correcting capabilities with an increase in the number of iterations and one can set it as a percentage for errors of various multiplicities. A distribution of error syndromes to separate iterations has been performed, which makes it possible to reduce the length of a check word in a code. As a result, this leads to an increase in a rate of iterative codes in comparison with the traditional correction codes. A comparative analysis of IDCC and LDPC codes has been carried out to determine a scope of their optimal use
Highlights
The error correcting coding theory has passed a complex and controversial path in its development
LDPC codes are best for large code lengths, for which approximate estimates of their correcting capabilities are possible with a validation less than 100 %, and iteratively decoded cyclic codes (IDCC) are best for small lengths of codes with accurate estimates of their error-correcting capabilities
We performed mathematical substantiation of permutation decoding of cyclic codes
Summary
The error correcting coding theory has passed a complex and controversial path in its development. The entire 70-year period has been the search for answers to the main questions: which code is the best and how to construct it? Error correction codes have found wide application in various technical fields, such as satellite and mobile communications, data storage and archiving systems, etc. Several directions emerged in error correction coding. They develop separately and interact with each other rarely. The main trend in development of modern communication systems is a constant increase in transmission rates and practical development of the terabyte range. The main reserve in improvement of the quality and rate of transmission is the use of error correction coding. Iterative error correction codes are very promising for solution of such problems
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