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The Scientific Issues of Ternopil Volodymyr Hnatiuk National Pedagogical University Series pedagogy

Результати дослідження коректувальної здатності ітеративних завадостійких кодів

Semerenko, Vasyl

Published Feb 21, 2019• 0 citations
OpenAlex

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

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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
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
This paper deals with the irregular binary low-density parity-check (LDPC) codes and two iterative low-complexity decoding algorithms. The first one is the majority error-correcting decoding algorithm, and the second one is iterative erasure-correcting decoding algorithm. The lower bounds on correcting capabilities (the guaranteed corrected error and erasure fraction respectively) of irregular LDPC code under decoding (error and erasure correcting respectively) algorithms with low-complexity were represented. These lower bounds were obtained as a result of analysis of Tanner graph representation of irregular LDPC code. The numerical results, obtained at the end of the paper for proposed lower-bounds achieved similar results for the previously known best lower-bounds for regular LDPC codes and were represented for the first time for the irregular LDPC codes.
This paper deals with the irregular binary low-density parity-check (LDPC) codes with the constituent single parity check (SPC) codes and the error-correcting iterative low-complex decoding algorithm. The lower bound on the error fraction, guaranteed corrected by the considered iterative algorithm, was obtained for the irregular LDPC code for the first time in this paper. This lower bound was obtained as a result of analysis of Tanner graph representation of irregular LDPC code. The number of decoding iterations, required to correct the errors, is a logarithmic function of the code length. The numerical results, obtained at the end of the paper for proposed lower bound achieved similar results for the previously known best lower-bounds for regular LDPC codes and were represented for the first time for the irregular LDPC codes.
The focus of this paper is directed towards the investigation of the characteristics of symbol-by-symbol iterative decoding algorithms for error-correcting block product-codes (block turbo-codes) which enable to reliable information transfer at relatively low received signal/noise and provide high power efficiency. Specific feature of investigated product codes is construction with usage of low-density parity-check codes (LDPC) and these code constructions are in the class of LDPC too. According to this fact the considered code constructions have symbol-by-symbol decoding algorithms developed for total class LDPC codes, namely BP (belief propagation) and its modification MIN_SUM_BP. The BP decoding algorithm is iterative and for implementation the signal/noise is required, for implementation of MIN_SUM_BP decoding algorithm the signal/noise is not required. The resulted characteristics of product codes constructed with usage of LDPC based on project geometry (length of code words, information volume, code rate, error performances) are presented in this paper. These component LDPC codes are cyclic and have encoding and decoding algorithms with low complexity implementation. The computer simulations for encoding and iterative symbol-by-symbol decoding algorithms for the number of turbo-codes with different code rate and information volumes are performed. The results of computer simulations have shown that MIN_SUM_BP decoding algorithm is more effective than BP decoding algorithm for channel with additive white gaussian noise concerning error-performances.
Binary cyclic codes achieve good error correction performance and allow the implementation of very simpleencoder and decoder circuits. Among them, BCH codesrepresent a very important class of t-error correcting codes, with known structural properties and error correction capability. Decoding of binary cyclic codes is often accomplished through hard-decision decoders, although it is recognized that softdecision decoding algorithms can produce significant coding gain with respect to hard-decision techniques. Several approaches have been proposed to implement iterative soft-decision decoding of binary cyclic codes. We study the technique based on “extended parity-check matrices”, and show that such method is not suitable for high rates or long codes. We propose a new approach, based on “reduced parity-check matrices” and “spread parity-check matrices”, that can achieve better correction performance in many practical cases, without increasing the complexity.
Among the classes of LDPC codes that have been constructed and designed, the only class of LDPC codes that are cyclic is the class of codes constructed based on the incidence vectors of lines of finite geometries, called finite geometry LDPC codes. Cyclic finite geometry LDPC codes are known to have large minimum distances and can provide good error performance with very low error-floors using iterative decoding based on belief propagation. Their cyclic structure allows them to be efficiently and systematically encoded with simple shift-registers in linear time with linear complexity. An obvious question is whether, besides cyclic finite geometry LDPC codes, there are other cyclic codes with large minimum distances that can be efficiently decoded iteratively using channel soft information. In this paper, we present one such class of cyclic codes. Codes in this class are two-step majority-logic decodable and they are also constructed based on finite geometries. Two iterative decoding algorithms are devised for this class of cyclic codes and they provide significant coding gain over the two-step majority-logic decoding of codes in this class.