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Construction of matched distance function for simple Markov channel

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The problem of error correction in communication channel may be solved by finding the most probable error vector in the channel. The equivalent in some cases problem may be formulated as finding the vector of least weight. To perform this, the distance function is needed matched to communication channel. Hamming and Euclid metrics are traditionally used in classical coding theory, but for many channels the correspondent matched distance functions are unknown. Finding such functions would allow decoding error probability decreasing, and it is actual task. In this paper the problem of decoding function development is solved, providing maximum likelihood decoding in simple Markov channel. Analysis of vectors probability in simple Markov channel is performed. The developed function is presented as sum of coefficients from the set depending on channel parameters. The way of coefficient computation is mentioned, providing matching the function with channel. Some approximations of coefficients are given for the case when channel parameters are unknown or uncertain. Affect of this function and its approximations on error probability is estimated experimentally using convolutional code. The decoding rule is proposed providing maximum likelihood decoding in simple Markov channel. Proposed function is matched with the channel for all code lengths, as opposed to known Markov metrics. The selection of coefficients for the decoding rule function is considered, simplifying computations by cost of possible losing the matching property. Error probability of maximum likelihood decoding using proposed function is estimated experimentally for convolutional code in simple Markov channel. The affect of coefficients approximation on decoding error probability increasing is estimated. The comparison with the class of known Markov metrics is performed. Experiments show that both proposed matched function and its simplifications provide significant gain in decoding error probability comparing to Hamming metric, and comparing to known Markov metric in area of low a priori channel bit error probabilities. Usage of quantized values of proposed function practically does not increase the error probability comparing to maximum likelihood decoding. The method based on analysis of error probability in two-state channels may be used to develop decoding functions for more complex Gilbert and Gilbert–Elliott channel models. Such functions would allow significant increasing in data transmission reliability in channels with complicated noise structure and provide maximum likelihood decoding in Markov channel with memory, instead of traditional approach which uses decorrelation of the channel and significantly reduces capacity.

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