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Построение согласованной функции расстояния для простого марковского канала
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.