Detection and correction of communication channel errors.
Let us assume that random distortions of individual characters are possible in the sequence of the communication channel. Everywhere below in this paper, distortions in the sequence of the communication channel will be considered random. Purposeful distortions will not be considered.
The distorted sequence will be denoted
E'2M = {e'1,e'2,…,e'2M}, (3.3.1)
where
e'i = ei + δi (3.3.2)
In (3.3.2), δi is the distortion introduced into the communication channel sequence due to some external random causes.
From the distorted sequence of the communication channel (3.3.1) we obtain the distorted plaintext sequence
B'2M = {b'1,b'2,…,b'2M} (3.3.3)
wherein
b'1 = e'1, b'2 = e'2 - π(e'1), b'i = e'i - (π(e'i-2 + e'i-1) - π(e 'i-1)) (3.3.4)
A characteristic feature of the encoded sequence is the dependency distance, which means that when decoding the next value bi of the decoded text depends on only three values in the encoded text: ei, ei-1 and ei-2. Thus, if there is an error in the i-th value e’i of the encoded sequence, this error during decoding can only affect 3 values when calculating the decoded sequence:
b'i = e'i - (π (ei-2 + ei-1) - π (ei-1))
b’i+1 = ei+1 - (π(ei-1 + e’i) - π(e’i))
b’i+2 = ei+2 - (π(e’i + ei+1) - π(ei+1))
All subsequent values of the decoded text, starting from bi + 3, are not affected by the e'i error.
Throughout the rest of this paper, the phrase “the absence of other distortions within the dependency distance” for the distorted value e’k = ek + δk, δk ≠ 0, will mean that two conditions are satisfied
1. e'k-2 = ek-2 and e'k-1 = ek-1,
2. e'k+2 = ek+2 and e'k+1 = ek+1,
those. there is no distortion two steps before and two steps after the distorted value.
Let k be the first value at which δk ≠ 0. Since in this case
e'k-2 = ek-2 and e'k-1 = ek-1,
then it follows from (3.3.4) that in this case
b'k ≠ bk,
where bk is the value from the decoded undistorted sequence (3.2.5). Suppose that k ϵ {3,4,…,2M}. The case when k ϵ {1,2} will be considered later.
Let us further consider ways to detect and correct some errors in the communication channel. Note that the only way to detect errors is to have non-zero values in even positions during decoding.
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