Lecture
Binary code Golay.
[C 0 23 + C 1 23 + C 2 23 + C 3 23 ] * 2 12 = 2 23 .
(23, 12)
From the table of binomial coefficients observed equality. This equality is a necessary but sufficient condition for the existence of a perfect, correcting 3-fold error (23, 12) Golay code.
1) the number of points inside the decoding sphere is 2 11
2) in total there are 2 12 decoding spheres
3) and the whole space is 2 23 points
This code corrects 3 errors, is perfect, which means all the spheres of some equal radius around the code words without intersecting, cover the whole space of points, i.e. there are no points between the spheres.
N = q m -1 / q-1 - the Hamming code having such a length is also perfect. (q - the base of the number system)
Codes with constant weight.
A code with a constant weight in each code combination contains the same amount of “1”.
(5.2)
Codes with constant weight are built on a very simple algorithm and are effective for establishing the fact of a single error. Corrective power is 0, detecting power is 1.
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Information and Coding Theory
Terms: Information and Coding Theory