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Binary Golay Code

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