Binary Golay Code

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.