Mathematical Definition
In mathematical terms, the extended binary Golay code consists of a 12-dimensional subspace W of the space V=F224 of 24-bit words such that any two distinct elements of W differ in at least eight coordinates. Equivalently, any non-zero element of W has at least eight non-zero coordinates.
- The possible sets of non-zero coordinates as w ranges over W are called code words. In the extended binary Golay code, all code words have the Hamming weights of 0, 8, 12, 16, or 24.
- Up to relabeling coordinates, W is unique.
The perfect binary Golay code is a perfect code. That is, the spheres of radius three around code words form a partition of the vector space.
The automorphism group of the binary Golay code is the Mathieu group . The automorphism group of the extended binary Golay code is the Mathieu group . The other Mathieu groups occur as stabilizers of one or several elements of W.
The Golay code words of weight eight are elements of the S(5,8,24) Steiner system.
Read more about this topic: Binary Golay Code
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