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.
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