Editing Binary Golay code (section)

== Mathematical definition ==
In mathematical terms, the extended binary Golay code ''G''<sub>24</sub> consists of a 12-dimensional [[linear subspace]] ''W'' of the space {{nowrap|1=''V'' = '''F'''{{su|b=2|p=24}}}} of 24-bit words such that any two distinct elements of ''W'' differ in at least 8 coordinates.  ''W'' is called a linear code because it is a vector space. In all, ''W'' comprises {{nowrap|1=4096 = 2<sup>12</sup>}} elements.
* The elements of ''W'' are called ''[[Code word (communication)|code words]]''. They can also be described as subsets of a set of 24 elements, where addition is defined as taking the symmetric difference of the subsets.
* In the extended binary Golay code, all code words have [[Hamming weight]]s of 0, 8, 12, 16, or 24. Code words of weight 8 are called '''octads''' and code words of weight 12 are called '''dodecads'''.
* Octads of the code ''G''<sub>24</sub> are elements of the S(5,8,24) [[Steiner system]]. There are {{nowrap|1=759 = 3 × 11 × 23}} octads and 759 complements thereof. It follows that there are {{nowrap|1=2576 = 2<sup>4</sup> × 7 × 23}} dodecads.
* Two octads intersect (have 1's in common) in 0, 2, or 4 coordinates in the binary vector representation (these are the possible intersection sizes in the subset representation). An octad and a dodecad intersect at 2, 4, or 6 coordinates.
* Up to relabeling coordinates, ''W'' is unique.

The binary Golay code, ''G''<sub>23</sub> is a [[perfect code]]. That is, the spheres of radius three around code words form a partition of the vector space. ''G''<sub>23</sub> is a 12-dimensional [[linear subspace|subspace]] of the space '''F'''{{su|b=2|p=23}}.

The automorphism group of the perfect binary Golay code ''G''<sub>23</sub> (meaning the subgroup of the group ''S<sub>23</sub>'' of permutations of the coordinates of '''F'''{{su|b=2|p=23}} which leave ''G''<sub>23</sub> invariant), is the [[Mathieu group]] <math>M_{23}</math>. The [[automorphism group]] of the extended binary Golay code is the [[Mathieu group]] <math>M_{24}</math>, of order {{nowrap|2<sup>10</sup> × 3<sup>3</sup> × 5 × 7 × 11 × 23}}. <math>M_{24}</math> is transitive on octads and on dodecads. The other Mathieu groups occur as [[Group action (mathematics)|stabilizer]]s of one or several elements of ''W''.

There is a single word of weight 24, which is a 1-dimensional invariant subspace. <math>M_{24}</math> therefore has an 11-dimensional irreducible representation on the field with 2 elements. In addition, since the binary golay code is a 12-dimensional subspace of a 24-dimensional space, <math>M_{24}</math> also acts on the 12-dimensional [[Quotient space (linear algebra)|quotient space]], called the ''binary Golay cocode''. A word in the cocode is in the same [[coset]] as a word of length 0, 1, 2, 3, or 4. In the last case, 6 (disjoint) cocode words all lie in the same coset. There is an 11-dimensional invariant subspace, consisting of cocode words with odd weight, which gives <math>M_{24}</math> a second 11-dimensional representation on the field with 2 elements.