Editing Binary Golay code (section)

== Constructions ==
* [[Lexicode|Lexicographic code]]:  Order the vectors in ''V'' lexicographically (i.e., interpret them as unsigned 24-bit binary integers and take the usual ordering).  Starting with ''w''<sub>0</sub> = 0, define ''w''<sub>1</sub>, ''w''<sub>2</sub>, ..., ''w''<sub>12</sub> by the rule that ''w''<sub>''n''</sub> is the smallest integer which differs from all linear combinations of previous elements in at least eight coordinates.  Then ''W'' can be defined as the span of ''w''<sub>1</sub>, ..., ''w''<sub>12</sub>.
* [[Mathieu group]]: Witt in 1938 published a construction of the largest Mathieu group that can be used to construct the extended binary Golay code.<ref>{{Cite journal |url=http://scholarworks.sjsu.edu/etd_theses/4053 |title=Construction and Simplicity of the Large Mathieu Groups |last= Hansen |first=Robert Peter |journal=Master's Theses|date=2011 |doi=10.31979/etd.qnhv-a5us }}</ref>
* [[Quadratic residue code]]:  Consider the set ''N'' of quadratic non-residues (mod 23).  This is an 11-element subset of the [[cyclic group]] '''Z'''/23'''Z'''.  Consider the translates ''t''+''N'' of this subset.  Augment each translate to a 12-element set ''S''<sub>''t''</sub> by adding an element ∞.  Then labeling the basis elements of ''V'' by 0, 1, 2, ..., 22, ∞, ''W'' can be defined as the span of the words ''S''<sub>''t''</sub> together with the word consisting of all basis vectors.  (The perfect code is obtained by leaving out ∞.)
* As a [[cyclic code]]: The perfect G<sub>23</sub> code can be constructed via the factorization of <math>x^{23}+1</math> over the binary field [[GF(2)]]: <math display="block">x^{23} + 1 = (x+1)(x^{11} + x^9+x^7+x^6+x^5+x+1)(x^{11}+x^{10}+x^6+x^5+x^4+x^2+1).</math> It is the code generated by <math>\left(x^{11}+x^{10}+x^6+x^5+x^4+x^2+1\right)</math>.<ref>{{harvnb|Roman|1996|loc=p. 324 Example 7.4.3}}</ref> Either of degree 11 irreducible factors can be used to generate the code.<ref>{{harvnb|Pless|1998|loc=p. 114}}</ref>
* Turyn's construction of 1967, "A Simple Construction of the Binary Golay Code," that starts from the [[Hamming code]] of length 8 and does not use the quadratic residues mod 23.<ref>{{harvnb|Turyn|1967|loc=Section VI}}</ref>
* From the [[Steiner system#The Steiner system S.285.2C 8.2C 24.29|Steiner System S(5,8,24)]], consisting of 759 subsets of a 24-set. If one interprets the support of each subset as a 0-1-codeword of length 24 (with Hamming-weight 8), these are the "octads" in the binary Golay code. The entire Golay code can be obtained by repeatedly taking the [[symmetric difference]]s of subsets, i.e. binary addition. An easier way to write down the Steiner system resp. the octads is the [[Miracle Octad Generator]] of R. T. Curtis, that uses a particular 1:1-correspondence between the 35 partitions of an 8-set into two 4-sets and the 35 partitions of the finite vector space <math>\mathbb{F}_2^4</math> into 4 planes.<ref>{{Cite web |url=http://finitegeometry.org/sc/24/MOG.html |title=The Miracle Octad Generator |last= Cullinane |first=Steven H. |website=Finite Geometry of the Square and Cube}}</ref> Nowadays often the compact approach of Conway's hexacode, that uses a 4&times;6 array of square cells, is used.  
* Winning positions in the [[mathematical game]] of Mogul: a position in Mogul is a row of 24 coins. Each turn consists of flipping from one to seven coins such that the leftmost of the flipped coins goes from head to tail. The losing positions are those with no legal move. If heads are interpreted as 1 and tails as 0 then moving to a codeword from the extended binary Golay code guarantees it will be possible to force a win.
* A [[generator matrix]] for the binary Golay code is '''I A''', where '''I''' is the 12×12 identity matrix, and '''A''' is the complement of the [[adjacency matrix]] of the [[icosahedron]].

===A convenient representation===
It is convenient to use the "[[Miracle Octad Generator]]" format, with coordinates in an array of 4 rows, 6 columns. Addition is taking the symmetric difference. All 6 columns have the same parity, which equals that of the top row.

A partition of the 6 columns into 3 pairs of adjacent ones constitutes a [[Mathieu group M24#Trio subgroup|trio]]. This is a partition into 3 octad sets. A subgroup, the [[projective special linear group]] PSL(2,7) x S<sub>3</sub> of a trio subgroup of M<sub>24</sub> is useful for generating a basis. PSL(2,7) permutes the octads internally, in parallel. S<sub>3</sub> permutes the 3 octads bodily.

The basis begins with octad T:
 0 1 1 1 1 1
 1 0 0 0 0 0
 1 0 0 0 0 0
 1 0 0 0 0 0

and 5 similar octads. The sum '''N''' of all 6 of these code words consists of all 1's. Adding N to a code word produces its complement.

Griess (p.&nbsp;59) uses the labeling:
 ∞ 0 | ∞ 0 | ∞ 0
 3 2 | 3 2 | 3 2
 5 1 | 5 1 | 5 1
 6 4 | 6 4 | 6 4

PSL(2,7) is naturally the linear fractional group generated by (0123456) and (0∞)(16)(23)(45). The 7-cycle acts on T to give a subspace including also the basis elements
 0 1 1 0 1 0 
 0 0 0 0 0 0
 0 1 0 1 0 1
 1 1 0 0 0 0

and
 0 1 1 0 1 0
 0 1 0 1 0 1
 1 1 0 0 0 0
 0 0 0 0 0 0

The resulting 7-dimensional subspace has a 3-dimensional quotient space upon ignoring the latter 2 octads.

There are 4 other code words of similar structure that complete the basis of 12 code words for this representation of W.

W has a subspace of dimension 4, symmetric under PSL(2,7) x S<sub>3</sub>, spanned by N and 3 dodecads formed of subsets {0,3,5,6}, {0,1,4,6}, and {0,1,2,5}.