Editing Binary Golay code (section)

===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}.