The Golay code is a perfect linear error-correcting code. There are two essentially distinct versions of the Golay code: a binary version and a ternary version.
The binary version 
is a 
binary linear code
consisting of 
codewords of length 23 and minimum distance 7. The ternary
version is a 
ternary linear code, consisting of 
codewords of length 11
with minimum distance 5.
A parity check matrix for the binary Golay code is given by the matrix 
,
where 
is the 
identity matrix
and 
is the 
matrix
![M=[1 0 0 1 1 1 0 0 0 1 1 1; 1 0 1 0 1 1 0 1 1 0 0 1; 1 0 1 1 0 1 1 0 1 0 1 0; 1 0 1 1 1 0 1 1 0 1 0 0; 1 1 0 0 1 1 1 0 1 1 0 0; 1 1 0 1 0 1 1 1 0 0 0 1; 1 1 0 1 1 0 0 1 1 0 1 0; 1 1 1 0 0 1 0 1 0 1 1 0; 1 1 1 0 1 0 1 0 0 0 1 1; 1 1 1 1 0 0 0 0 1 1 0 1; 0 1 1 1 1 1 1 1 1 1 1 1].](/images/equations/GolayCode/NumberedEquation1.svg) |
By adding a parity check bit to each codeword in 
, the extended Golay code 
, which is a nearly perfect ![[24,12,8]](/images/equations/GolayCode/Inline13.svg)
binary linear code, is obtained. The automorphism
group of 
is the Mathieu group 
.
A second 
generator is the adjacency matrix for the icosahedron, with 
appended, where 
is a unit matrix and 
is an identity
matrix.
A third 
generator begins a list with the 24-bit 0 word (000...000) and repeatedly appends
first 24-bit word that has eight or more differences from all words in the list.
Conway and Sloane list many further methods.
Amazingly, Golay's original paper was barely a half-page long but has proven to have deep connections to group theory, graph theory, number theory, combinatorics, game theory, multidimensional geometry, and even particle physics.
