Golay Code
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].](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/GolayCode/NumberedEquation1.gif) |
By adding a parity check bit to each codeword in 
, the extended Golay code 
, which is
a nearly perfect ![[24,12,8]](/National_Library/20160521004321im_/http://mathworld.wolfram.com/images/equations/GolayCode/Inline13.gif)
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.
Contact the MathWorld Team
golay code
