<s>
In	O
mathematics	O
and	O
electronics	O
engineering	O
,	O
a	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
is	O
a	O
type	O
of	O
linear	O
error-correcting	B-Error_Name
code	I-Error_Name
used	O
in	O
digital	O
communications	O
.	O
</s>
<s>
The	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
,	O
along	O
with	O
the	O
ternary	O
Golay	O
code	O
,	O
has	O
a	O
particularly	O
deep	O
and	O
interesting	O
connection	O
to	O
the	O
theory	O
of	O
finite	O
sporadic	O
groups	O
in	O
mathematics	O
.	O
</s>
<s>
There	O
are	O
two	O
closely	O
related	O
binary	B-Error_Name
Golay	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
The	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
,	O
G24	O
(	O
sometimes	O
just	O
called	O
the	O
"	O
Golay	O
code	O
"	O
in	O
finite	O
group	O
theory	O
)	O
encodes	O
12	O
bits	O
of	O
data	O
in	O
a	O
24-bit	O
word	O
in	O
such	O
a	O
way	O
that	O
any	O
3-bit	O
errors	O
can	O
be	O
corrected	O
or	O
any	O
7-bit	O
errors	O
can	O
be	O
detected	O
.	O
</s>
<s>
The	O
other	O
,	O
the	O
perfect	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
,	O
G23	O
,	O
has	O
codewords	O
of	O
length	O
23	O
and	O
is	O
obtained	O
from	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
by	O
deleting	O
one	O
coordinate	O
position	O
(	O
conversely	O
,	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
is	O
obtained	O
from	O
the	O
perfect	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
by	O
adding	O
a	O
parity	B-Error_Name
bit	I-Error_Name
)	O
.	O
</s>
<s>
In	O
mathematical	O
terms	O
,	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
G24	O
consists	O
of	O
a	O
12-dimensional	O
linear	O
subspace	O
W	O
of	O
the	O
space	O
of	O
24-bit	O
words	O
such	O
that	O
any	O
two	O
distinct	O
elements	O
of	O
W	O
differ	O
in	O
at	O
least	O
8	O
coordinates	O
.	O
</s>
<s>
In	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
,	O
all	O
code	O
words	O
have	O
Hamming	O
weights	O
of	O
0	O
,	O
8	O
,	O
12	O
,	O
16	O
,	O
or	O
24	O
.	O
</s>
<s>
The	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
,	O
G23	O
is	O
a	O
perfect	O
code	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
the	O
perfect	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
G23	O
(	O
meaning	O
the	O
subgroup	O
of	O
the	O
group	O
S23	O
of	O
permutations	O
of	O
the	O
coordinates	O
of	O
F	O
which	O
leave	O
G23	O
invariant	O
)	O
,	O
is	O
the	O
Mathieu	O
group	O
.	O
</s>
<s>
The	O
automorphism	B-Algorithm
group	I-Algorithm
of	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
is	O
the	O
Mathieu	O
group	O
,	O
of	O
order	O
.	O
</s>
<s>
Lexicographic	B-Error_Name
code	I-Error_Name
:	O
Order	O
the	O
vectors	O
in	O
V	O
lexicographically	O
(	O
i.e.	O
,	O
interpret	O
them	O
as	O
unsigned	O
24-bit	O
binary	O
integers	O
and	O
take	O
the	O
usual	O
ordering	O
)	O
.	O
</s>
<s>
Mathieu	O
group	O
:	O
Witt	O
in	O
1938	O
published	O
a	O
construction	O
of	O
the	O
largest	O
Mathieu	O
group	O
that	O
can	O
be	O
used	O
to	O
construct	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Turyn	O
's	O
construction	O
of	O
1967	O
,	O
"	O
A	O
Simple	O
Construction	O
of	O
the	O
Binary	B-Error_Name
Golay	I-Error_Name
Code	I-Error_Name
,	O
"	O
that	O
starts	O
from	O
the	O
Hamming	B-Error_Name
code	I-Error_Name
of	O
length	O
8	O
and	O
does	O
not	O
use	O
the	O
quadratic	O
residues	O
mod	O
23	O
.	O
</s>
<s>
If	O
one	O
interprets	O
the	O
support	O
of	O
each	O
subset	O
as	O
a	O
0-1-codeword	O
of	O
length	O
24	O
(	O
with	O
Hamming-weight	O
8	O
)	O
,	O
these	O
are	O
the	O
"	O
octads	O
"	O
in	O
the	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
If	O
heads	O
are	O
interpreted	O
as	O
1	O
and	O
tails	O
as	O
0	O
then	O
moving	O
to	O
a	O
codeword	O
from	O
the	O
extended	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
guarantees	O
it	O
will	O
be	O
possible	O
to	O
force	O
a	O
win	O
.	O
</s>
<s>
A	O
generator	O
matrix	O
for	O
the	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
is	O
I	O
A	O
,	O
where	O
I	O
is	O
the	O
12×12	O
identity	O
matrix	O
,	O
and	O
A	O
is	O
the	O
complement	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
icosahedron	O
.	O
</s>
<s>
All	O
6	O
columns	O
have	O
the	O
same	O
parity	B-Error_Name
,	O
which	O
equals	O
that	O
of	O
the	O
top	O
row	O
.	O
</s>
<s>
Hundreds	O
of	O
color	O
pictures	O
of	O
Jupiter	O
and	O
Saturn	B-Device
in	O
their	O
1979	O
,	O
1980	O
,	O
and	O
1981	O
fly-bys	O
would	O
be	O
transmitted	O
within	O
a	O
constrained	O
telecommunications	O
bandwidth	O
.	O
</s>
<s>
The	O
MIL-STD-188	O
American	O
military	O
standards	O
for	O
automatic	B-Device
link	I-Device
establishment	I-Device
in	O
high	B-Algorithm
frequency	I-Algorithm
radio	O
systems	O
specify	O
the	O
use	O
of	O
an	O
extended	O
(	O
24	O
,	O
12	O
)	O
Golay	O
code	O
for	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
.	O
</s>
