<s>
Lexicographic	B-Error_Name
codes	I-Error_Name
or	O
lexicodes	B-Error_Name
are	O
greedily	O
generated	O
error-correcting	B-Error_Name
codes	I-Error_Name
with	O
remarkably	O
good	O
properties	O
.	O
</s>
<s>
The	O
binary	O
lexicographic	B-Error_Name
codes	I-Error_Name
are	O
linear	O
codes	O
,	O
and	O
include	O
the	O
Hamming	B-Error_Name
codes	I-Error_Name
and	O
the	O
binary	B-Error_Name
Golay	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
A	O
lexicode	B-Error_Name
of	O
length	O
n	O
and	O
minimum	O
distance	O
d	O
over	O
a	O
finite	O
field	O
is	O
generated	O
by	O
starting	O
with	O
the	O
all-zero	O
vector	O
and	O
iteratively	O
adding	O
the	O
next	O
vector	O
(	O
in	O
lexicographic	O
order	O
)	O
of	O
minimum	O
Hamming	O
distance	O
d	O
from	O
the	O
vectors	O
added	O
so	O
far	O
.	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
length-3	O
lexicode	B-Error_Name
of	O
minimum	O
distance	O
2	O
would	O
consist	O
of	O
the	O
vectors	O
marked	O
by	O
an	O
"	O
X	O
"	O
in	O
the	O
following	O
example	O
:	O
</s>
<s>
Here	O
is	O
a	O
table	O
of	O
all	O
n-bit	O
lexicode	B-Error_Name
by	O
d-bit	O
minimal	O
hamming	O
distance	O
,	O
resulting	O
of	O
maximum	O
2m	O
codewords	O
dictionnary	O
.	O
</s>
<s>
For	O
example	O
,	O
F4	O
code	O
(	O
n	O
=	O
4	O
,	O
d	O
=	O
2	O
,	O
m	O
=3	O
)	O
,	O
extended	O
Hamming	B-Error_Name
code	I-Error_Name
(	O
n	O
=	O
8	O
,	O
d	O
=	O
4	O
,	O
m	O
=	O
4	O
)	O
and	O
especially	O
Golay	O
code	O
(	O
n	O
=	O
24	O
,	O
d	O
=	O
8	O
,	O
m	O
=	O
12	O
)	O
shows	O
exceptional	O
compactness	O
compared	O
to	O
neighbors	O
.	O
</s>
<s>
Since	O
lexicodes	B-Error_Name
are	O
linear	O
,	O
they	O
can	O
also	O
be	O
constructed	O
by	O
means	O
of	O
their	O
basis	O
.	O
</s>
<s>
Following	O
C	O
generate	O
lexicographic	B-Error_Name
code	I-Error_Name
and	O
parameters	O
are	O
set	O
for	O
the	O
Golay	O
code	O
(	O
N	O
=	O
24	O
,	O
D=	O
8	O
)	O
.	O
</s>
<s>
The	O
theory	O
of	O
lexicographic	B-Error_Name
codes	I-Error_Name
is	O
closely	O
connected	O
to	O
combinatorial	O
game	O
theory	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
codewords	O
in	O
a	O
binary	O
lexicographic	B-Error_Name
code	I-Error_Name
of	O
distance	O
d	O
encode	O
the	O
winning	O
positions	O
in	O
a	O
variant	O
of	O
Grundy	O
's	O
game	O
,	O
played	O
on	O
a	O
collection	O
of	O
heaps	O
of	O
stones	O
,	O
in	O
which	O
each	O
move	O
consists	O
of	O
replacing	O
any	O
one	O
heap	O
by	O
at	O
most	O
d	O
1	O
smaller	O
heaps	O
,	O
and	O
the	O
goal	O
is	O
to	O
take	O
the	O
last	O
stone	O
.	O
</s>
