<s>
In	O
theoretical	O
computer	O
science	O
and	O
coding	B-Error_Name
theory	I-Error_Name
,	O
the	O
long	B-Error_Name
code	I-Error_Name
is	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
that	O
is	O
locally	B-Error_Name
decodable	I-Error_Name
.	O
</s>
<s>
Long	B-Error_Name
codes	I-Error_Name
have	O
an	O
extremely	O
poor	O
rate	O
,	O
but	O
play	O
a	O
fundamental	O
role	O
in	O
the	O
theory	O
of	O
hardness	O
of	O
approximation	O
.	O
</s>
<s>
Then	O
the	O
long	B-Error_Name
code	I-Error_Name
encoding	O
of	O
a	O
message	O
is	O
the	O
string	O
where	O
denotes	O
concatenation	O
of	O
strings	O
.	O
</s>
<s>
The	O
Walsh-Hadamard	B-Error_Name
code	I-Error_Name
is	O
a	O
subcode	O
of	O
the	O
long	B-Error_Name
code	I-Error_Name
,	O
and	O
can	O
be	O
obtained	O
by	O
only	O
using	O
functions	O
that	O
are	O
linear	O
functions	O
when	O
interpreted	O
as	O
functions	O
on	O
the	O
finite	O
field	O
with	O
two	O
elements	O
.	O
</s>
<s>
Since	O
there	O
are	O
only	O
such	O
functions	O
,	O
the	O
block	O
length	O
of	O
the	O
Walsh-Hadamard	B-Error_Name
code	I-Error_Name
is	O
.	O
</s>
<s>
An	O
equivalent	O
definition	O
of	O
the	O
long	B-Error_Name
code	I-Error_Name
is	O
as	O
follows	O
:	O
</s>
<s>
The	O
Long	B-Error_Name
code	I-Error_Name
encoding	O
of	O
is	O
defined	O
to	O
be	O
the	O
truth	O
table	O
of	O
the	O
Boolean	O
dictatorship	O
function	O
on	O
the	O
th	O
coordinate	O
,	O
i.e.	O
,	O
the	O
truth	O
table	O
of	O
with	O
.	O
</s>
<s>
Thus	O
,	O
the	O
Long	B-Error_Name
code	I-Error_Name
encodes	O
a	O
-bit	O
string	O
as	O
a	O
-bit	O
string	O
.	O
</s>
<s>
The	O
long	B-Error_Name
code	I-Error_Name
does	O
not	O
contain	O
repetitions	O
,	O
in	O
the	O
sense	O
that	O
the	O
function	O
computing	O
the	O
th	O
bit	O
of	O
the	O
output	O
is	O
different	O
from	O
any	O
function	O
computing	O
the	O
th	O
bit	O
of	O
the	O
output	O
for	O
.	O
</s>
<s>
Among	O
all	O
codes	O
that	O
do	O
not	O
contain	O
repetitions	O
,	O
the	O
long	B-Error_Name
code	I-Error_Name
has	O
the	O
longest	O
possible	O
output	O
.	O
</s>
