<s>
A	O
locally	B-Error_Name
decodable	I-Error_Name
code	I-Error_Name
(	O
LDC	O
)	O
is	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
that	O
allows	O
a	O
single	O
bit	O
of	O
the	O
original	O
message	O
to	O
be	O
decoded	O
with	O
high	O
probability	O
by	O
only	O
examining	O
(	O
or	O
querying	O
)	O
a	O
small	O
number	O
of	O
bits	O
of	O
a	O
possibly	O
corrupted	O
codeword	O
.	O
</s>
<s>
Note	O
that	O
locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
are	O
not	O
a	O
subset	O
of	O
locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
,	O
though	O
there	O
is	O
some	O
overlap	O
between	O
the	O
two	O
.	O
</s>
<s>
Locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
can	O
also	O
be	O
concatenated	O
,	O
where	O
a	O
message	O
is	O
encoded	O
first	O
using	O
one	O
scheme	O
,	O
and	O
the	O
resulting	O
codeword	O
is	O
encoded	O
again	O
using	O
a	O
different	O
scheme	O
.	O
</s>
<s>
(	O
Note	O
that	O
,	O
in	O
this	O
context	O
,	O
concatenation	O
is	O
the	O
term	O
used	O
by	O
scholars	O
to	O
refer	O
to	O
what	O
is	O
usually	O
called	O
composition	B-Application
;	O
see	O
)	O
.	O
</s>
<s>
The	O
final	O
encoding	O
is	O
still	O
locally	B-Error_Name
decodable	I-Error_Name
,	O
and	O
requires	O
additional	O
steps	O
to	O
decode	O
both	O
layers	O
of	O
encoding	O
.	O
</s>
<s>
Locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
have	O
applications	O
to	O
data	O
transmission	O
and	O
storage	O
,	O
complexity	O
theory	O
,	O
data	O
structures	O
,	O
derandomization	O
,	O
theory	O
of	O
fault	O
tolerant	O
computation	O
,	O
and	O
private	O
information	O
retrieval	O
schemes	O
.	O
</s>
<s>
Locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
are	O
especially	O
useful	O
for	O
data	O
transmission	O
over	O
noisy	O
channels	O
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
(	O
a	O
special	O
case	O
of	O
Reed	O
Muller	O
codes	O
)	O
was	O
used	O
in	O
1971	O
by	O
Mariner	O
9	O
to	O
transmit	O
pictures	O
of	O
Mars	O
back	O
to	O
Earth	O
.	O
</s>
<s>
(	O
The	O
Hadamard	B-Error_Name
code	I-Error_Name
falls	O
under	O
the	O
general	O
umbrella	O
of	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
,	O
and	O
just	O
happens	O
to	O
be	O
locally	B-Error_Name
decodable	I-Error_Name
;	O
the	O
actual	O
algorithm	O
used	O
to	O
decode	O
the	O
transmission	O
from	O
Mars	O
was	O
a	O
generic	O
error-correction	O
scheme	O
.	O
)	O
</s>
<s>
One	O
of	O
the	O
applications	O
of	O
locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
in	O
complexity	O
theory	O
is	O
hardness	O
amplification	O
.	O
</s>
<s>
We	O
can	O
use	O
a	O
polynomial	O
length	O
locally	B-Error_Name
decodable	I-Error_Name
code	I-Error_Name
with	O
polylogarithmic	O
query	O
complexity	O
that	O
tolerates	O
some	O
constant	O
fraction	O
of	O
errors	O
to	O
encode	O
the	O
string	O
that	O
represents	O
to	O
create	O
a	O
new	O
string	O
of	O
length	O
.	O
</s>
<s>
One	O
can	O
easily	O
see	O
that	O
locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
have	O
applications	O
in	O
this	O
setting	O
.	O
</s>
<s>
A	O
general	O
procedure	O
to	O
produce	O
a	O
-server	O
private	O
information	O
scheme	O
from	O
a	O
perfectly	O
smooth	O
-query	O
locally	B-Error_Name
decodable	I-Error_Name
code	I-Error_Name
is	O
as	O
follows	O
:	O
</s>
<s>
The	O
Hadamard	O
(	O
or	O
Walsh-Hadamard	O
)	O
code	O
is	O
an	O
example	O
of	O
a	O
simple	O
locally	B-Error_Name
decodable	I-Error_Name
code	I-Error_Name
that	O
maps	O
a	O
string	O
of	O
length	O
to	O
a	O
codeword	O
of	O
length	O
.	O
</s>
