<s>
A	O
locally	B-Error_Name
testable	I-Error_Name
code	I-Error_Name
is	O
a	O
type	O
of	O
error-correcting	B-Error_Name
code	I-Error_Name
for	O
which	O
it	O
can	O
be	O
determined	O
if	O
a	O
string	O
is	O
a	O
word	O
in	O
that	O
code	O
by	O
looking	O
at	O
a	O
small	O
(	O
frequently	O
constant	O
)	O
number	O
of	O
bits	O
of	O
the	O
string	O
.	O
</s>
<s>
In	O
contrast	O
,	O
locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
use	O
a	O
small	O
number	O
of	O
bits	O
of	O
the	O
codeword	O
to	O
probabilistically	O
recover	O
the	O
original	O
information	O
.	O
</s>
<s>
The	O
fraction	O
of	O
errors	O
determines	O
how	O
likely	O
it	O
is	O
that	O
the	O
decoder	O
correctly	O
recovers	O
the	O
original	O
bit	O
;	O
however	O
,	O
not	O
all	O
locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
are	O
locally	O
testable	O
.	O
</s>
<s>
A	O
code	O
is	O
called	O
-local	O
-testable	O
if	O
there	O
exists	O
a	O
Turing	O
machine	O
M	O
given	O
random	B-General_Concept
access	I-General_Concept
to	O
an	O
input	O
that	O
makes	O
at	O
most	O
non-adaptive	O
queries	O
of	O
and	O
satisfies	O
the	O
following	O
:	O
</s>
<s>
It	O
remains	O
an	O
open	O
question	O
whether	O
there	O
are	O
any	O
locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
of	O
linear	O
size	O
,	O
but	O
there	O
are	O
several	O
constructions	O
that	O
are	O
considered	O
"	O
nearly	O
linear	O
"	O
:	O
</s>
<s>
In	O
November	O
2021	O
two	O
preprints	O
have	O
reported	O
the	O
first	O
polynomial-time	O
construction	O
of	O
"	O
-LTCs	O
"	O
namely	O
locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
with	O
constant	O
rate	O
,	O
constant	O
distance	O
and	O
constant	O
locality	O
.	O
</s>
<s>
Locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
have	O
a	O
lot	O
in	O
common	O
with	O
probabilistically	O
checkable	O
proofs	O
(	O
PCPs	O
)	O
.	O
</s>
<s>
Locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
,	O
on	O
the	O
other	O
hand	O
,	O
accept	O
if	O
it	O
is	O
part	O
of	O
the	O
code	O
.	O
</s>
<s>
Many	O
things	O
can	O
go	O
wrong	O
in	O
assuming	O
a	O
PCP	O
proof	O
encodes	O
a	O
locally	B-Error_Name
testable	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Despite	O
this	O
difference	O
,	O
locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
and	O
PCPs	O
are	O
similar	O
enough	O
that	O
frequently	O
to	O
construct	O
one	O
,	O
a	O
prover	O
will	O
construct	O
the	O
other	O
along	O
the	O
way	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
famous	O
error-correcting	B-Error_Name
codes	I-Error_Name
,	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
,	O
is	O
a	O
locally	B-Error_Name
testable	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
A	O
codeword	O
x	O
is	O
encoded	O
in	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
to	O
be	O
the	O
linear	O
function	O
(	O
mod	O
2	O
)	O
.	O
</s>
<s>
To	O
test	O
if	O
a	O
string	O
w	O
is	O
a	O
codeword	O
of	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
,	O
all	O
we	O
have	O
to	O
do	O
is	O
test	O
if	O
the	O
function	O
it	O
encodes	O
is	O
linear	O
.	O
</s>
<s>
The	O
Long	B-Error_Name
code	I-Error_Name
is	O
another	O
code	O
with	O
very	O
large	O
blowup	O
which	O
is	O
close	O
to	O
locally	O
testable	O
.	O
</s>
<s>
Other	O
locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
include	O
Reed-Muller	O
codes	O
(	O
see	O
locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
for	O
a	O
decoding	O
algorithm	O
)	O
,	O
Reed-Solomon	B-Error_Name
codes	I-Error_Name
,	O
and	O
the	O
short	O
code	O
.	O
</s>
