<s>
In	O
coding	B-Error_Name
theory	I-Error_Name
,	O
expander	B-Error_Name
codes	I-Error_Name
form	O
a	O
class	O
of	O
error-correcting	B-Error_Name
codes	I-Error_Name
that	O
are	O
constructed	O
from	O
bipartite	O
expander	O
graphs	O
.	O
</s>
<s>
Along	O
with	O
Justesen	O
codes	O
,	O
expander	B-Error_Name
codes	I-Error_Name
are	O
of	O
particular	O
interest	O
since	O
they	O
have	O
a	O
constant	O
positive	O
rate	O
,	O
a	O
constant	O
positive	O
relative	O
distance	O
,	O
and	O
a	O
constant	O
alphabet	O
size	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
alphabet	O
contains	O
only	O
two	O
elements	O
,	O
so	O
expander	B-Error_Name
codes	I-Error_Name
belong	O
to	O
the	O
class	O
of	O
binary	O
codes	O
.	O
</s>
<s>
Furthermore	O
,	O
expander	B-Error_Name
codes	I-Error_Name
can	O
be	O
both	O
encoded	O
and	O
decoded	O
in	O
time	O
proportional	O
to	O
the	O
block	O
length	O
of	O
the	O
code	O
.	O
</s>
<s>
In	O
coding	B-Error_Name
theory	I-Error_Name
,	O
an	O
expander	B-Error_Name
code	I-Error_Name
is	O
a	O
linear	O
block	O
code	O
whose	O
parity	O
check	O
matrix	O
is	O
the	O
adjacency	O
matrix	O
of	O
a	O
bipartite	O
expander	O
graph	O
.	O
</s>
<s>
Let	O
be	O
an	O
error-correcting	B-Error_Name
code	O
of	O
block	O
length	O
.	O
</s>
<s>
The	O
expander	B-Error_Name
code	I-Error_Name
is	O
the	O
code	O
of	O
block	O
length	O
whose	O
codewords	O
are	O
the	O
words	O
such	O
that	O
,	O
for	O
,	O
is	O
a	O
codeword	O
of	O
.	O
</s>
<s>
Then	O
the	O
distance	O
of	O
a	O
expander	B-Error_Name
code	I-Error_Name
is	O
at	O
least	O
.	O
</s>
<s>
The	O
encoding	O
time	O
for	O
an	O
expander	B-Error_Name
code	I-Error_Name
is	O
upper	O
bounded	O
by	O
that	O
of	O
a	O
general	O
linear	O
code	O
-	O
by	O
matrix	O
multiplication	O
.	O
</s>
<s>
Decoding	O
of	O
expander	B-Error_Name
codes	I-Error_Name
is	O
possible	O
in	O
time	O
when	O
using	O
the	O
following	O
algorithm	O
.	O
</s>
