<s>
In	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
telecommunication	O
,	O
information	O
theory	O
,	O
and	O
searching	O
theory	O
,	O
error-correcting	B-Error_Name
codes	I-Error_Name
with	I-Error_Name
feedback	I-Error_Name
are	O
error	B-Error_Name
correcting	I-Error_Name
codes	I-Error_Name
designed	O
to	O
work	O
in	O
the	O
presence	O
of	O
feedback	O
from	O
the	O
receiver	O
to	O
the	O
sender	O
.	O
</s>
<s>
An	O
error-correcting	B-Error_Name
code	I-Error_Name
is	O
a	O
way	O
of	O
encoding	B-Error_Name
x	O
as	O
a	O
message	O
such	O
that	O
Bob	O
will	O
successfully	O
understand	O
the	O
value	O
x	O
as	O
intended	O
by	O
Alice	O
,	O
even	O
if	O
the	O
message	O
Alice	O
sends	O
and	O
the	O
message	O
Bob	O
receives	O
differ	O
.	O
</s>
<s>
In	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
with	O
feedback	O
,	O
the	O
channel	O
is	O
two-way	O
:	O
Bob	O
can	O
send	O
feedback	O
to	O
Alice	O
about	O
the	O
message	O
he	O
received	O
.	O
</s>
<s>
In	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
without	O
noisy	O
feedback	O
,	O
the	O
feedback	O
received	O
by	O
the	O
sender	O
is	O
always	O
free	O
of	O
errors	O
.	O
</s>
<s>
In	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
with	O
noisy	O
feedback	O
,	O
errors	O
can	O
occur	O
in	O
the	O
feedback	O
,	O
as	O
well	O
as	O
in	O
the	O
message	O
.	O
</s>
<s>
An	O
error-correcting	B-Error_Name
code	I-Error_Name
with	O
noiseless	O
feedback	O
is	O
equivalent	O
to	O
an	O
adaptive	B-Algorithm
search	B-Application
strategy	O
with	O
errors	O
.	O
</s>
<s>
In	O
his	O
1964	O
dissertation	O
,	O
Elwyn	O
Berlekamp	O
considered	O
error	B-Error_Name
correcting	I-Error_Name
codes	I-Error_Name
with	O
noiseless	O
feedback	O
.	O
</s>
