<s>
In	O
coding	B-Error_Name
theory	I-Error_Name
,	O
fountain	B-Error_Name
codes	I-Error_Name
(	O
also	O
known	O
as	O
rateless	B-Error_Name
erasure	I-Error_Name
codes	I-Error_Name
)	O
are	O
a	O
class	O
of	O
erasure	O
codes	O
with	O
the	O
property	O
that	O
a	O
potentially	O
limitless	O
sequence	O
of	O
encoding	B-Algorithm
symbols	O
can	O
be	O
generated	O
from	O
a	O
given	O
set	O
of	O
source	O
symbols	O
such	O
that	O
the	O
original	O
source	O
symbols	O
can	O
ideally	O
be	O
recovered	O
from	O
any	O
subset	O
of	O
the	O
encoding	B-Algorithm
symbols	O
of	O
size	O
equal	O
to	O
or	O
only	O
slightly	O
larger	O
than	O
the	O
number	O
of	O
source	O
symbols	O
.	O
</s>
<s>
A	O
fountain	B-Error_Name
code	I-Error_Name
is	O
optimal	O
if	O
the	O
original	O
k	O
source	O
symbols	O
can	O
be	O
recovered	O
from	O
any	O
k	O
successfully	O
received	O
encoding	B-Algorithm
symbols	O
(	O
i.e.	O
,	O
excluding	O
those	O
that	O
were	O
erased	O
)	O
.	O
</s>
<s>
Fountain	B-Error_Name
codes	I-Error_Name
are	O
known	O
that	O
have	O
efficient	O
encoding	B-Algorithm
and	O
decoding	O
algorithms	O
and	O
that	O
allow	O
the	O
recovery	O
of	O
the	O
original	O
k	O
source	O
symbols	O
from	O
any	O
k’	O
of	O
the	O
encoding	B-Algorithm
symbols	O
with	O
high	O
probability	O
,	O
where	O
k’	O
is	O
just	O
slightly	O
larger	O
than	O
k	O
.	O
</s>
<s>
LT	O
codes	O
were	O
the	O
first	O
practical	O
realization	O
of	O
fountain	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Raptor	O
codes	O
and	O
online	O
codes	O
were	O
subsequently	O
introduced	O
,	O
and	O
achieve	O
linear	O
time	O
encoding	B-Algorithm
and	O
decoding	O
complexity	B-General_Concept
through	O
a	O
pre-coding	O
stage	O
of	O
the	O
input	O
symbols	O
.	O
</s>
<s>
Fountain	B-Error_Name
codes	I-Error_Name
are	O
flexibly	O
applicable	O
at	O
a	O
fixed	O
code	O
rate	O
,	O
or	O
where	O
a	O
fixed	O
code	O
rate	O
cannot	O
be	O
determined	O
a	O
priori	O
,	O
and	O
where	O
efficient	O
encoding	B-Algorithm
and	O
decoding	O
of	O
large	O
amounts	O
of	O
data	O
is	O
required	O
.	O
</s>
<s>
One	O
example	O
is	O
that	O
of	O
a	O
data	B-Device
carousel	I-Device
,	O
where	O
some	O
large	O
file	O
is	O
continuously	O
broadcast	O
to	O
a	O
set	O
of	O
receivers	O
.	O
</s>
<s>
Using	O
a	O
fixed-rate	O
erasure	O
code	O
,	O
a	O
receiver	O
missing	O
a	O
source	O
symbol	O
(	O
due	O
to	O
a	O
transmission	O
error	O
)	O
faces	O
the	O
coupon	O
collector	O
's	O
problem	O
:	O
it	O
must	O
successfully	O
receive	O
an	O
encoding	B-Algorithm
symbol	O
which	O
it	O
does	O
not	O
already	O
have	O
.	O
</s>
<s>
This	O
problem	O
becomes	O
much	O
more	O
apparent	O
when	O
using	O
a	O
traditional	O
short-length	O
erasure	O
code	O
,	O
as	O
the	O
file	O
must	O
be	O
split	O
into	O
several	O
blocks	O
,	O
each	O
being	O
separately	O
encoded	B-Algorithm
:	O
the	O
receiver	O
must	O
now	O
collect	O
the	O
required	O
number	O
of	O
missing	O
encoding	B-Algorithm
symbols	O
for	O
each	O
block	O
.	O
</s>
<s>
Using	O
a	O
fountain	B-Error_Name
code	I-Error_Name
,	O
it	O
suffices	O
for	O
a	O
receiver	O
to	O
retrieve	O
any	O
subset	O
of	O
encoding	B-Algorithm
symbols	O
of	O
size	O
slightly	O
larger	O
than	O
the	O
set	O
of	O
source	O
symbols	O
.	O
</s>
<s>
(	O
In	O
practice	O
,	O
the	O
broadcast	O
is	O
typically	O
scheduled	O
for	O
a	O
fixed	O
period	O
of	O
time	O
by	O
an	O
operator	O
based	O
on	O
characteristics	O
of	O
the	O
network	O
and	O
receivers	O
and	O
desired	O
delivery	O
reliability	O
,	O
and	O
thus	O
the	O
fountain	B-Error_Name
code	I-Error_Name
is	O
used	O
at	O
a	O
code	O
rate	O
that	O
is	O
determined	O
dynamically	O
at	O
the	O
time	O
when	O
the	O
file	O
is	O
scheduled	O
to	O
be	O
broadcast	O
.	O
)	O
</s>
<s>
Another	O
application	O
is	O
that	O
of	O
hybrid	B-Error_Name
ARQ	I-Error_Name
in	O
reliable	B-Operating_System
multicast	I-Operating_System
scenarios	O
:	O
parity	O
information	O
that	O
is	O
requested	O
by	O
a	O
receiver	O
can	O
potentially	O
be	O
useful	O
for	O
all	O
receivers	O
in	O
the	O
multicast	O
group	O
.	O
</s>
<s>
Raptor	O
codes	O
are	O
the	O
most	O
efficient	O
fountain	B-Error_Name
codes	I-Error_Name
at	O
this	O
time	O
,	O
having	O
very	O
efficient	O
linear	O
time	O
encoding	B-Algorithm
and	O
decoding	O
algorithms	O
,	O
and	O
requiring	O
only	O
a	O
small	O
constant	O
number	O
of	O
XOR	O
operations	O
per	O
generated	O
symbol	O
for	O
both	O
encoding	B-Algorithm
and	O
decoding	O
.	O
</s>
<s>
IETF	O
RFC	O
5053	O
specifies	O
in	O
detail	O
a	O
systematic	O
Raptor	O
code	O
,	O
which	O
has	O
been	O
adopted	O
into	O
multiple	O
standards	O
beyond	O
the	O
IETF	O
,	O
such	O
as	O
within	O
the	O
3GPP	O
MBMS	O
standard	O
for	O
broadcast	O
file	O
delivery	O
and	O
streaming	O
services	O
,	O
the	O
DVB-H	B-Protocol
IPDC	O
standard	O
for	O
delivering	O
IP	O
services	O
over	O
DVB	B-Application
networks	O
,	O
and	O
DVB-IPTV	O
for	O
delivering	O
commercial	O
TV	O
services	O
over	O
an	O
IP	O
network	O
.	O
</s>
<s>
This	O
code	O
can	O
be	O
used	O
with	O
up	O
to	O
8,192	O
source	O
symbols	O
in	O
a	O
source	O
block	O
,	O
and	O
a	O
total	O
of	O
up	O
to	O
65,536	O
encoded	B-Algorithm
symbols	O
generated	O
for	O
a	O
source	O
block	O
.	O
</s>
<s>
The	O
relative	O
reception	O
overhead	O
is	O
defined	O
as	O
the	O
extra	O
encoding	B-Algorithm
data	O
required	O
beyond	O
the	O
length	O
of	O
the	O
source	O
data	O
to	O
recover	O
the	O
original	O
source	O
data	O
,	O
measured	O
as	O
a	O
percentage	O
of	O
the	O
size	O
of	O
the	O
source	O
data	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
relative	O
reception	O
overhead	O
is	O
0.2	O
%	O
,	O
then	O
this	O
means	O
that	O
source	O
data	O
of	O
size	O
1megabyte	O
can	O
be	O
recovered	O
from	O
1.002megabytes	O
of	O
encoding	B-Algorithm
data	O
.	O
</s>
<s>
This	O
code	O
is	O
able	O
to	O
recover	O
a	O
source	O
block	O
from	O
any	O
set	O
of	O
encoded	B-Algorithm
symbols	O
equal	O
to	O
the	O
number	O
of	O
source	O
symbols	O
in	O
the	O
source	O
block	O
with	O
high	O
probability	O
,	O
and	O
in	O
rare	O
cases	O
from	O
slightly	O
more	O
than	O
the	O
number	O
of	O
source	O
symbols	O
in	O
the	O
source	O
block	O
.	O
</s>
<s>
In	O
that	O
respect	O
,	O
fountain	B-Error_Name
codes	I-Error_Name
are	O
expected	O
to	O
allow	O
efficient	O
repair	O
process	O
in	O
case	O
of	O
a	O
failure	O
:	O
When	O
a	O
single	O
encoded	B-Algorithm
symbol	O
is	O
lost	O
,	O
it	O
should	O
not	O
require	O
too	O
much	O
communication	O
and	O
computation	O
among	O
other	O
encoded	B-Algorithm
symbols	O
in	O
order	O
to	O
resurrect	O
the	O
lost	O
symbol	O
.	O
</s>
<s>
Repairable	O
fountain	B-Error_Name
codes	I-Error_Name
are	O
projected	O
to	O
address	O
fountain	B-Error_Name
code	I-Error_Name
design	O
objectives	O
for	O
storage	O
systems	O
.	O
</s>
<s>
A	O
detailed	O
survey	O
about	O
fountain	B-Error_Name
codes	I-Error_Name
and	O
their	O
applications	O
can	O
be	O
found	O
at	O
.	O
</s>
<s>
Liquid	O
Cloud	O
Storage	O
is	O
based	O
on	O
using	O
a	O
large	O
erasure	O
code	O
such	O
as	O
the	O
RaptorQ	O
code	O
specified	O
in	O
IETF	O
RFC	O
6330	O
(	O
which	O
provides	O
significantly	O
better	O
data	O
protection	O
than	O
other	O
systems	O
)	O
,	O
using	O
a	O
background	O
repair	O
process	O
(	O
which	O
significantly	O
reduces	O
the	O
repair	O
bandwidth	O
requirements	O
compared	O
to	O
other	O
systems	O
)	O
,	O
and	O
using	O
a	O
stream	O
data	O
organization	O
(	O
which	O
allows	O
fast	O
access	O
to	O
data	O
even	O
when	O
not	O
all	O
encoded	B-Algorithm
symbols	O
are	O
available	O
)	O
.	O
</s>
