<s>
In	O
information	O
theory	O
and	O
coding	B-Error_Name
theory	I-Error_Name
with	O
applications	O
in	O
computer	B-General_Concept
science	I-General_Concept
and	O
telecommunication	O
,	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
(	O
EDAC	O
)	O
or	O
error	B-Error_Name
control	I-Error_Name
are	O
techniques	O
that	O
enable	O
reliable	B-Protocol
delivery	I-Protocol
of	O
digital	B-General_Concept
data	I-General_Concept
over	O
unreliable	O
communication	O
channels	O
.	O
</s>
<s>
Error	B-Error_Name
detection	I-Error_Name
techniques	O
allow	O
detecting	O
such	O
errors	O
,	O
while	O
error	B-Error_Name
correction	I-Error_Name
enables	O
reconstruction	O
of	O
the	O
original	O
data	O
in	O
many	O
cases	O
.	O
</s>
<s>
Error	B-Error_Name
detection	I-Error_Name
is	O
the	O
detection	O
of	O
errors	O
caused	O
by	O
noise	O
or	O
other	O
impairments	O
during	O
transmission	O
from	O
the	O
transmitter	O
to	O
the	O
receiver	O
.	O
</s>
<s>
Error	B-Error_Name
correction	I-Error_Name
is	O
the	O
detection	O
of	O
errors	O
and	O
reconstruction	O
of	O
the	O
original	O
,	O
error-free	O
data	O
.	O
</s>
<s>
The	O
effectiveness	O
of	O
their	O
error	B-Error_Name
correction	I-Error_Name
method	O
was	O
verified	O
by	O
the	O
accuracy	O
of	O
copying	O
through	O
the	O
centuries	O
demonstrated	O
by	O
discovery	O
of	O
the	O
Dead	O
Sea	O
Scrolls	O
in	O
1947	O
–	O
1956	O
,	O
dating	O
from	O
.	O
</s>
<s>
The	O
modern	O
development	O
of	O
error	B-Error_Name
correction	I-Error_Name
codes	I-Error_Name
is	O
credited	O
to	O
Richard	O
Hamming	O
in	O
1947	O
.	O
</s>
<s>
A	O
description	O
of	O
Hamming	B-Error_Name
's	I-Error_Name
code	I-Error_Name
appeared	O
in	O
Claude	O
Shannon	O
's	O
A	O
Mathematical	O
Theory	O
of	O
Communication	O
and	O
was	O
quickly	O
generalized	O
by	O
Marcel	O
J	O
.	O
E	O
.	O
Golay	O
.	O
</s>
<s>
All	O
error-detection	O
and	O
correction	O
schemes	O
add	O
some	O
redundancy	B-General_Concept
(	O
i.e.	O
,	O
some	O
extra	O
data	O
)	O
to	O
a	O
message	O
,	O
which	O
receivers	O
can	O
use	O
to	O
check	O
consistency	O
of	O
the	O
delivered	O
message	O
,	O
and	O
to	O
recover	O
data	O
that	O
has	O
been	O
determined	O
to	O
be	O
corrupted	O
.	O
</s>
<s>
If	O
error	B-Error_Name
detection	I-Error_Name
is	O
required	O
,	O
a	O
receiver	O
can	O
simply	O
apply	O
the	O
same	O
algorithm	O
to	O
the	O
received	O
data	O
bits	O
and	O
compare	O
its	O
output	O
with	O
the	O
received	O
check	O
bits	O
;	O
if	O
the	O
values	O
do	O
not	O
match	O
,	O
an	O
error	O
has	O
occurred	O
at	O
some	O
point	O
during	O
the	O
transmission	O
.	O
</s>
<s>
If	O
error	B-Error_Name
correction	I-Error_Name
is	O
required	O
,	O
a	O
receiver	O
can	O
apply	O
the	O
decoding	O
algorithm	O
to	O
the	O
received	O
data	O
bits	O
and	O
the	O
received	O
check	O
bits	O
to	O
recover	O
the	O
original	O
error-free	O
data	O
.	O
</s>
<s>
Good	O
error	B-Error_Name
control	I-Error_Name
performance	O
requires	O
the	O
scheme	O
to	O
be	O
selected	O
based	O
on	O
the	O
characteristics	O
of	O
the	O
communication	O
channel	O
.	O
</s>
<s>
This	O
is	O
known	O
as	O
automatic	B-Error_Name
repeat	I-Error_Name
request	I-Error_Name
(	O
ARQ	O
)	O
,	O
and	O
is	O
most	O
notably	O
used	O
in	O
the	O
Internet	O
.	O
</s>
<s>
An	O
alternate	O
approach	O
for	O
error	B-Error_Name
control	I-Error_Name
is	O
hybrid	B-Error_Name
automatic	I-Error_Name
repeat	I-Error_Name
request	I-Error_Name
(	O
HARQ	B-Error_Name
)	O
,	O
which	O
is	O
a	O
combination	O
of	O
ARQ	O
and	O
error-correction	B-Error_Name
coding	O
.	O
</s>
<s>
There	O
are	O
three	O
major	O
types	O
of	O
error	B-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
Automatic	B-Error_Name
repeat	I-Error_Name
request	I-Error_Name
(	O
ARQ	O
)	O
is	O
an	O
error	B-Error_Name
control	I-Error_Name
method	O
for	O
data	O
transmission	O
that	O
makes	O
use	O
of	O
error-detection	O
codes	O
,	O
acknowledgment	O
and/or	O
negative	O
acknowledgment	O
messages	O
,	O
and	O
timeouts	O
to	O
achieve	O
reliable	O
data	O
transmission	O
.	O
</s>
<s>
An	O
acknowledgment	O
is	O
a	O
message	O
sent	O
by	O
the	O
receiver	O
to	O
indicate	O
that	O
it	O
has	O
correctly	O
received	O
a	O
data	B-Protocol
frame	I-Protocol
.	O
</s>
<s>
Usually	O
,	O
when	O
the	O
transmitter	O
does	O
not	O
receive	O
the	O
acknowledgment	O
before	O
the	O
timeout	O
occurs	O
(	O
i.e.	O
,	O
within	O
a	O
reasonable	O
amount	O
of	O
time	O
after	O
sending	O
the	O
data	B-Protocol
frame	I-Protocol
)	O
,	O
it	O
retransmits	O
the	O
frame	O
until	O
it	O
is	O
either	O
correctly	O
received	O
or	O
the	O
error	O
persists	O
beyond	O
a	O
predetermined	O
number	O
of	O
retransmissions	O
.	O
</s>
<s>
Three	O
types	O
of	O
ARQ	B-Error_Name
protocols	I-Error_Name
are	O
Stop-and-wait	B-Error_Name
ARQ	I-Error_Name
,	O
Go-Back-N	B-Error_Name
ARQ	I-Error_Name
,	O
and	O
Selective	B-Error_Name
Repeat	I-Error_Name
ARQ	I-Error_Name
.	O
</s>
<s>
However	O
,	O
ARQ	O
requires	O
the	O
availability	O
of	O
a	O
back	B-Protocol
channel	I-Protocol
,	O
results	O
in	O
possibly	O
increased	O
latency	O
due	O
to	O
retransmissions	O
,	O
and	O
requires	O
the	O
maintenance	O
of	O
buffers	O
and	O
timers	O
for	O
retransmissions	O
,	O
which	O
in	O
the	O
case	O
of	O
network	B-Protocol
congestion	I-Protocol
can	O
put	O
a	O
strain	O
on	O
the	O
server	O
and	O
overall	O
network	O
capacity	O
.	O
</s>
<s>
For	O
example	O
,	O
ARQ	O
is	O
used	O
on	O
shortwave	O
radio	O
data	O
links	O
in	O
the	O
form	O
of	O
ARQ-E	B-Device
,	O
or	O
combined	O
with	O
multiplexing	O
as	O
ARQ-M	B-Device
.	O
</s>
<s>
Forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
(	O
FEC	O
)	O
is	O
a	O
process	O
of	O
adding	O
redundant	B-General_Concept
data	I-General_Concept
such	O
as	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
(	O
ECC	O
)	O
to	O
a	O
message	O
so	O
that	O
it	O
can	O
be	O
recovered	O
by	O
a	O
receiver	O
even	O
when	O
a	O
number	O
of	O
errors	O
(	O
up	O
to	O
the	O
capability	O
of	O
the	O
code	O
being	O
used	O
)	O
are	O
introduced	O
,	O
either	O
during	O
the	O
process	O
of	O
transmission	O
or	O
on	O
storage	O
.	O
</s>
<s>
Since	O
the	O
receiver	O
does	O
not	O
have	O
to	O
ask	O
the	O
sender	O
for	O
retransmission	O
of	O
the	O
data	O
,	O
a	O
backchannel	B-Protocol
is	O
not	O
required	O
in	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
Error-correcting	B-Error_Name
codes	I-Error_Name
are	O
used	O
in	O
lower-layer	B-Application
communication	O
such	O
as	O
cellular	O
network	O
,	O
high-speed	O
fiber-optic	B-Application
communication	I-Application
and	O
Wi-Fi	O
,	O
as	O
well	O
as	O
for	O
reliable	O
storage	O
in	O
media	O
such	O
as	O
flash	B-Device
memory	I-Device
,	O
hard	B-Device
disk	I-Device
and	O
RAM	B-General_Concept
.	O
</s>
<s>
Error-correcting	B-Error_Name
codes	I-Error_Name
are	O
usually	O
distinguished	O
between	O
convolutional	B-Error_Name
codes	I-Error_Name
and	O
block	O
codes	O
:	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
are	O
processed	O
on	O
a	O
bit-by-bit	O
basis	O
.	O
</s>
<s>
Early	O
examples	O
of	O
block	O
codes	O
are	O
repetition	B-Error_Name
codes	I-Error_Name
,	O
Hamming	B-Error_Name
codes	I-Error_Name
and	O
multidimensional	B-Error_Name
parity-check	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
They	O
were	O
followed	O
by	O
a	O
number	O
of	O
efficient	O
codes	O
,	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
being	O
the	O
most	O
notable	O
due	O
to	O
their	O
current	O
widespread	O
use	O
.	O
</s>
<s>
Turbo	B-Error_Name
codes	I-Error_Name
and	O
low-density	B-Error_Name
parity-check	I-Error_Name
codes	I-Error_Name
(	O
LDPC	B-Error_Name
)	O
are	O
relatively	O
new	O
constructions	O
that	O
can	O
provide	O
almost	O
optimal	O
efficiency	O
.	O
</s>
<s>
Shannon	O
's	O
theorem	O
is	O
an	O
important	O
theorem	O
in	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
,	O
and	O
describes	O
the	O
maximum	O
information	O
rate	O
at	O
which	O
reliable	O
communication	O
is	O
possible	O
over	O
a	O
channel	O
that	O
has	O
a	O
certain	O
error	O
probability	O
or	O
signal-to-noise	O
ratio	O
(	O
SNR	O
)	O
.	O
</s>
<s>
The	O
actual	O
maximum	O
code	O
rate	O
allowed	O
depends	O
on	O
the	O
error-correcting	B-Error_Name
code	I-Error_Name
used	O
,	O
and	O
may	O
be	O
lower	O
.	O
</s>
<s>
Hybrid	B-Error_Name
ARQ	I-Error_Name
is	O
a	O
combination	O
of	O
ARQ	O
and	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
Messages	O
are	O
always	O
transmitted	O
with	O
FEC	O
parity	O
data	O
(	O
and	O
error-detection	O
redundancy	B-General_Concept
)	O
.	O
</s>
<s>
The	O
latter	O
approach	O
is	O
particularly	O
attractive	O
on	O
an	O
erasure	O
channel	O
when	O
using	O
a	O
rateless	B-Error_Name
erasure	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Error	B-Error_Name
detection	I-Error_Name
is	O
most	O
commonly	O
realized	O
using	O
a	O
suitable	O
hash	B-Error_Name
function	I-Error_Name
(	O
or	O
specifically	O
,	O
a	O
checksum	B-Protocol
,	O
cyclic	O
redundancy	B-General_Concept
check	O
or	O
other	O
algorithm	O
)	O
.	O
</s>
<s>
A	O
hash	B-Error_Name
function	I-Error_Name
adds	O
a	O
fixed-length	O
tag	O
to	O
a	O
message	O
,	O
which	O
enables	O
receivers	O
to	O
verify	O
the	O
delivered	O
message	O
by	O
recomputing	O
the	O
tag	O
and	O
comparing	O
it	O
with	O
the	O
one	O
provided	O
.	O
</s>
<s>
There	O
exists	O
a	O
vast	O
variety	O
of	O
different	O
hash	B-Error_Name
function	I-Error_Name
designs	O
.	O
</s>
<s>
However	O
,	O
some	O
are	O
of	O
particularly	O
widespread	O
use	O
because	O
of	O
either	O
their	O
simplicity	O
or	O
their	O
suitability	O
for	O
detecting	O
certain	O
kinds	O
of	O
errors	O
(	O
e.g.	O
,	O
the	O
cyclic	O
redundancy	B-General_Concept
check	O
's	O
performance	O
in	O
detecting	O
burst	O
errors	O
)	O
.	O
</s>
<s>
A	O
repetition	B-Error_Name
code	I-Error_Name
is	O
a	O
coding	O
scheme	O
that	O
repeats	O
the	O
bits	O
across	O
a	O
channel	O
to	O
achieve	O
error-free	O
communication	O
.	O
</s>
<s>
A	O
repetition	B-Error_Name
code	I-Error_Name
is	O
very	O
inefficient	O
,	O
and	O
can	O
be	O
susceptible	O
to	O
problems	O
if	O
the	O
error	O
occurs	O
in	O
exactly	O
the	O
same	O
place	O
for	O
each	O
group	O
(	O
e.g.	O
,	O
"	O
1010	O
1010	O
1010	O
"	O
in	O
the	O
previous	O
example	O
would	O
be	O
detected	O
as	O
correct	O
)	O
.	O
</s>
<s>
The	O
advantage	O
of	O
repetition	B-Error_Name
codes	I-Error_Name
is	O
that	O
they	O
are	O
extremely	O
simple	O
,	O
and	O
are	O
in	O
fact	O
used	O
in	O
some	O
transmissions	O
of	O
numbers	O
stations	O
.	O
</s>
<s>
Parity	O
bits	O
added	O
to	O
each	O
"	O
word	O
"	O
sent	O
are	O
called	O
transverse	B-Error_Name
redundancy	I-Error_Name
checks	I-Error_Name
,	O
while	O
those	O
added	O
at	O
the	O
end	O
of	O
a	O
stream	O
of	O
"	O
words	O
"	O
are	O
called	O
longitudinal	B-Error_Name
redundancy	I-Error_Name
checks	I-Error_Name
.	O
</s>
<s>
With	O
more	O
error	B-Error_Name
correction	I-Error_Name
bits	O
,	O
more	O
errors	O
can	O
be	O
detected	O
and	O
in	O
some	O
cases	O
corrected	O
.	O
</s>
<s>
A	O
checksum	B-Protocol
of	O
a	O
message	O
is	O
a	O
modular	O
arithmetic	O
sum	O
of	O
message	O
code	O
words	O
of	O
a	O
fixed	O
word	O
length	O
(	O
e.g.	O
,	O
byte	O
values	O
)	O
.	O
</s>
<s>
Checksum	B-Protocol
schemes	O
include	O
parity	O
bits	O
,	O
check	B-Error_Name
digits	I-Error_Name
,	O
and	O
longitudinal	B-Error_Name
redundancy	I-Error_Name
checks	I-Error_Name
.	O
</s>
<s>
Some	O
checksum	B-Protocol
schemes	O
,	O
such	O
as	O
the	O
Damm	O
algorithm	O
,	O
the	O
Luhn	B-Algorithm
algorithm	I-Algorithm
,	O
and	O
the	O
Verhoeff	B-Error_Name
algorithm	I-Error_Name
,	O
are	O
specifically	O
designed	O
to	O
detect	O
errors	O
commonly	O
introduced	O
by	O
humans	O
in	O
writing	O
down	O
or	O
remembering	O
identification	O
numbers	O
.	O
</s>
<s>
A	O
cyclic	O
redundancy	B-General_Concept
check	O
(	O
CRC	O
)	O
is	O
a	O
non-secure	O
hash	B-Error_Name
function	I-Error_Name
designed	O
to	O
detect	O
accidental	O
changes	O
to	O
digital	B-General_Concept
data	I-General_Concept
in	O
computer	B-Architecture
networks	I-Architecture
.	O
</s>
<s>
CRCs	O
are	O
particularly	O
easy	O
to	O
implement	O
in	O
hardware	O
and	O
are	O
therefore	O
commonly	O
used	O
in	O
computer	B-Architecture
networks	I-Architecture
and	O
storage	O
devices	O
such	O
as	O
hard	B-Device
disk	I-Device
drives	I-Device
.	O
</s>
<s>
The	O
output	O
of	O
a	O
cryptographic	O
hash	B-Error_Name
function	I-Error_Name
,	O
also	O
known	O
as	O
a	O
message	O
digest	O
,	O
can	O
provide	O
strong	O
assurances	B-Protocol
about	O
data	O
integrity	O
,	O
whether	O
changes	O
of	O
the	O
data	O
are	O
accidental	O
(	O
e.g.	O
,	O
due	O
to	O
transmission	O
errors	O
)	O
or	O
maliciously	O
introduced	O
.	O
</s>
<s>
Any	O
modification	O
to	O
the	O
data	O
will	O
likely	O
be	O
detected	O
through	O
a	O
mismatching	O
hash	B-Error_Name
value	I-Error_Name
.	O
</s>
<s>
Furthermore	O
,	O
given	O
some	O
hash	B-Error_Name
value	I-Error_Name
,	O
it	O
is	O
typically	O
infeasible	O
to	O
find	O
some	O
input	O
data	O
(	O
other	O
than	O
the	O
one	O
given	O
)	O
that	O
will	O
yield	O
the	O
same	O
hash	B-Error_Name
value	I-Error_Name
.	O
</s>
<s>
If	O
an	O
attacker	O
can	O
change	O
not	O
only	O
the	O
message	O
but	O
also	O
the	O
hash	B-Error_Name
value	I-Error_Name
,	O
then	O
a	O
keyed	O
hash	B-Error_Name
or	O
message	B-Algorithm
authentication	I-Algorithm
code	I-Algorithm
(	O
MAC	O
)	O
can	O
be	O
used	O
for	O
additional	O
security	O
.	O
</s>
<s>
Without	O
knowing	O
the	O
key	O
,	O
it	O
is	O
not	O
possible	O
for	O
the	O
attacker	O
to	O
easily	O
or	O
conveniently	O
calculate	O
the	O
correct	O
keyed	O
hash	B-Error_Name
value	I-Error_Name
for	O
a	O
modified	O
message	O
.	O
</s>
<s>
Any	O
error-correcting	B-Error_Name
code	I-Error_Name
can	O
be	O
used	O
for	O
error	B-Error_Name
detection	I-Error_Name
.	O
</s>
<s>
Using	O
minimum-distance-based	O
error-correcting	B-Error_Name
codes	I-Error_Name
for	O
error	B-Error_Name
detection	I-Error_Name
can	O
be	O
suitable	O
if	O
a	O
strict	O
limit	O
on	O
the	O
minimum	O
number	O
of	O
errors	O
to	O
be	O
detected	O
is	O
desired	O
.	O
</s>
<s>
Codes	O
with	O
minimum	O
Hamming	O
distance	O
d	O
=	O
2	O
are	O
degenerate	O
cases	O
of	O
error-correcting	B-Error_Name
codes	I-Error_Name
,	O
and	O
can	O
be	O
used	O
to	O
detect	O
single	O
errors	O
.	O
</s>
<s>
Applications	O
that	O
require	O
low	O
latency	O
(	O
such	O
as	O
telephone	O
conversations	O
)	O
cannot	O
use	O
automatic	B-Error_Name
repeat	I-Error_Name
request	I-Error_Name
(	O
ARQ	O
)	O
;	O
they	O
must	O
use	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
(	O
FEC	O
)	O
.	O
</s>
<s>
Reliability	O
and	O
inspection	O
engineering	O
also	O
make	O
use	O
of	O
the	O
theory	O
of	O
error-correcting	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
In	O
a	O
typical	O
TCP/IP	B-Protocol
stack	I-Protocol
,	O
error	B-Error_Name
control	I-Error_Name
is	O
performed	O
at	O
multiple	O
levels	O
:	O
</s>
<s>
Each	O
Ethernet	B-Protocol
frame	I-Protocol
uses	O
CRC-32	O
error	B-Error_Name
detection	I-Error_Name
.	O
</s>
<s>
The	O
IPv4	B-Protocol
header	O
contains	O
a	O
checksum	B-Protocol
protecting	O
the	O
contents	O
of	O
the	O
header	O
.	O
</s>
<s>
Packets	B-Protocol
with	O
incorrect	O
checksums	B-Algorithm
are	O
dropped	O
within	O
the	O
network	O
or	O
at	O
the	O
receiver	O
.	O
</s>
<s>
The	O
checksum	B-Protocol
was	O
omitted	O
from	O
the	O
IPv6	B-Protocol
header	O
in	O
order	O
to	O
minimize	O
processing	O
costs	O
in	O
network	B-Protocol
routing	I-Protocol
and	O
because	O
current	O
link	B-Protocol
layer	I-Protocol
technology	O
is	O
assumed	O
to	O
provide	O
sufficient	O
error	B-Error_Name
detection	I-Error_Name
(	O
see	O
also	O
RFC	O
3819	O
)	O
.	O
</s>
<s>
UDP	B-Protocol
has	O
an	O
optional	O
checksum	B-Protocol
covering	O
the	O
payload	O
and	O
addressing	O
information	O
in	O
the	O
UDP	B-Protocol
and	O
IP	O
headers	O
.	O
</s>
<s>
Packets	B-Protocol
with	O
incorrect	O
checksums	B-Algorithm
are	O
discarded	O
by	O
the	O
network	B-Protocol
stack	I-Protocol
.	O
</s>
<s>
The	O
checksum	B-Protocol
is	O
optional	O
under	O
IPv4	B-Protocol
,	O
and	O
required	O
under	O
IPv6	B-Protocol
.	O
</s>
<s>
TCP	B-Protocol
provides	O
a	O
checksum	B-Protocol
for	O
protecting	O
the	O
payload	O
and	O
addressing	O
information	O
in	O
the	O
TCP	B-Protocol
and	O
IP	O
headers	O
.	O
</s>
<s>
Packets	B-Protocol
with	O
incorrect	O
checksums	B-Algorithm
are	O
discarded	O
by	O
the	O
network	B-Protocol
stack	I-Protocol
,	O
and	O
eventually	O
get	O
retransmitted	O
using	O
ARQ	O
,	O
either	O
explicitly	O
(	O
such	O
as	O
through	O
three-way	O
handshake	O
)	O
or	O
implicitly	O
due	O
to	O
a	O
timeout	O
.	O
</s>
<s>
The	O
development	O
of	O
error-correction	B-Error_Name
codes	O
was	O
tightly	O
coupled	O
with	O
the	O
history	O
of	O
deep-space	O
missions	O
due	O
to	O
the	O
extreme	O
dilution	O
of	O
signal	O
power	O
over	O
interplanetary	O
distances	O
,	O
and	O
the	O
limited	O
power	O
availability	O
aboard	O
space	O
probes	O
.	O
</s>
<s>
Whereas	O
early	O
missions	O
sent	O
their	O
data	O
uncoded	O
,	O
starting	O
in	O
1968	O
,	O
digital	O
error	B-Error_Name
correction	I-Error_Name
was	O
implemented	O
in	O
the	O
form	O
of	O
(	O
sub-optimally	O
decoded	O
)	O
convolutional	B-Error_Name
codes	I-Error_Name
and	O
Reed	O
–	O
Muller	O
codes	O
.	O
</s>
<s>
The	O
Voyager	O
1	O
and	O
Voyager	O
2	O
missions	O
,	O
which	O
started	O
in	O
1977	O
,	O
were	O
designed	O
to	O
deliver	O
color	O
imaging	O
and	O
scientific	O
information	O
from	O
Jupiter	O
and	O
Saturn	B-Device
.	O
</s>
<s>
This	O
resulted	O
in	O
increased	O
coding	O
requirements	O
,	O
and	O
thus	O
,	O
the	O
spacecraft	O
were	O
supported	O
by	O
(	O
optimally	O
Viterbi-decoded	O
)	O
convolutional	B-Error_Name
codes	I-Error_Name
that	O
could	O
be	O
concatenated	O
with	O
an	O
outer	O
Golay	O
(	O
24	O
,	O
12	O
,	O
8	O
)	O
code	O
.	O
</s>
<s>
The	O
Voyager	O
2	O
craft	O
additionally	O
supported	O
an	O
implementation	O
of	O
a	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
.	O
</s>
<s>
The	O
concatenated	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
–	O
Viterbi	O
(	O
RSV	O
)	O
code	O
allowed	O
for	O
very	O
powerful	O
error	B-Error_Name
correction	I-Error_Name
,	O
and	O
enabled	O
the	O
spacecraft	O
's	O
extended	O
journey	O
to	O
Uranus	O
and	O
Neptune	B-Application
.	O
</s>
<s>
The	O
Consultative	O
Committee	O
for	O
Space	O
Data	O
Systems	O
currently	O
recommends	O
usage	O
of	O
error	B-Error_Name
correction	I-Error_Name
codes	I-Error_Name
with	O
performance	O
similar	O
to	O
the	O
Voyager	O
2	O
RSV	O
code	O
as	O
a	O
minimum	O
.	O
</s>
<s>
Concatenated	O
codes	O
are	O
increasingly	O
falling	O
out	O
of	O
favor	O
with	O
space	O
missions	O
,	O
and	O
are	O
replaced	O
by	O
more	O
powerful	O
codes	O
such	O
as	O
Turbo	B-Error_Name
codes	I-Error_Name
or	O
LDPC	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
The	O
different	O
kinds	O
of	O
deep	O
space	O
and	O
orbital	O
missions	O
that	O
are	O
conducted	O
suggest	O
that	O
trying	O
to	O
find	O
a	O
one-size-fits-all	O
error	B-Error_Name
correction	I-Error_Name
system	O
will	O
be	O
an	O
ongoing	O
problem	O
.	O
</s>
<s>
The	O
demand	O
for	O
satellite	O
transponder	B-Architecture
bandwidth	O
continues	O
to	O
grow	O
,	O
fueled	O
by	O
the	O
desire	O
to	O
deliver	O
television	O
(	O
including	O
new	O
channels	O
and	O
high-definition	B-Device
television	I-Device
)	O
and	O
IP	O
data	O
.	O
</s>
<s>
Transponder	B-Architecture
availability	O
and	O
bandwidth	O
constraints	O
have	O
limited	O
this	O
growth	O
.	O
</s>
<s>
Transponder	B-Architecture
capacity	O
is	O
determined	O
by	O
the	O
selected	O
modulation	B-Application
scheme	O
and	O
the	O
proportion	O
of	O
capacity	O
consumed	O
by	O
FEC	O
.	O
</s>
<s>
Error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
codes	O
are	O
often	O
used	O
to	O
improve	O
the	O
reliability	O
of	O
data	O
storage	O
media	O
.	O
</s>
<s>
A	O
parity	O
track	O
capable	O
of	O
detecting	O
single-bit	O
errors	O
was	O
present	O
on	O
the	O
first	O
magnetic	B-Architecture
tape	I-Architecture
data	I-Architecture
storage	I-Architecture
in	O
1951	O
.	O
</s>
<s>
The	O
optimal	O
rectangular	O
code	O
used	O
in	O
group	B-Device
coded	I-Device
recording	I-Device
tapes	O
not	O
only	O
detects	O
but	O
also	O
corrects	O
single-bit	O
errors	O
.	O
</s>
<s>
Some	O
file	O
formats	O
,	O
particularly	O
archive	B-General_Concept
formats	I-General_Concept
,	O
include	O
a	O
checksum	B-Protocol
(	O
most	O
often	O
CRC32	O
)	O
to	O
detect	O
corruption	O
and	O
truncation	O
and	O
can	O
employ	O
redundancy	B-General_Concept
or	O
parity	B-General_Concept
files	I-General_Concept
to	O
recover	O
portions	O
of	O
corrupted	O
data	O
.	O
</s>
<s>
Reed-Solomon	B-Error_Name
codes	I-Error_Name
are	O
used	O
in	O
compact	O
discs	O
to	O
correct	O
errors	O
caused	O
by	O
scratches	O
.	O
</s>
<s>
Modern	O
hard	B-Device
drives	I-Device
use	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
to	O
detect	O
and	O
correct	O
minor	O
errors	O
in	O
sector	O
reads	O
,	O
and	O
to	O
recover	O
corrupted	O
data	O
from	O
failing	O
sectors	O
and	O
store	O
that	O
data	O
in	O
the	O
spare	O
sectors	O
.	O
</s>
<s>
RAID	B-Architecture
systems	O
use	O
a	O
variety	O
of	O
error	B-Error_Name
correction	I-Error_Name
techniques	O
to	O
recover	O
data	O
when	O
a	O
hard	B-Device
drive	I-Device
completely	O
fails	O
.	O
</s>
<s>
Filesystems	O
such	O
as	O
ZFS	B-Application
or	O
Btrfs	B-Operating_System
,	O
as	O
well	O
as	O
some	O
RAID	B-Architecture
implementations	O
,	O
support	O
data	B-Error_Name
scrubbing	I-Error_Name
and	O
resilvering	O
,	O
which	O
allows	O
bad	O
blocks	O
to	O
be	O
detected	O
and	O
(	O
hopefully	O
)	O
recovered	O
before	O
they	O
are	O
used	O
.	O
</s>
<s>
Dynamic	O
random-access	O
memory	O
(	O
DRAM	O
)	O
may	O
provide	O
stronger	O
protection	O
against	O
soft	O
errors	O
by	O
relying	O
on	O
error-correcting	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Such	O
error-correcting	B-Error_Name
memory	O
,	O
known	O
as	O
ECC	O
or	O
EDAC-protected	O
memory	O
,	O
is	O
particularly	O
desirable	O
for	O
mission-critical	O
applications	O
,	O
such	O
as	O
scientific	O
computing	O
,	O
financial	O
,	O
medical	O
,	O
etc	O
.	O
</s>
<s>
Error-correcting	B-Error_Name
memory	O
controllers	O
traditionally	O
use	O
Hamming	B-Error_Name
codes	I-Error_Name
,	O
although	O
some	O
use	O
triple	B-Error_Name
modular	I-Error_Name
redundancy	I-Error_Name
.	O
</s>
<s>
As	O
long	O
as	O
a	O
single-event	O
upset	O
(	O
SEU	O
)	O
does	O
not	O
exceed	O
the	O
error	O
threshold	O
(	O
e.g.	O
,	O
a	O
single	O
error	O
)	O
in	O
any	O
particular	O
word	O
between	O
accesses	O
,	O
it	O
can	O
be	O
corrected	O
(	O
e.g.	O
,	O
by	O
a	O
single-bit	O
error-correcting	B-Error_Name
code	I-Error_Name
)	O
,	O
and	O
the	O
illusion	O
of	O
an	O
error-free	O
memory	O
system	O
may	O
be	O
maintained	O
.	O
</s>
<s>
In	O
addition	O
to	O
hardware	O
providing	O
features	O
required	O
for	O
ECC	B-General_Concept
memory	I-General_Concept
to	O
operate	O
,	O
operating	B-General_Concept
systems	I-General_Concept
usually	O
contain	O
related	O
reporting	O
facilities	O
that	O
are	O
used	O
to	O
provide	O
notifications	O
when	O
soft	O
errors	O
are	O
transparently	O
recovered	O
.	O
</s>
<s>
One	O
example	O
is	O
the	O
Linux	B-Operating_System
kernel	I-Operating_System
's	O
EDAC	O
subsystem	O
(	O
previously	O
known	O
as	O
Bluesmoke	O
)	O
,	O
which	O
collects	O
the	O
data	O
from	O
error-checking-enabled	O
components	O
inside	O
a	O
computer	O
system	O
;	O
besides	O
collecting	O
and	O
reporting	O
back	O
the	O
events	O
related	O
to	O
ECC	B-General_Concept
memory	I-General_Concept
,	O
it	O
also	O
supports	O
other	O
checksumming	O
errors	O
,	O
including	O
those	O
detected	O
on	O
the	O
PCI	B-Protocol
bus	I-Protocol
.	O
</s>
<s>
A	O
few	O
systems	O
also	O
support	O
memory	B-General_Concept
scrubbing	I-General_Concept
to	O
catch	O
and	O
correct	O
errors	O
early	O
before	O
they	O
become	O
unrecoverable	O
.	O
</s>
