<s>
In	O
computing	O
,	O
telecommunication	O
,	O
information	O
theory	O
,	O
and	O
coding	B-Error_Name
theory	I-Error_Name
,	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
(	O
FEC	O
)	O
or	O
channel	O
coding	O
is	O
a	O
technique	O
used	O
for	O
controlling	B-Error_Name
errors	I-Error_Name
in	O
data	O
transmission	O
over	O
unreliable	O
or	O
noisy	O
communication	O
channels	O
.	O
</s>
<s>
The	O
central	O
idea	O
is	O
that	O
the	O
sender	O
encodes	O
the	O
message	O
in	O
a	O
redundant	B-General_Concept
way	O
,	O
most	O
often	O
by	O
using	O
an	O
error	B-Error_Name
correction	I-Error_Name
code	I-Error_Name
or	O
error	B-Error_Name
correcting	I-Error_Name
code	I-Error_Name
,	O
(	O
ECC	B-General_Concept
)	O
.	O
</s>
<s>
The	O
redundancy	B-General_Concept
allows	O
the	O
receiver	O
not	O
only	O
to	O
detect	B-Error_Name
errors	I-Error_Name
that	O
may	O
occur	O
anywhere	O
in	O
the	O
message	O
,	O
but	O
often	O
to	O
correct	O
a	O
limited	O
number	O
of	O
errors	O
.	O
</s>
<s>
The	O
American	O
mathematician	O
Richard	O
Hamming	B-Error_Name
pioneered	O
this	O
field	O
in	O
the	O
1940s	O
and	O
invented	O
the	O
first	O
error-correcting	B-Error_Name
code	I-Error_Name
in	O
1950	O
:	O
the	O
Hamming	B-Error_Name
(	O
7	O
,	O
4	O
)	O
code	O
.	O
</s>
<s>
FEC	O
is	O
widely	O
used	O
in	O
modems	B-Application
and	O
in	O
cellular	O
networks	O
,	O
as	O
well	O
.	O
</s>
<s>
The	O
Viterbi	O
decoder	O
implements	O
a	O
soft-decision	B-Error_Name
algorithm	I-Error_Name
to	O
demodulate	O
digital	O
data	O
from	O
an	O
analog	O
signal	O
corrupted	O
by	O
noise	O
.	O
</s>
<s>
FEC	O
information	O
is	O
added	O
to	O
mass	B-Device
storage	I-Device
(	O
magnetic	O
,	O
optical	O
and	O
solid	O
state/flash	O
based	O
)	O
devices	O
to	O
enable	O
recovery	O
of	O
corrupted	O
data	O
,	O
and	O
is	O
used	O
as	O
ECC	B-General_Concept
computer	B-General_Concept
memory	I-General_Concept
on	O
systems	O
that	O
require	O
special	O
provisions	O
for	O
reliability	O
.	O
</s>
<s>
The	O
maximum	O
proportion	O
of	O
errors	O
or	O
missing	O
bits	O
that	O
can	O
be	O
corrected	O
is	O
determined	O
by	O
the	O
design	O
of	O
the	O
ECC	B-General_Concept
,	O
so	O
different	O
forward	O
error	B-Error_Name
correcting	I-Error_Name
codes	I-Error_Name
are	O
suitable	O
for	O
different	O
conditions	O
.	O
</s>
<s>
In	O
general	O
,	O
a	O
stronger	O
code	O
induces	O
more	O
redundancy	B-General_Concept
that	O
needs	O
to	O
be	O
transmitted	O
using	O
the	O
available	O
bandwidth	O
,	O
which	O
reduces	O
the	O
effective	O
bit-rate	O
while	O
improving	O
the	O
received	O
effective	O
signal-to-noise	O
ratio	O
.	O
</s>
<s>
After	O
years	O
of	O
research	O
,	O
some	O
advanced	O
FEC	O
systems	O
like	O
polar	B-Error_Name
code	I-Error_Name
come	O
very	O
close	O
to	O
the	O
theoretical	O
maximum	O
given	O
by	O
the	O
Shannon	O
channel	O
capacity	O
under	O
the	O
hypothesis	O
of	O
an	O
infinite	O
length	O
frame	O
.	O
</s>
<s>
ECC	B-General_Concept
is	O
accomplished	O
by	O
adding	O
redundancy	B-General_Concept
to	O
the	O
transmitted	O
information	O
using	O
an	O
algorithm	O
.	O
</s>
<s>
A	O
redundant	B-General_Concept
bit	O
may	O
be	O
a	O
complicated	O
function	O
of	O
many	O
original	O
information	O
bits	O
.	O
</s>
<s>
A	O
simplistic	O
example	O
of	O
ECC	B-General_Concept
is	O
to	O
transmit	O
each	O
data	O
bit	O
3	O
times	O
,	O
which	O
is	O
known	O
as	O
a	O
(	O
3	O
,	O
1	O
)	O
repetition	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
The	O
correcting	O
ability	O
of	O
this	O
ECC	B-General_Concept
is	O
:	O
</s>
<s>
Though	O
simple	O
to	O
implement	O
and	O
widely	O
used	O
,	O
this	O
triple	B-Error_Name
modular	I-Error_Name
redundancy	I-Error_Name
is	O
a	O
relatively	O
inefficient	O
ECC	B-General_Concept
.	O
</s>
<s>
Better	O
ECC	B-General_Concept
codes	O
typically	O
examine	O
the	O
last	O
several	O
tens	O
or	O
even	O
the	O
last	O
several	O
hundreds	O
of	O
previously	O
received	O
bits	O
to	O
determine	O
how	O
to	O
decode	O
the	O
current	O
small	O
handful	O
of	O
bits	O
(	O
typically	O
in	O
groups	O
of	O
2	O
to	O
8	O
bits	O
)	O
.	O
</s>
<s>
ECC	B-General_Concept
could	O
be	O
said	O
to	O
work	O
by	O
"	O
averaging	O
noise	O
"	O
;	O
since	O
each	O
data	O
bit	O
affects	O
many	O
transmitted	O
symbols	O
,	O
the	O
corruption	O
of	O
some	O
symbols	O
by	O
noise	O
usually	O
allows	O
the	O
original	O
user	O
data	O
to	O
be	O
extracted	O
from	O
the	O
other	O
,	O
uncorrupted	O
received	O
symbols	O
that	O
also	O
depend	O
on	O
the	O
same	O
user	O
data	O
.	O
</s>
<s>
Because	O
of	O
this	O
"	O
risk-pooling	O
"	O
effect	O
,	O
digital	O
communication	O
systems	O
that	O
use	O
ECC	B-General_Concept
tend	O
to	O
work	O
well	O
above	O
a	O
certain	O
minimum	O
signal-to-noise	O
ratio	O
and	O
not	O
at	O
all	O
below	O
it	O
.	O
</s>
<s>
Interleaving	O
ECC	B-General_Concept
coded	O
data	O
can	O
reduce	O
the	O
all	O
or	O
nothing	O
properties	O
of	O
transmitted	O
ECC	B-General_Concept
codes	O
when	O
the	O
channel	O
errors	O
tend	O
to	O
occur	O
in	O
bursts	O
.	O
</s>
<s>
However	O
,	O
some	O
systems	O
adapt	O
to	O
the	O
given	O
channel	O
error	O
conditions	O
:	O
some	O
instances	O
of	O
hybrid	B-Error_Name
automatic	I-Error_Name
repeat-request	I-Error_Name
use	O
a	O
fixed	O
ECC	B-General_Concept
method	O
as	O
long	O
as	O
the	O
ECC	B-General_Concept
can	O
handle	O
the	O
error	O
rate	O
,	O
then	O
switch	O
to	O
ARQ	B-Error_Name
when	O
the	O
error	O
rate	O
gets	O
too	O
high	O
;	O
</s>
<s>
adaptive	B-Protocol
modulation	I-Protocol
and	I-Protocol
coding	I-Protocol
uses	O
a	O
variety	O
of	O
ECC	B-General_Concept
rates	O
,	O
adding	O
more	O
error-correction	B-Error_Name
bits	O
per	O
packet	O
when	O
there	O
are	O
higher	O
error	O
rates	O
in	O
the	O
channel	O
,	O
or	O
taking	O
them	O
out	O
when	O
they	O
are	O
not	O
needed	O
.	O
</s>
<s>
The	O
two	O
main	O
categories	O
of	O
ECC	B-General_Concept
codes	O
are	O
block	O
codes	O
and	O
convolutional	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Convolutional	B-Error_Name
codes	I-Error_Name
work	O
on	O
bit	O
or	O
symbol	O
streams	O
of	O
arbitrary	O
length	O
.	O
</s>
<s>
They	O
are	O
most	O
often	O
soft	O
decoded	O
with	O
the	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
,	O
though	O
other	O
algorithms	O
are	O
sometimes	O
used	O
.	O
</s>
<s>
Viterbi	O
decoding	O
allows	O
asymptotically	O
optimal	O
decoding	O
efficiency	O
with	O
increasing	O
constraint	B-Error_Name
length	I-Error_Name
of	O
the	O
convolutional	B-Error_Name
code	I-Error_Name
,	O
but	O
at	O
the	O
expense	O
of	O
exponentially	O
increasing	O
complexity	O
.	O
</s>
<s>
A	O
convolutional	B-Error_Name
code	I-Error_Name
that	O
is	O
terminated	O
is	O
also	O
a	O
'	O
block	O
code	O
 '	O
in	O
that	O
it	O
encodes	O
a	O
block	O
of	O
input	O
data	O
,	O
but	O
the	O
block	O
size	O
of	O
a	O
convolutional	B-Error_Name
code	I-Error_Name
is	O
generally	O
arbitrary	O
,	O
while	O
block	O
codes	O
have	O
a	O
fixed	O
size	O
dictated	O
by	O
their	O
algebraic	O
characteristics	O
.	O
</s>
<s>
Types	O
of	O
termination	O
for	O
convolutional	B-Error_Name
codes	I-Error_Name
include	O
"	O
tail-biting	O
"	O
and	O
"	O
bit-flushing	O
"	O
.	O
</s>
<s>
There	O
are	O
many	O
types	O
of	O
block	O
codes	O
;	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	I-Error_Name
is	O
noteworthy	O
for	O
its	O
widespread	O
use	O
in	O
compact	O
discs	O
,	O
DVDs	O
,	O
and	O
hard	O
disk	O
drives	O
.	O
</s>
<s>
Other	O
examples	O
of	O
classical	O
block	O
codes	O
include	O
Golay	O
,	O
BCH	O
,	O
Multidimensional	B-Error_Name
parity	I-Error_Name
,	O
and	O
Hamming	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Hamming	B-Error_Name
ECC	B-General_Concept
is	O
commonly	O
used	O
to	O
correct	O
NAND	O
flash	O
memory	O
errors	O
.	O
</s>
<s>
This	O
provides	O
single-bit	O
error	B-Error_Name
correction	I-Error_Name
and	O
2-bit	O
error	B-Error_Name
detection	I-Error_Name
.	O
</s>
<s>
Hamming	B-Error_Name
codes	I-Error_Name
are	O
only	O
suitable	O
for	O
more	O
reliable	O
single-level	O
cell	O
(	O
SLC	O
)	O
NAND	O
.	O
</s>
<s>
Denser	O
multi-level	B-Device
cell	I-Device
(	O
MLC	O
)	O
NAND	O
may	O
use	O
multi-bit	O
correcting	O
ECC	B-General_Concept
such	O
as	O
BCH	O
or	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
.	O
</s>
<s>
NOR	O
Flash	O
typically	O
does	O
not	O
use	O
any	O
error	B-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
In	O
contrast	O
,	O
convolutional	B-Error_Name
codes	I-Error_Name
are	O
typically	O
decoded	O
using	O
soft-decision	O
algorithms	O
like	O
the	O
Viterbi	O
,	O
MAP	O
or	O
BCJR	B-Error_Name
algorithms	I-Error_Name
,	O
which	O
process	O
(	O
discretized	O
)	O
analog	O
signals	O
,	O
and	O
which	O
allow	O
for	O
much	O
higher	O
error-correction	B-Error_Name
performance	O
than	O
hard-decision	O
decoding	O
.	O
</s>
<s>
In	O
contrast	O
to	O
classical	O
block	O
codes	O
that	O
often	O
specify	O
an	O
error-detecting	O
or	O
error-correcting	B-Error_Name
ability	O
,	O
many	O
modern	O
block	O
codes	O
such	O
as	O
LDPC	B-Error_Name
codes	I-Error_Name
lack	O
such	O
guarantees	O
.	O
</s>
<s>
Most	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
codes	O
correct	O
only	O
bit-flips	O
,	O
but	O
not	O
bit-insertions	O
or	O
bit-deletions	O
.	O
</s>
<s>
In	O
this	O
setting	O
,	O
the	O
Hamming	B-Error_Name
distance	O
is	O
the	O
appropriate	O
way	O
to	O
measure	O
the	O
bit	O
error	O
rate	O
.	O
</s>
<s>
A	O
few	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
codes	O
are	O
designed	O
to	O
correct	O
bit-insertions	O
and	O
bit-deletions	O
,	O
such	O
as	O
Marker	O
Codes	O
and	O
Watermark	O
Codes	O
.	O
</s>
<s>
The	O
fundamental	O
principle	O
of	O
ECC	B-General_Concept
is	O
to	O
add	O
redundant	B-General_Concept
bits	O
in	O
order	O
to	O
help	O
the	O
decoder	O
to	O
find	O
out	O
the	O
true	O
message	O
that	O
was	O
encoded	O
by	O
the	O
transmitter	O
.	O
</s>
<s>
The	O
code-rate	O
of	O
a	O
given	O
ECC	B-General_Concept
system	O
is	O
defined	O
as	O
the	O
ratio	O
between	O
the	O
number	O
of	O
information	O
bits	O
and	O
the	O
total	O
number	O
of	O
bits	O
(	O
i.e.	O
,	O
information	O
plus	O
redundancy	B-General_Concept
bits	O
)	O
in	O
a	O
given	O
communication	O
package	O
.	O
</s>
<s>
A	O
low	O
code-rate	O
close	O
to	O
zero	O
implies	O
a	O
strong	O
code	O
that	O
uses	O
many	O
redundant	B-General_Concept
bits	O
to	O
achieve	O
a	O
good	O
performance	O
,	O
while	O
a	O
large	O
code-rate	O
close	O
to	O
1	O
implies	O
a	O
weak	O
code	O
.	O
</s>
<s>
The	O
redundant	B-General_Concept
bits	O
that	O
protect	O
the	O
information	O
have	O
to	O
be	O
transferred	O
using	O
the	O
same	O
communication	O
resources	O
that	O
they	O
are	O
trying	O
to	O
protect	O
.	O
</s>
<s>
On	O
the	O
other	O
extreme	O
,	O
not	O
using	O
any	O
ECC	B-General_Concept
(	O
i.e.	O
,	O
a	O
code-rate	O
equal	O
to	O
1	O
)	O
uses	O
the	O
full	O
channel	O
for	O
information	O
transfer	O
purposes	O
,	O
at	O
the	O
cost	O
of	O
leaving	O
the	O
bits	O
without	O
any	O
additional	O
protection	O
.	O
</s>
<s>
One	O
interesting	O
question	O
is	O
the	O
following	O
:	O
how	O
efficient	O
in	O
terms	O
of	O
information	O
transfer	O
can	O
an	O
ECC	B-General_Concept
be	O
that	O
has	O
a	O
negligible	O
decoding	O
error	O
rate	O
?	O
</s>
<s>
This	O
question	O
was	O
answered	O
by	O
Claude	O
Shannon	O
with	O
his	O
second	O
theorem	O
,	O
which	O
says	O
that	O
the	O
channel	O
capacity	O
is	O
the	O
maximum	O
bit	O
rate	O
achievable	O
by	O
any	O
ECC	B-General_Concept
whose	O
error	O
rate	O
tends	O
to	O
zero	O
:	O
His	O
proof	O
relies	O
on	O
Gaussian	O
random	O
coding	O
,	O
which	O
is	O
not	O
suitable	O
to	O
real-world	O
applications	O
.	O
</s>
<s>
Classical	O
(	O
algebraic	O
)	O
block	O
codes	O
and	O
convolutional	B-Error_Name
codes	I-Error_Name
are	O
frequently	O
combined	O
in	O
concatenated	O
coding	O
schemes	O
in	O
which	O
a	O
short	O
constraint-length	O
Viterbi-decoded	O
convolutional	B-Error_Name
code	I-Error_Name
does	O
most	O
of	O
the	O
work	O
and	O
a	O
block	O
code	O
(	O
usually	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
)	O
with	O
larger	O
symbol	O
size	O
and	O
block	O
length	O
"	O
mops	O
up	O
"	O
any	O
errors	O
made	O
by	O
the	O
convolutional	O
decoder	O
.	O
</s>
<s>
Single	O
pass	O
decoding	O
with	O
this	O
family	O
of	O
error	B-Error_Name
correction	I-Error_Name
codes	I-Error_Name
can	O
yield	O
very	O
low	O
error	O
rates	O
,	O
but	O
for	O
long	O
range	O
transmission	O
conditions	O
(	O
like	O
deep	O
space	O
)	O
iterative	O
decoding	O
is	O
recommended	O
.	O
</s>
<s>
They	O
can	O
provide	O
performance	O
very	O
close	O
to	O
the	O
channel	O
capacity	O
(	O
the	O
theoretical	O
maximum	O
)	O
using	O
an	O
iterated	O
soft-decision	B-Error_Name
decoding	I-Error_Name
approach	O
,	O
at	O
linear	O
time	O
complexity	O
in	O
terms	O
of	O
their	O
block	O
length	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
were	O
first	O
introduced	O
by	O
Robert	O
G	O
.	O
Gallager	O
in	O
his	O
PhD	O
thesis	O
in	O
1960	O
,	O
</s>
<s>
but	O
due	O
to	O
the	O
computational	O
effort	O
in	O
implementing	O
encoder	O
and	O
decoder	O
and	O
the	O
introduction	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
,	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
are	O
now	O
used	O
in	O
many	O
recent	O
high-speed	O
communication	O
standards	O
,	O
such	O
as	O
DVB-S2	O
(	O
Digital	O
Video	O
Broadcasting	O
–	O
Satellite	O
–	O
Second	O
Generation	O
)	O
,	O
WiMAX	O
(	O
IEEE	O
802.16e	O
standard	O
for	O
microwave	O
communications	O
)	O
,	O
High-Speed	O
Wireless	O
LAN	O
(	O
IEEE	O
802.11n	O
)	O
,	O
10GBase-T	O
Ethernet	O
(	O
802.3an	O
)	O
and	O
G.hn/G.9960	B-General_Concept
(	O
ITU-T	O
Standard	O
for	O
networking	O
over	O
power	O
lines	O
,	O
phone	O
lines	O
and	O
coaxial	O
cable	O
)	O
.	O
</s>
<s>
Other	O
LDPC	B-Error_Name
codes	I-Error_Name
are	O
standardized	O
for	O
wireless	O
communication	O
standards	O
within	O
3GPP	O
MBMS	O
(	O
see	O
fountain	B-Error_Name
codes	I-Error_Name
)	O
.	O
</s>
<s>
Turbo	B-Error_Name
coding	I-Error_Name
is	O
an	O
iterated	O
soft-decoding	O
scheme	O
that	O
combines	O
two	O
or	O
more	O
relatively	O
simple	O
convolutional	B-Error_Name
codes	I-Error_Name
and	O
an	O
interleaver	O
to	O
produce	O
a	O
block	O
code	O
that	O
can	O
perform	O
to	O
within	O
a	O
fraction	O
of	O
a	O
decibel	O
of	O
the	O
Shannon	O
limit	O
.	O
</s>
<s>
Predating	O
LDPC	B-Error_Name
codes	I-Error_Name
in	O
terms	O
of	O
practical	O
application	O
,	O
they	O
now	O
provide	O
similar	O
performance	O
.	O
</s>
<s>
One	O
of	O
the	O
earliest	O
commercial	O
applications	O
of	O
turbo	B-Error_Name
coding	I-Error_Name
was	O
the	O
CDMA2000	O
1x	O
(	O
TIA	O
IS-2000	O
)	O
digital	O
cellular	O
technology	O
developed	O
by	O
Qualcomm	O
and	O
sold	O
by	O
Verizon	B-Application
Wireless	I-Application
,	O
Sprint	O
,	O
and	O
other	O
carriers	O
.	O
</s>
<s>
Like	O
1x	O
,	O
EV-DO	O
was	O
developed	O
by	O
Qualcomm	O
,	O
and	O
is	O
sold	O
by	O
Verizon	B-Application
Wireless	I-Application
,	O
Sprint	O
,	O
and	O
other	O
carriers	O
(	O
Verizon	B-Application
's	O
marketing	O
name	O
for	O
1xEV-DO	O
is	O
Broadband	O
Access	O
,	O
Sprint	O
's	O
consumer	O
and	O
business	O
marketing	O
names	O
for	O
1xEV-DO	O
are	O
Power	O
Vision	O
and	O
Mobile	O
Broadband	O
,	O
respectively	O
)	O
.	O
</s>
<s>
Locally	B-Error_Name
decodable	I-Error_Name
codes	I-Error_Name
are	O
error-correcting	B-Error_Name
codes	I-Error_Name
for	O
which	O
single	O
bits	O
of	O
the	O
message	O
can	O
be	O
probabilistically	O
recovered	O
by	O
only	O
looking	O
at	O
a	O
small	O
(	O
say	O
constant	O
)	O
number	O
of	O
positions	O
of	O
a	O
codeword	O
,	O
even	O
after	O
the	O
codeword	O
has	O
been	O
corrupted	O
at	O
some	O
constant	O
fraction	O
of	O
positions	O
.	O
</s>
<s>
Locally	B-Error_Name
testable	I-Error_Name
codes	I-Error_Name
are	O
error-correcting	B-Error_Name
codes	I-Error_Name
for	O
which	O
it	O
can	O
be	O
checked	O
probabilistically	O
whether	O
a	O
signal	O
is	O
close	O
to	O
a	O
codeword	O
by	O
only	O
looking	O
at	O
a	O
small	O
number	O
of	O
positions	O
of	O
the	O
signal	O
.	O
</s>
<s>
Interleaving	O
is	O
frequently	O
used	O
in	O
digital	O
communication	O
and	O
storage	O
systems	O
to	O
improve	O
the	O
performance	O
of	O
forward	O
error	B-Error_Name
correcting	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
If	O
the	O
number	O
of	O
errors	O
within	O
a	O
code	O
word	O
exceeds	O
the	O
error-correcting	B-Error_Name
code	I-Error_Name
's	O
capability	O
,	O
it	O
fails	O
to	O
recover	O
the	O
original	O
code	O
word	O
.	O
</s>
<s>
Therefore	O
,	O
interleaving	O
is	O
widely	O
used	O
for	O
burst	B-Error_Name
error-correction	I-Error_Name
.	O
</s>
<s>
The	O
analysis	O
of	O
modern	O
iterated	O
codes	O
,	O
like	O
turbo	B-Error_Name
codes	I-Error_Name
and	O
LDPC	B-Error_Name
codes	I-Error_Name
,	O
typically	O
assumes	O
an	O
independent	O
distribution	O
of	O
errors	O
.	O
</s>
<s>
Systems	O
using	O
LDPC	B-Error_Name
codes	I-Error_Name
therefore	O
typically	O
employ	O
additional	O
interleaving	O
across	O
the	O
symbols	O
within	O
a	O
code	O
word	O
.	O
</s>
<s>
For	O
turbo	B-Error_Name
codes	I-Error_Name
,	O
an	O
interleaver	O
is	O
an	O
integral	O
component	O
and	O
its	O
proper	O
design	O
is	O
crucial	O
for	O
good	O
performance	O
.	O
</s>
<s>
a	O
contention-free	O
quadratic	O
permutation	B-Algorithm
polynomial	I-Algorithm
(	O
QPP	O
)	O
.	O
</s>
<s>
In	O
multi-carrier	O
communication	O
systems	O
,	O
interleaving	O
across	O
carriers	O
may	O
be	O
employed	O
to	O
provide	O
frequency	O
diversity	B-Protocol
,	O
e.g.	O
,	O
to	O
mitigate	O
frequency-selective	O
fading	O
or	O
narrowband	O
interference	O
.	O
</s>
<s>
Here	O
,	O
each	O
group	O
of	O
the	O
same	O
letter	O
represents	O
a	O
4-bit	O
one-bit	O
error-correcting	B-Error_Name
codeword	O
.	O
</s>
<s>
In	O
each	O
of	O
the	O
codewords	O
""	O
,	O
""	O
,	O
""	O
,	O
and	O
""	O
,	O
only	O
one	O
bit	O
is	O
altered	O
,	O
so	O
one-bit	O
error-correcting	B-Error_Name
code	I-Error_Name
will	O
decode	O
everything	O
correctly	O
.	O
</s>
<s>
An	O
example	O
of	O
such	O
an	O
algorithm	O
is	O
based	O
on	O
neural	B-Architecture
network	I-Architecture
structures	O
.	O
</s>
<s>
Simulating	O
the	O
behaviour	O
of	O
error-correcting	B-Error_Name
codes	I-Error_Name
(	O
ECCs	O
)	O
in	O
software	O
is	O
a	O
common	O
practice	O
to	O
design	O
,	O
validate	O
and	O
improve	O
ECCs	O
.	O
</s>
<s>
The	O
upcoming	O
wireless	O
5G	O
standard	O
raises	O
a	O
new	O
range	O
of	O
applications	O
for	O
the	O
software	O
ECCs	O
:	O
the	O
Cloud	O
Radio	O
Access	O
Networks	O
(	O
C-RAN	B-General_Concept
)	O
in	O
a	O
Software-defined	B-Architecture
radio	I-Architecture
(	O
SDR	O
)	O
context	O
.	O
</s>
<s>
(	O
A	O
Fast	O
Forward	B-Error_Name
Error	I-Error_Name
Correction	I-Error_Name
Toolbox	O
)	O
:	O
a	O
full	O
communication	O
chain	O
in	O
C++	O
(	O
many	O
supported	O
codes	O
like	O
Turbo	O
,	O
LDPC	B-Error_Name
,	O
Polar	B-Error_Name
codes	I-Error_Name
,	O
etc	O
.	O
</s>
<s>
IT++	B-Language
:	O
a	O
C++	O
library	O
of	O
classes	O
and	O
functions	O
for	O
linear	O
algebra	O
,	O
numerical	O
optimization	O
,	O
signal	O
processing	O
,	O
communications	O
,	O
and	O
statistics	O
.	O
</s>
<s>
Barker	B-Protocol
code	I-Protocol
used	O
for	O
radar	O
,	O
telemetry	O
,	O
ultra	O
sound	O
,	O
Wifi	O
,	O
DSSS	O
mobile	O
phone	O
networks	O
,	O
GPS	O
etc	O
.	O
</s>
