<s>
In	O
information	O
theory	O
,	O
a	O
low-density	O
parity-check	O
(	O
LDPC	B-Error_Name
)	O
code	O
is	O
a	O
linear	O
error	B-Error_Name
correcting	I-Error_Name
code	I-Error_Name
,	O
a	O
method	O
of	O
transmitting	O
a	O
message	O
over	O
a	O
noisy	O
transmission	O
channel	O
.	O
</s>
<s>
An	O
LDPC	B-Error_Name
code	I-Error_Name
is	O
constructed	O
using	O
a	O
sparse	O
Tanner	O
graph	O
(	O
subclass	O
of	O
the	O
bipartite	O
graph	O
)	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
are	O
capacity-approaching	O
codes	O
,	O
which	O
means	O
that	O
practical	O
constructions	O
exist	O
that	O
allow	O
the	O
noise	O
threshold	O
to	O
be	O
set	O
very	O
close	O
to	O
the	O
theoretical	O
maximum	O
(	O
the	O
Shannon	O
limit	O
)	O
for	O
a	O
symmetric	O
memoryless	O
channel	O
.	O
</s>
<s>
Using	O
iterative	O
belief	O
propagation	O
techniques	O
,	O
LDPC	B-Error_Name
codes	I-Error_Name
can	O
be	O
decoded	O
in	O
time	O
linear	O
to	O
their	O
block	O
length	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
are	O
finding	O
increasing	O
use	O
in	O
applications	O
requiring	O
reliable	O
and	O
highly	O
efficient	O
information	O
transfer	O
over	O
bandwidth-constrained	O
or	O
return-channel-constrained	O
links	O
in	O
the	O
presence	O
of	O
corrupting	O
noise	O
.	O
</s>
<s>
Implementation	O
of	O
LDPC	B-Error_Name
codes	I-Error_Name
has	O
lagged	O
behind	O
that	O
of	O
other	O
codes	O
,	O
notably	O
turbo	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
The	O
fundamental	O
patent	O
for	O
turbo	B-Error_Name
codes	I-Error_Name
expired	O
on	O
August	O
29	O
,	O
2013	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
are	O
also	O
known	O
as	O
Gallager	B-Error_Name
codes	I-Error_Name
,	O
in	O
honor	O
of	O
Robert	O
G	O
.	O
Gallager	O
,	O
who	O
developed	O
the	O
LDPC	B-Error_Name
concept	O
in	O
his	O
doctoral	O
dissertation	O
at	O
the	O
Massachusetts	O
Institute	O
of	O
Technology	O
in	O
1960	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
have	O
also	O
been	O
shown	O
to	O
have	O
ideal	O
combinatorial	O
properties	O
.	O
</s>
<s>
In	O
his	O
dissertation	O
,	O
Gallager	O
showed	O
that	O
LDPC	B-Error_Name
codes	I-Error_Name
achieve	O
the	O
Gilbert	O
–	O
Varshamov	O
bound	O
for	O
linear	O
codes	O
over	O
binary	O
fields	O
with	O
high	O
probability	O
.	O
</s>
<s>
In	O
2020	O
it	O
was	O
shown	O
that	O
Gallager	O
's	O
LDPC	B-Error_Name
codes	I-Error_Name
achieve	O
list	O
decoding	O
capacity	O
and	O
also	O
achieve	O
the	O
Gilbert	O
–	O
Varshamov	O
bound	O
for	O
linear	O
codes	O
over	O
general	O
fields	O
.	O
</s>
<s>
Impractical	O
to	O
implement	O
when	O
first	O
developed	O
by	O
Gallager	O
in	O
1963	O
,	O
LDPC	B-Error_Name
codes	I-Error_Name
were	O
forgotten	O
until	O
his	O
work	O
was	O
rediscovered	O
in	O
1996	O
.	O
</s>
<s>
Turbo	B-Error_Name
codes	I-Error_Name
,	O
another	O
class	O
of	O
capacity-approaching	O
codes	O
discovered	O
in	O
1993	O
,	O
became	O
the	O
coding	O
scheme	O
of	O
choice	O
in	O
the	O
late	O
1990s	O
,	O
used	O
for	O
applications	O
such	O
as	O
the	O
Deep	O
Space	O
Network	O
and	O
satellite	O
communications	O
.	O
</s>
<s>
However	O
,	O
the	O
advances	O
in	O
low-density	B-Error_Name
parity-check	I-Error_Name
codes	I-Error_Name
have	O
seen	O
them	O
surpass	O
turbo	B-Error_Name
codes	I-Error_Name
in	O
terms	O
of	O
error	B-Error_Name
floor	I-Error_Name
and	O
performance	O
in	O
the	O
higher	O
code	O
rate	O
range	O
,	O
leaving	O
turbo	B-Error_Name
codes	I-Error_Name
better	O
suited	O
for	O
the	O
lower	O
code	O
rates	O
only	O
.	O
</s>
<s>
In	O
2003	O
,	O
an	O
irregular	O
repeat	O
accumulate	O
(	O
IRA	O
)	O
style	O
LDPC	B-Error_Name
code	I-Error_Name
beat	O
six	O
turbo	B-Error_Name
codes	I-Error_Name
to	O
become	O
the	O
error-correcting	B-Error_Name
code	I-Error_Name
in	O
the	O
new	O
DVB-S2	O
standard	O
for	O
digital	B-Application
television	I-Application
.	O
</s>
<s>
The	O
DVB-S2	O
selection	O
committee	O
made	O
decoder	O
complexity	O
estimates	O
for	O
the	O
turbo	B-Error_Name
code	I-Error_Name
proposals	O
using	O
a	O
much	O
less	O
efficient	O
serial	O
decoder	O
architecture	O
rather	O
than	O
a	O
parallel	O
decoder	O
architecture	O
.	O
</s>
<s>
This	O
forced	O
the	O
turbo	B-Error_Name
code	I-Error_Name
proposals	O
to	O
use	O
frame	O
sizes	O
on	O
the	O
order	O
of	O
one	O
half	O
the	O
frame	O
size	O
of	O
the	O
LDPC	B-Error_Name
proposals	O
.	O
</s>
<s>
In	O
2008	O
,	O
LDPC	B-Error_Name
beat	O
convolutional	O
turbo	B-Error_Name
codes	I-Error_Name
as	O
the	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
(	O
FEC	O
)	O
system	O
for	O
the	O
ITU-T	B-General_Concept
G.hn	I-General_Concept
standard	O
.	O
</s>
<s>
G.hn	B-General_Concept
chose	O
LDPC	B-Error_Name
codes	I-Error_Name
over	O
turbo	B-Error_Name
codes	I-Error_Name
because	O
of	O
their	O
lower	O
decoding	O
complexity	O
(	O
especially	O
when	O
operating	O
at	O
data	O
rates	O
close	O
to	O
1.0	O
Gbit/s	O
)	O
and	O
because	O
the	O
proposed	O
turbo	B-Error_Name
codes	I-Error_Name
exhibited	O
a	O
significant	O
error	B-Error_Name
floor	I-Error_Name
at	O
the	O
desired	O
range	O
of	O
operation	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
are	O
also	O
used	O
for	O
10GBASE-T	O
Ethernet	O
,	O
which	O
sends	O
data	O
at	O
10	O
gigabits	O
per	O
second	O
over	O
twisted-pair	O
cables	O
.	O
</s>
<s>
As	O
of	O
2009	O
,	O
LDPC	B-Error_Name
codes	I-Error_Name
are	O
also	O
part	O
of	O
the	O
Wi-Fi	O
802.11	O
standard	O
as	O
an	O
optional	O
part	O
of	O
802.11n	O
and	O
802.11ac	O
,	O
in	O
the	O
High	O
Throughput	O
(	O
HT	O
)	O
PHY	O
specification	O
.	O
</s>
<s>
LDPC	B-Error_Name
is	O
a	O
mandatory	O
part	O
of	O
802.11ax	O
(	O
Wi-Fi	O
6	O
)	O
.	O
</s>
<s>
Some	O
OFDM	B-Protocol
systems	O
add	O
an	O
additional	O
outer	O
error	O
correction	O
that	O
fixes	O
the	O
occasional	O
errors	O
(	O
the	O
"	O
error	B-Error_Name
floor	I-Error_Name
"	O
)	O
that	O
get	O
past	O
the	O
LDPC	B-Error_Name
correction	O
inner	O
code	O
even	O
at	O
low	O
bit	O
error	O
rates	O
.	O
</s>
<s>
The	O
Reed-Solomon	B-Error_Name
code	I-Error_Name
with	O
LDPC	B-Error_Name
Coded	O
Modulation	O
(	O
RS-LCM	O
)	O
uses	O
a	O
Reed-Solomon	B-Error_Name
outer	O
code	O
.	O
</s>
<s>
The	O
DVB-S2	O
,	O
the	O
DVB-T2	B-Protocol
and	O
the	O
DVB-C2	B-Device
standards	O
all	O
use	O
a	O
BCH	O
code	O
outer	O
code	O
to	O
mop	O
up	O
residual	O
errors	O
after	O
LDPC	B-Error_Name
decoding	O
.	O
</s>
<s>
5G	O
NR	O
uses	O
polar	B-Error_Name
code	I-Error_Name
for	O
the	O
control	O
channels	O
and	O
LDPC	B-Error_Name
for	O
the	O
data	O
channels	O
.	O
</s>
<s>
Although	O
LDPC	B-Error_Name
code	I-Error_Name
has	O
had	O
its	O
success	O
in	O
commercial	O
hard	O
disk	O
drives	O
,	O
to	O
fully	O
exploit	O
its	O
error	O
correction	O
capability	O
in	O
SSDs	B-Device
demands	O
unconventional	O
fine-grained	O
flash	O
memory	O
sensing	O
,	O
leading	O
to	O
an	O
increased	O
memory	O
read	O
latency	O
.	O
</s>
<s>
LDPC-in-SSD	O
is	O
an	O
effective	O
approach	O
to	O
deploy	O
LDPC	B-Error_Name
in	O
SSD	B-Device
with	O
a	O
very	O
small	O
latency	O
increase	O
,	O
which	O
turns	O
LDPC	B-Error_Name
in	O
SSD	B-Device
into	O
a	O
reality	O
.	O
</s>
<s>
Since	O
then	O
,	O
LDPC	B-Error_Name
has	O
been	O
widely	O
adopted	O
in	O
commercial	O
SSDs	B-Device
in	O
both	O
customer-grades	O
and	O
enterprise-grades	O
by	O
major	O
storage	O
venders	O
.	O
</s>
<s>
Many	O
TLC	O
(	O
and	O
later	O
)	O
SSDs	B-Device
are	O
using	O
LDPC	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
functionally	O
are	O
defined	O
by	O
a	O
sparse	O
parity-check	O
matrix	O
.	O
</s>
<s>
This	O
sparse	B-Algorithm
matrix	I-Algorithm
is	O
often	O
randomly	O
generated	O
,	O
subject	O
to	O
the	O
sparsity	B-Algorithm
constraints	O
—	O
LDPC	B-Error_Name
code	I-Error_Name
construction	O
is	O
discussed	O
later	O
.	O
</s>
<s>
Below	O
is	O
a	O
graph	O
fragment	O
of	O
an	O
example	O
LDPC	B-Error_Name
code	I-Error_Name
using	O
Forney	O
's	O
factor	O
graph	O
notation	O
.	O
</s>
<s>
This	O
is	O
a	O
popular	O
way	O
of	O
graphically	O
representing	O
an	O
(	O
n	O
,	O
k	O
)	O
LDPC	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
This	O
LDPC	B-Error_Name
code	I-Error_Name
fragment	O
represents	O
a	O
three-bit	O
message	O
encoded	O
as	O
six	O
bits	O
.	O
</s>
<s>
Figure	O
1	O
illustrates	O
the	O
functional	O
components	O
of	O
most	O
LDPC	B-Error_Name
encoders	O
.	O
</s>
<s>
The	O
first	O
4680	O
data	O
bits	O
are	O
repeated	O
13	O
times	O
(	O
used	O
in	O
13	O
parity	O
codes	O
)	O
,	O
while	O
the	O
remaining	O
data	O
bits	O
are	O
used	O
in	O
3	O
parity	O
codes	O
(	O
irregular	O
LDPC	B-Error_Name
code	I-Error_Name
)	O
.	O
</s>
<s>
For	O
comparison	O
,	O
classic	O
turbo	B-Error_Name
codes	I-Error_Name
typically	O
use	O
two	O
constituent	O
codes	O
configured	O
in	O
parallel	O
,	O
each	O
of	O
which	O
encodes	O
the	O
entire	O
input	O
block	O
(	O
K	O
)	O
of	O
data	O
bits	O
.	O
</s>
<s>
The	O
LDPC	B-Error_Name
code	I-Error_Name
,	O
in	O
contrast	O
,	O
uses	O
many	O
low	O
depth	O
constituent	O
codes	O
(	O
accumulators	O
)	O
in	O
parallel	O
,	O
each	O
of	O
which	O
encode	O
only	O
a	O
small	O
portion	O
of	O
the	O
input	O
frame	O
.	O
</s>
<s>
The	O
repeat	O
and	O
distribute	O
operations	O
perform	O
the	O
function	O
of	O
the	O
interleaver	O
in	O
the	O
turbo	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
The	O
ability	O
to	O
more	O
precisely	O
manage	O
the	O
connections	O
of	O
the	O
various	O
constituent	O
codes	O
and	O
the	O
level	O
of	O
redundancy	O
for	O
each	O
input	O
bit	O
give	O
more	O
flexibility	O
in	O
the	O
design	O
of	O
LDPC	B-Error_Name
codes	I-Error_Name
,	O
which	O
can	O
lead	O
to	O
better	O
performance	O
than	O
turbo	B-Error_Name
codes	I-Error_Name
in	O
some	O
instances	O
.	O
</s>
<s>
Turbo	B-Error_Name
codes	I-Error_Name
still	O
seem	O
to	O
perform	O
better	O
than	O
LDPCs	B-Error_Name
at	O
low	O
code	O
rates	O
,	O
or	O
at	O
least	O
the	O
design	O
of	O
well	O
performing	O
low	O
rate	O
codes	O
is	O
easier	O
for	O
turbo	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
As	O
with	O
other	O
codes	O
,	O
the	O
maximum	O
likelihood	O
decoding	O
of	O
an	O
LDPC	B-Error_Name
code	I-Error_Name
on	O
the	O
binary	O
symmetric	O
channel	O
is	O
an	O
NP-complete	O
problem	O
.	O
</s>
<s>
The	O
sub-optimal	O
decoding	O
techniques	O
view	O
each	O
parity	O
check	O
that	O
makes	O
up	O
the	O
LDPC	B-Error_Name
as	O
an	O
independent	O
single	O
parity	O
check	O
(	O
SPC	O
)	O
code	O
.	O
</s>
<s>
Each	O
SPC	O
code	O
is	O
decoded	O
separately	O
using	O
soft-in-soft-out	B-Error_Name
(	O
SISO	O
)	O
techniques	O
such	O
as	O
SOVA	O
,	O
BCJR	B-Error_Name
,	O
MAP	B-General_Concept
,	O
and	O
other	O
derivates	O
thereof	O
.	O
</s>
<s>
In	O
a	O
practical	O
LDPC	B-Error_Name
decoder	O
implementation	O
,	O
sets	O
of	O
SPC	O
codes	O
are	O
decoded	O
in	O
parallel	O
to	O
increase	O
throughput	O
.	O
</s>
<s>
While	O
illustrative	O
,	O
this	O
erasure	O
example	O
does	O
not	O
show	O
the	O
use	O
of	O
soft-decision	O
decoding	O
or	O
soft-decision	O
message	O
passing	O
,	O
which	O
is	O
used	O
in	O
virtually	O
all	O
commercial	O
LDPC	B-Error_Name
decoders	O
.	O
</s>
<s>
The	O
original	O
technique	O
that	O
was	O
used	O
for	O
decoding	O
LDPC	B-Error_Name
codes	I-Error_Name
was	O
known	O
as	O
flooding	O
.	O
</s>
<s>
Highly	O
reliable	O
nodes	O
,	O
whose	O
log-likelihood	B-General_Concept
ratio	I-General_Concept
(	O
LLR	O
)	O
magnitude	O
is	O
large	O
and	O
does	O
not	O
change	O
significantly	O
from	O
one	O
update	O
to	O
the	O
next	O
,	O
do	O
not	O
require	O
updates	O
with	O
the	O
same	O
frequency	O
as	O
other	O
nodes	O
,	O
whose	O
sign	O
and	O
magnitude	O
fluctuate	O
more	O
widely	O
.	O
</s>
<s>
These	O
scheduling	O
algorithms	O
show	O
greater	O
speed	O
of	O
convergence	O
and	O
lower	O
error	B-Error_Name
floors	I-Error_Name
than	O
those	O
that	O
use	O
flooding	O
.	O
</s>
<s>
These	O
lower	O
error	B-Error_Name
floors	I-Error_Name
are	O
achieved	O
by	O
the	O
ability	O
of	O
the	O
Informed	O
Dynamic	O
Scheduling	O
(	O
IDS	O
)	O
algorithm	O
to	O
overcome	O
trapping	O
sets	O
of	O
near	O
codewords	O
.	O
</s>
<s>
For	O
an	O
(	O
n	O
,	O
k	O
)	O
LDPC	B-Error_Name
code	I-Error_Name
of	O
rate	O
k/n	O
,	O
a	O
full	O
iteration	O
occurs	O
when	O
n	O
variable	O
and	O
nk	O
constraint	O
nodes	O
have	O
been	O
updated	O
,	O
no	O
matter	O
the	O
order	O
in	O
which	O
they	O
were	O
updated	O
.	O
</s>
<s>
For	O
large	O
block	O
sizes	O
,	O
LDPC	B-Error_Name
codes	I-Error_Name
are	O
commonly	O
constructed	O
by	O
first	O
studying	O
the	O
behaviour	O
of	O
decoders	O
.	O
</s>
<s>
As	O
the	O
block	O
size	O
tends	O
to	O
infinity	O
,	O
LDPC	B-Error_Name
decoders	O
can	O
be	O
shown	O
to	O
have	O
a	O
noise	O
threshold	O
below	O
which	O
decoding	O
is	O
reliably	O
achieved	O
,	O
and	O
above	O
which	O
decoding	O
is	O
not	O
achieved	O
,	O
colloquially	O
referred	O
to	O
as	O
the	O
cliff	O
effect	O
.	O
</s>
<s>
An	O
approximate	O
graphical	O
approach	O
to	O
visualising	O
this	O
threshold	O
is	O
an	O
EXIT	B-Error_Name
chart	I-Error_Name
.	O
</s>
<s>
The	O
construction	O
of	O
a	O
specific	O
LDPC	B-Error_Name
code	I-Error_Name
after	O
this	O
optimization	O
falls	O
into	O
two	O
main	O
types	O
of	O
techniques	O
:	O
</s>
<s>
Combinatorial	O
approaches	O
can	O
be	O
used	O
to	O
optimize	O
the	O
properties	O
of	O
small	O
block-size	O
LDPC	B-Error_Name
codes	I-Error_Name
or	O
to	O
create	O
codes	O
with	O
simple	O
encoders	O
.	O
</s>
<s>
Some	O
LDPC	B-Error_Name
codes	I-Error_Name
are	O
based	O
on	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
,	O
such	O
as	O
the	O
RS-LDPC	O
code	O
used	O
in	O
the	O
10	O
Gigabit	O
Ethernet	O
standard	O
.	O
</s>
<s>
Compared	O
to	O
randomly	O
generated	O
LDPC	B-Error_Name
codes	I-Error_Name
,	O
structured	O
LDPC	B-Error_Name
codes	I-Error_Name
—	O
such	O
as	O
the	O
LDPC	B-Error_Name
code	I-Error_Name
used	O
in	O
the	O
DVB-S2	O
standard	O
—	O
can	O
have	O
simpler	O
and	O
therefore	O
lower-cost	O
hardware	O
—	O
in	O
particular	O
,	O
codes	O
constructed	O
such	O
that	O
the	O
H	O
matrix	O
is	O
a	O
circulant	O
matrix	O
.	O
</s>
<s>
Yet	O
another	O
way	O
of	O
constructing	O
LDPC	B-Error_Name
codes	I-Error_Name
is	O
to	O
use	O
finite	O
geometries	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
can	O
be	O
compared	O
with	O
other	O
powerful	O
coding	O
schemes	O
,	O
e.g.	O
</s>
<s>
turbo	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
In	O
one	O
hand	O
,	O
BER	O
performance	O
of	O
turbo	B-Error_Name
codes	I-Error_Name
is	O
influenced	O
by	O
low	O
codes	O
limitations	O
.	O
</s>
<s>
LDPC	B-Error_Name
codes	I-Error_Name
have	O
no	O
limitations	O
of	O
minimum	O
distance	O
,	O
that	O
indirectly	O
means	O
that	O
LDPC	B-Error_Name
codes	I-Error_Name
may	O
be	O
more	O
efficient	O
on	O
relatively	O
large	O
code	O
rates	O
(	O
e.g.	O
</s>
<s>
3/4	O
,	O
5/6	O
,	O
7/8	O
)	O
than	O
turbo	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
However	O
,	O
LDPC	B-Error_Name
codes	I-Error_Name
are	O
not	O
the	O
complete	O
replacement	O
:	O
turbo	B-Error_Name
codes	I-Error_Name
are	O
the	O
best	O
solution	O
at	O
the	O
lower	O
code	O
rates	O
(	O
e.g.	O
</s>
