<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
a	O
group	O
of	O
error-correcting	B-Error_Name
codes	I-Error_Name
that	O
were	O
introduced	O
by	O
Irving	O
S	O
.	O
Reed	O
and	O
Gustave	O
Solomon	O
in	O
1960	O
.	O
</s>
<s>
They	O
have	O
many	O
applications	O
,	O
the	O
most	O
prominent	O
of	O
which	O
include	O
consumer	O
technologies	O
such	O
as	O
MiniDiscs	O
,	O
CDs	B-Device
,	O
DVDs	O
,	O
Blu-ray	O
discs	O
,	O
QR	O
codes	O
,	O
data	O
transmission	O
technologies	O
such	O
as	O
DSL	O
and	O
WiMAX	O
,	O
broadcast	B-Operating_System
systems	O
such	O
as	O
satellite	O
communications	O
,	O
DVB	B-Application
and	O
ATSC	B-Device
,	O
and	O
storage	O
systems	O
such	O
as	O
RAID	O
6	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
operate	O
on	O
a	O
block	O
of	O
data	O
treated	O
as	O
a	O
set	O
of	O
finite-field	O
elements	O
called	O
symbols	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
able	O
to	O
detect	O
and	O
correct	O
multiple	O
symbol	O
errors	O
.	O
</s>
<s>
By	O
adding	O
=	O
−	O
check	O
symbols	O
to	O
the	O
data	O
,	O
a	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
can	O
detect	O
(	O
but	O
not	O
correct	O
)	O
any	O
combination	O
of	O
up	O
to	O
erroneous	O
symbols	O
,	O
or	O
locate	O
and	O
correct	O
up	O
to	O
erroneous	O
symbols	O
at	O
unknown	O
locations	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
also	O
suitable	O
as	O
multiple-burst	O
bit-error	O
correcting	O
codes	O
,	O
since	O
a	O
sequence	O
of	O
consecutive	O
bit	O
errors	O
can	O
affect	O
at	O
most	O
two	O
symbols	O
of	O
size	O
.	O
</s>
<s>
There	O
are	O
two	O
basic	O
types	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
original	O
view	O
and	O
BCH	O
view	O
with	O
BCH	O
view	O
being	O
the	O
most	O
common	O
,	O
as	O
BCH	O
view	O
decoders	O
are	O
faster	O
and	O
require	O
less	O
working	O
storage	O
than	O
original	O
view	O
decoders	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
were	O
developed	O
in	O
1960	O
by	O
Irving	O
S	O
.	O
Reed	O
and	O
Gustave	O
Solomon	O
,	O
who	O
were	O
then	O
staff	O
members	O
of	O
MIT	O
Lincoln	O
Laboratory	O
.	O
</s>
<s>
The	O
result	O
of	O
this	O
is	O
that	O
there	O
are	O
two	O
main	O
types	O
of	O
Reed	B-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
,	O
ones	O
that	O
use	O
the	O
original	O
encoding	O
scheme	O
,	O
and	O
ones	O
that	O
use	O
the	O
BCH	O
encoding	O
scheme	O
.	O
</s>
<s>
The	O
Gorenstein	O
–	O
Zierler	O
decoder	O
and	O
the	O
related	O
work	O
on	O
BCH	O
codes	O
are	O
described	O
in	O
a	O
book	O
Error	B-Error_Name
Correcting	I-Error_Name
Codes	I-Error_Name
by	O
W	O
.	O
Wesley	O
Peterson	O
(	O
1961	O
)	O
.	O
</s>
<s>
Stone	O
(	O
and	O
others	O
)	O
recognized	O
that	O
Reed	B-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
could	O
use	O
the	O
BCH	O
scheme	O
of	O
using	O
a	O
fixed	O
generator	O
polynomial	O
,	O
making	O
such	O
codes	O
a	O
special	O
class	O
of	O
BCH	O
codes	O
,	O
but	O
Reed	B-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
based	O
on	O
the	O
original	O
encoding	O
scheme	O
,	O
are	O
not	O
a	O
class	O
of	O
BCH	O
codes	O
,	O
and	O
depending	O
on	O
the	O
set	O
of	O
evaluation	O
points	O
,	O
they	O
are	O
not	O
even	O
cyclic	O
codes	O
.	O
</s>
<s>
In	O
1969	O
,	O
an	O
improved	O
BCH	O
scheme	O
decoder	O
was	O
developed	O
by	O
Elwyn	O
Berlekamp	O
and	O
James	O
Massey	O
,	O
and	O
has	O
since	O
been	O
known	O
as	O
the	O
Berlekamp	B-Algorithm
–	I-Algorithm
Massey	I-Algorithm
decoding	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
In	O
1977	O
,	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
were	O
implemented	O
in	O
the	O
Voyager	O
program	O
in	O
the	O
form	O
of	O
concatenated	O
error	B-Error_Name
correction	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
The	O
first	O
commercial	O
application	O
in	O
mass-produced	O
consumer	O
products	O
appeared	O
in	O
1982	O
with	O
the	O
compact	O
disc	O
,	O
where	O
two	O
interleaved	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
used	O
.	O
</s>
<s>
Today	O
,	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
widely	O
implemented	O
in	O
digital	B-General_Concept
storage	I-General_Concept
devices	O
and	O
digital	O
communication	O
standards	O
,	O
though	O
they	O
are	O
being	O
slowly	O
replaced	O
by	O
Bose	O
–	O
Chaudhuri	O
–	O
Hocquenghem	O
(	O
BCH	O
)	O
codes	O
.	O
</s>
<s>
For	O
example	O
,	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
used	O
in	O
the	O
Digital	B-Application
Video	I-Application
Broadcasting	I-Application
(	O
DVB	B-Application
)	O
standard	O
DVB-S	O
,	O
in	O
conjunction	O
with	O
a	O
convolutional	B-Error_Name
inner	O
code	O
,	O
but	O
BCH	O
codes	O
are	O
used	O
with	O
LDPC	B-Error_Name
in	O
its	O
successor	O
,	O
DVB-S2	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	O
is	O
a	O
key	O
component	O
of	O
the	O
compact	O
disc	O
.	O
</s>
<s>
In	O
the	O
CD	B-Device
,	O
two	O
layers	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	O
separated	O
by	O
a	O
28-way	O
convolutional	B-Error_Name
interleaver	O
yields	O
a	O
scheme	O
called	O
Cross-Interleaved	B-Error_Name
Reed	I-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
Coding	I-Error_Name
(	O
CIRC	B-Error_Name
)	O
.	O
</s>
<s>
The	O
first	O
element	O
of	O
a	O
CIRC	B-Error_Name
decoder	O
is	O
a	O
relatively	O
weak	O
inner	O
(	O
32	O
,	O
28	O
)	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
,	O
shortened	O
from	O
a	O
(	O
255,251	O
)	O
code	O
with	O
8-bit	O
symbols	O
.	O
</s>
<s>
The	O
result	O
is	O
a	O
CIRC	B-Error_Name
that	O
can	O
completely	O
correct	O
error	O
bursts	O
up	O
to	O
4000	O
bits	O
,	O
or	O
about	O
2.5mm	O
on	O
the	O
disc	O
surface	O
.	O
</s>
<s>
This	O
code	O
is	O
so	O
strong	O
that	O
most	O
CD	B-Device
playback	O
errors	O
are	O
almost	O
certainly	O
caused	O
by	O
tracking	O
errors	O
that	O
cause	O
the	O
laser	O
to	O
jump	O
track	O
,	O
not	O
by	O
uncorrectable	O
error	O
bursts	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
error	I-Error_Name
correction	I-Error_Name
is	O
also	O
used	O
in	O
parchive	B-General_Concept
files	O
which	O
are	O
commonly	O
posted	O
accompanying	O
multimedia	O
files	O
on	O
USENET	B-Application
.	O
</s>
<s>
The	O
distributed	O
online	O
storage	O
service	O
Wuala	B-Application
(	O
discontinued	O
in	O
2015	O
)	O
also	O
used	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
when	O
breaking	O
up	O
files	O
.	O
</s>
<s>
Almost	O
all	O
two-dimensional	O
bar	O
codes	O
such	O
as	O
PDF-417	O
,	O
MaxiCode	O
,	O
Datamatrix	O
,	O
QR	O
Code	O
,	O
and	O
Aztec	O
Code	O
use	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
error	I-Error_Name
correction	I-Error_Name
to	O
allow	O
correct	O
reading	O
even	O
if	O
a	O
portion	O
of	O
the	O
bar	O
code	O
is	O
damaged	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	O
is	O
less	O
common	O
in	O
one-dimensional	O
bar	O
codes	O
,	O
but	O
is	O
used	O
by	O
the	O
PostBar	O
symbology	O
.	O
</s>
<s>
Specialized	O
forms	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
,	O
specifically	O
Cauchy-RS	O
and	O
Vandermonde-RS	O
,	O
can	O
be	O
used	O
to	O
overcome	O
the	O
unreliable	O
nature	O
of	O
data	O
transmission	O
over	O
erasure	O
channels	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
are	O
also	O
used	O
in	O
xDSL	O
systems	O
and	O
CCSDS	O
's	O
Space	O
Communications	O
Protocol	O
Specifications	O
as	O
a	O
form	O
of	O
forward	B-Error_Name
error	I-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
One	O
significant	O
application	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	O
was	O
to	O
encode	O
the	O
digital	O
pictures	O
sent	O
back	O
by	O
the	O
Voyager	O
program	O
.	O
</s>
<s>
Voyager	O
introduced	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	O
concatenated	O
with	O
convolutional	B-Error_Name
codes	I-Error_Name
,	O
a	O
practice	O
that	O
has	O
since	O
become	O
very	O
widespread	O
in	O
deep	O
space	O
and	O
satellite	O
(	O
e.g.	O
,	O
direct	O
digital	O
broadcasting	B-Operating_System
)	O
communications	O
.	O
</s>
<s>
Correcting	O
these	O
burst	O
errors	O
is	O
a	O
job	O
best	O
done	O
by	O
short	O
or	O
simplified	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Modern	O
versions	O
of	O
concatenated	O
Reed	O
–	O
Solomon/Viterbi	O
-decoded	O
convolutional	B-Error_Name
coding	I-Error_Name
were	O
and	O
are	O
used	O
on	O
the	O
Mars	O
Pathfinder	O
,	O
Galileo	O
,	O
Mars	O
Exploration	O
Rover	O
and	O
Cassini	O
missions	O
,	O
where	O
they	O
perform	O
within	O
about	O
1	O
–	O
1.5	O
dB	O
of	O
the	O
ultimate	O
limit	O
,	O
the	O
Shannon	O
capacity	O
.	O
</s>
<s>
These	O
concatenated	O
codes	O
are	O
now	O
being	O
replaced	O
by	O
more	O
powerful	O
turbo	B-Error_Name
codes	I-Error_Name
:	O
</s>
<s>
"	O
The	O
development	O
of	O
turbo	O
and	O
LDPC	B-Error_Name
codes	I-Error_Name
for	O
deep-space	O
applications.	O
"	O
</s>
<s>
Years	O
Code	O
Mission(s )	O
1958	O
–	O
present	O
Uncoded	O
Explorer	O
,	O
Mariner	O
,	O
many	O
others	O
1968	O
–	O
1978	O
convolutional	B-Error_Name
codes	I-Error_Name
(	O
CC	B-Error_Name
)	O
(	O
25	O
,	O
1/2	O
)	O
Pioneer	O
,	O
Venus	O
1969	O
–	O
1975	O
Reed-Muller	O
code	O
(	O
32	O
,	O
6	O
)	O
Mariner	O
,	O
Viking	O
1977	O
–	O
present	O
Binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
Voyager	O
1977	O
–	O
present	O
RS(255, 223 )	O
+	O
CC( 7	O
,	O
1/2	O
)	O
Voyager	O
,	O
Galileo	O
,	O
many	O
others	O
1989	O
–	O
2003	O
RS(255, 223 )	O
+	O
CC( 7	O
,	O
1/3	O
)	O
Voyager	O
1989	O
–	O
2003	O
RS(255, 223 )	O
+	O
CC( 14	O
,	O
1/4	O
)	O
Galileo	O
1996	O
–	O
present	O
RS	O
+	O
CC	B-Error_Name
(	O
15	O
,	O
1/6	O
)	O
Cassini	O
,	O
Mars	O
Pathfinder	O
,	O
others	O
2004	O
–	O
present	O
Turbo	B-Error_Name
codes	I-Error_Name
Messenger	O
,	O
Stereo	O
,	O
MRO	O
,	O
others	O
est	O
.	O
</s>
<s>
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
actually	O
a	O
family	O
of	O
codes	O
,	O
where	O
every	O
code	O
is	O
characterised	O
by	O
three	O
parameters	O
:	O
an	O
alphabet	O
size	O
q	O
,	O
a	O
block	O
length	O
n	O
,	O
and	O
a	O
message	O
length	O
k	O
,	O
with	O
k	O
<	O
n	O
≤	O
q	O
.	O
</s>
<s>
In	O
the	O
most	O
useful	O
parameterizations	O
of	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
,	O
the	O
block	O
length	O
is	O
usually	O
some	O
constant	O
multiple	O
of	O
the	O
message	O
length	O
,	O
that	O
is	O
,	O
the	O
rate	O
is	O
some	O
constant	O
,	O
and	O
furthermore	O
,	O
the	O
block	O
length	O
is	O
equal	O
to	O
or	O
one	O
less	O
than	O
the	O
alphabet	O
size	O
,	O
that	O
is	O
,	O
or	O
.	O
</s>
<s>
There	O
are	O
different	O
encoding	O
procedures	O
for	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
,	O
and	O
thus	O
,	O
there	O
are	O
different	O
ways	O
to	O
describe	O
the	O
set	O
of	O
all	O
codewords	O
.	O
</s>
<s>
In	O
the	O
original	O
view	O
of	O
,	O
every	O
codeword	O
of	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
a	O
sequence	O
of	O
function	O
values	O
of	O
a	O
polynomial	O
of	O
degree	O
less	O
than	O
k	O
.	O
In	O
order	O
to	O
obtain	O
a	O
codeword	O
of	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
,	O
the	O
message	O
symbols	O
(	O
each	O
within	O
the	O
q-sized	O
alphabet	O
)	O
are	O
treated	O
as	O
the	O
coefficients	O
of	O
a	O
polynomial	O
p	O
of	O
degree	O
less	O
than	O
k	O
,	O
over	O
the	O
finite	O
field	O
F	O
with	O
q	O
elements	O
.	O
</s>
<s>
Formally	O
,	O
the	O
set	O
of	O
codewords	O
of	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Since	O
any	O
two	O
distinct	O
polynomials	O
of	O
degree	O
less	O
than	O
agree	O
in	O
at	O
most	O
points	O
,	O
this	O
means	O
that	O
any	O
two	O
codewords	O
of	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
disagree	O
in	O
at	O
least	O
positions	O
.	O
</s>
<s>
Furthermore	O
,	O
there	O
are	O
two	O
polynomials	O
that	O
do	O
agree	O
in	O
points	O
but	O
are	O
not	O
equal	O
,	O
and	O
thus	O
,	O
the	O
distance	O
of	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
exactly	O
.	O
</s>
<s>
Being	O
a	O
code	O
that	O
achieves	O
this	O
optimal	O
trade-off	O
,	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
belongs	O
to	O
the	O
class	O
of	O
maximum	O
distance	O
separable	O
codes	O
.	O
</s>
<s>
Thus	O
the	O
classical	O
encoding	O
function	O
for	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
This	O
function	O
is	O
a	O
linear	B-Architecture
mapping	I-Architecture
,	O
that	O
is	O
,	O
it	O
satisfies	O
for	O
the	O
following	O
-matrix	O
with	O
elements	O
from	O
:	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
a	O
linear	O
code	O
,	O
and	O
in	O
the	O
classical	O
encoding	O
procedure	O
,	O
its	O
generator	O
matrix	O
is	O
.	O
</s>
<s>
There	O
is	O
an	O
alternative	O
encoding	O
procedure	O
that	O
also	O
produces	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
,	O
but	O
that	O
does	O
so	O
in	O
a	O
systematic	O
way	O
.	O
</s>
<s>
The	O
alternative	O
encoding	O
function	O
for	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
then	O
again	O
just	O
the	O
sequence	O
of	O
values	O
:	O
</s>
<s>
Since	O
Lagrange	O
interpolation	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
is	O
a	O
linear	B-Architecture
mapping	I-Architecture
.	O
</s>
<s>
The	O
encoding	O
procedure	O
for	O
the	O
BCH	O
view	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
can	O
be	O
modified	O
to	O
yield	O
a	O
systematic	O
encoding	O
procedure	O
,	O
in	O
which	O
each	O
codeword	O
contains	O
the	O
message	O
as	O
a	O
prefix	O
,	O
and	O
simply	O
appends	O
error	O
correcting	O
symbols	O
as	O
a	O
suffix	O
.	O
</s>
<s>
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
a	O
[	O
n	O
,	O
k	O
,	O
n	O
k	O
+	O
1 ]	O
code	O
;	O
in	O
other	O
words	O
,	O
it	O
is	O
a	O
linear	O
block	O
code	O
of	O
length	O
n	O
(	O
over	O
F	O
)	O
with	O
dimension	O
k	O
and	O
minimum	O
Hamming	O
distance	O
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
optimal	O
in	O
the	O
sense	O
that	O
the	O
minimum	O
distance	O
has	O
the	O
maximum	O
value	O
possible	O
for	O
a	O
linear	O
code	O
of	O
size	O
(	O
n	O
,	O
k	O
)	O
;	O
this	O
is	O
known	O
as	O
the	O
Singleton	O
bound	O
.	O
</s>
<s>
The	O
error-correcting	O
ability	O
of	O
a	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
determined	O
by	O
its	O
minimum	O
distance	O
,	O
or	O
equivalently	O
,	O
by	O
,	O
the	O
measure	O
of	O
redundancy	O
in	O
the	O
block	O
.	O
</s>
<s>
If	O
the	O
locations	O
of	O
the	O
error	O
symbols	O
are	O
not	O
known	O
in	O
advance	O
,	O
then	O
a	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
can	O
correct	O
up	O
to	O
erroneous	O
symbols	O
,	O
i.e.	O
,	O
it	O
can	O
correct	O
half	O
as	O
many	O
errors	O
as	O
there	O
are	O
redundant	O
symbols	O
added	O
to	O
the	O
block	O
.	O
</s>
<s>
Sometimes	O
error	O
locations	O
are	O
known	O
in	O
advance	O
(	O
e.g.	O
,	O
"	O
side	O
information	O
"	O
in	O
demodulator	B-Device
signal-to-noise	O
ratios	O
)	O
—	O
these	O
are	O
called	O
erasures	O
.	O
</s>
<s>
A	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
(	O
like	O
any	O
MDS	O
code	O
)	O
is	O
able	O
to	O
correct	O
twice	O
as	O
many	O
erasures	O
as	O
errors	O
,	O
and	O
any	O
combination	O
of	O
errors	O
and	O
erasures	O
can	O
be	O
corrected	O
as	O
long	O
as	O
the	O
relation	O
is	O
satisfied	O
,	O
where	O
is	O
the	O
number	O
of	O
errors	O
and	O
is	O
the	O
number	O
of	O
erasures	O
in	O
the	O
block	O
.	O
</s>
<s>
For	O
practical	O
uses	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
,	O
it	O
is	O
common	O
to	O
use	O
a	O
finite	O
field	O
with	O
elements	O
.	O
</s>
<s>
Thus	O
a	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
operating	O
on	O
8-bit	O
symbols	O
has	O
symbols	O
per	O
block	O
.	O
</s>
<s>
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
properties	O
discussed	O
above	O
make	O
them	O
especially	O
well-suited	O
to	O
applications	O
where	O
errors	O
occur	O
in	O
bursts	O
.	O
</s>
<s>
Conversely	O
,	O
if	O
a	O
data	O
stream	O
is	O
not	O
characterized	O
by	O
error	O
bursts	O
or	O
drop-outs	O
but	O
by	O
random	O
single	O
bit	O
errors	O
,	O
a	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
usually	O
a	O
poor	O
choice	O
compared	O
to	O
a	O
binary	O
code	O
.	O
</s>
<s>
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
,	O
like	O
the	O
convolutional	B-Error_Name
code	I-Error_Name
,	O
is	O
a	O
transparent	O
code	O
.	O
</s>
<s>
However	O
,	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
loses	O
its	O
transparency	O
when	O
the	O
code	O
is	O
shortened	O
.	O
</s>
<s>
For	O
this	O
reason	O
it	O
is	O
mandatory	O
that	O
the	O
sense	O
of	O
the	O
data	O
(	O
i.e.	O
,	O
true	O
or	O
complemented	O
)	O
be	O
resolved	O
before	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
decoding	O
.	O
</s>
<s>
Whether	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
is	O
cyclic	O
or	O
not	O
depends	O
on	O
subtle	O
details	O
of	O
the	O
construction	O
.	O
</s>
<s>
Since	O
the	O
function	O
is	O
also	O
a	O
polynomial	O
of	O
the	O
same	O
degree	O
,	O
this	O
function	O
gives	O
rise	O
to	O
a	O
codeword	O
;	O
since	O
holds	O
,	O
this	O
codeword	O
is	O
the	O
cyclic	B-Algorithm
left-shift	I-Algorithm
of	O
the	O
original	O
codeword	O
derived	O
from	O
.	O
</s>
<s>
So	O
choosing	O
a	O
sequence	O
of	O
primitive	O
root	O
powers	O
as	O
the	O
evaluation	O
points	O
makes	O
the	O
original	O
view	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
cyclic	O
.	O
</s>
<s>
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
in	O
the	O
BCH	O
view	O
are	O
always	O
cyclic	O
because	O
BCH	O
codes	O
are	O
cyclic	O
.	O
</s>
<s>
Designers	O
are	O
not	O
required	O
to	O
use	O
the	O
"	O
natural	O
"	O
sizes	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
blocks	O
.	O
</s>
<s>
The	O
Delsarte	O
–	O
Goethals	O
–	O
Seidel	O
theorem	O
illustrates	O
an	O
example	O
of	O
an	O
application	O
of	O
shortened	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
The	O
Gorenstein	O
–	O
Zierler	O
decoder	O
and	O
the	O
related	O
work	O
on	O
BCH	O
codes	O
are	O
described	O
in	O
a	O
book	O
Error	B-Error_Name
Correcting	I-Error_Name
Codes	I-Error_Name
by	O
W	O
.	O
Wesley	O
Peterson	O
(	O
1961	O
)	O
.	O
</s>
<s>
As	O
a	O
result	O
of	O
the	O
Reed-Solomon	B-Error_Name
encoding	O
procedure	O
,	O
s(x )	O
is	O
divisible	O
by	O
the	O
generator	O
polynomial	O
g(x )	O
:	O
</s>
<s>
Chien	B-Error_Name
search	I-Error_Name
is	O
an	O
efficient	O
implementation	O
of	O
this	O
step	O
.	O
</s>
<s>
This	O
can	O
be	O
done	O
by	O
direct	O
solution	O
for	O
Yk	O
in	O
the	O
error	O
equations	O
matrix	O
given	O
above	O
,	O
or	O
using	O
the	O
Forney	B-Error_Name
algorithm	I-Error_Name
.	O
</s>
<s>
Using	O
Gaussian	B-Algorithm
elimination	I-Algorithm
:	O
</s>
<s>
To	O
calculate	O
the	O
error	O
values	O
,	O
apply	O
the	O
Forney	B-Error_Name
algorithm	I-Error_Name
.	O
</s>
<s>
The	O
Berlekamp	B-Algorithm
–	I-Algorithm
Massey	I-Algorithm
algorithm	I-Algorithm
is	O
an	O
alternate	O
iterative	O
procedure	O
for	O
finding	O
the	O
error	O
locator	O
polynomial	O
.	O
</s>
<s>
The	O
article	O
Berlekamp	B-Algorithm
–	I-Algorithm
Massey	I-Algorithm
algorithm	I-Algorithm
has	O
a	O
detailed	O
description	O
of	O
the	O
procedure	O
.	O
</s>
<s>
Since	O
r(x )	O
=	O
c(x )	O
+	O
e(x )	O
,	O
and	O
since	O
a	O
discrete	O
Fourier	O
transform	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
,	O
R(x )	O
=	O
C(x )	O
+	O
E(x )	O
.	O
</s>
<s>
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
achieves	O
this	O
bound	O
with	O
equality	O
,	O
and	O
can	O
thus	O
correct	O
up	O
to	O
errors	O
.	O
</s>
<s>
In	O
1999	O
,	O
Madhu	O
Sudan	O
and	O
Venkatesan	O
Guruswami	O
at	O
MIT	O
published	O
"	O
Improved	O
Decoding	O
of	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
and	O
Algebraic-Geometry	O
Codes	O
"	O
introducing	O
an	O
algorithm	O
that	O
allowed	O
for	O
the	O
correction	O
of	O
errors	O
beyond	O
half	O
the	O
minimum	O
distance	O
of	O
the	O
code	O
.	O
</s>
<s>
It	O
applies	O
to	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
and	O
more	O
generally	O
to	O
algebraic	O
geometric	O
codes	O
.	O
</s>
<s>
For	O
example	O
,	O
a	O
decoder	O
could	O
associate	O
with	O
each	O
symbol	O
an	O
additional	O
value	O
corresponding	O
to	O
the	O
channel	O
demodulator	B-Device
's	O
confidence	O
in	O
the	O
correctness	O
of	O
the	O
symbol	O
.	O
</s>
<s>
The	O
advent	O
of	O
LDPC	B-Error_Name
and	O
turbo	B-Error_Name
codes	I-Error_Name
,	O
which	O
employ	O
iterated	O
soft-decision	B-Error_Name
belief	O
propagation	O
decoding	O
methods	O
to	O
achieve	O
error-correction	O
performance	O
close	O
to	O
the	O
theoretical	O
limit	O
,	O
has	O
spurred	O
interest	O
in	O
applying	O
soft-decision	B-Error_Name
decoding	I-Error_Name
to	O
conventional	O
algebraic	O
codes	O
.	O
</s>
<s>
In	O
2003	O
,	O
Ralf	O
Koetter	O
and	O
Alexander	O
Vardy	O
presented	O
a	O
polynomial-time	O
soft-decision	B-Error_Name
algebraic	O
list-decoding	O
algorithm	O
for	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
,	O
which	O
was	O
based	O
upon	O
the	O
work	O
by	O
Sudan	O
and	O
Guruswami	O
.	O
</s>
<s>
In	O
2016	O
,	O
Steven	O
J	O
.	O
Franke	O
and	O
Joseph	O
H	O
.	O
Taylor	O
published	O
a	O
novel	O
soft-decision	B-Error_Name
decoder	I-Error_Name
.	O
</s>
<s>
The	O
decoders	O
described	O
in	O
this	O
section	O
use	O
the	O
Reed	B-Error_Name
Solomon	I-Error_Name
original	O
view	O
of	O
a	O
codeword	O
as	O
a	O
sequence	O
of	O
polynomial	O
values	O
where	O
the	O
polynomial	O
is	O
based	O
on	O
the	O
message	O
to	O
be	O
encoded	O
.	O
</s>
<s>
Using	O
Gaussian	B-Algorithm
elimination	I-Algorithm
:	O
</s>
