<s>
In	O
coding	B-Error_Name
theory	I-Error_Name
,	O
burst	B-Error_Name
error-correcting	I-Error_Name
codes	I-Error_Name
employ	O
methods	O
of	O
correcting	O
burst	O
errors	O
,	O
which	O
are	O
errors	O
that	O
occur	O
in	O
many	O
consecutive	O
bits	O
rather	O
than	O
occurring	O
in	O
bits	O
independently	O
of	O
each	O
other	O
.	O
</s>
<s>
Although	O
this	O
definition	O
is	O
sufficient	O
to	O
describe	O
what	O
a	O
burst	O
error	O
is	O
,	O
the	O
majority	O
of	O
the	O
tools	O
developed	O
for	O
burst	O
error	B-Error_Name
correction	I-Error_Name
rely	O
on	O
cyclic	O
codes	O
.	O
</s>
<s>
We	O
will	O
see	O
later	O
that	O
the	O
burst	O
error	B-Error_Name
detection	I-Error_Name
ability	O
of	O
any	O
code	O
is	O
bounded	O
from	O
above	O
by	O
.	O
</s>
<s>
Cyclic	O
codes	O
are	O
considered	O
optimal	O
for	O
burst	O
error	B-Error_Name
detection	I-Error_Name
since	O
they	O
meet	O
this	O
upper	O
bound	O
:	O
</s>
<s>
By	O
the	O
upper	O
bound	O
on	O
burst	O
error	B-Error_Name
detection	I-Error_Name
(	O
)	O
,	O
we	O
know	O
that	O
a	O
cyclic	O
code	O
can	O
not	O
detect	O
all	O
bursts	O
of	O
length	O
.	O
</s>
<s>
By	O
upper	O
bound	O
,	O
we	O
mean	O
a	O
limit	O
on	O
our	O
error	B-Error_Name
detection	I-Error_Name
ability	O
that	O
we	O
can	O
never	O
go	O
beyond	O
.	O
</s>
<s>
Now	O
,	O
we	O
repeat	O
the	O
same	O
question	O
but	O
for	O
error	B-Error_Name
correction	I-Error_Name
:	O
given	O
and	O
,	O
what	O
is	O
the	O
upper	O
bound	O
on	O
the	O
length	O
of	O
bursts	O
that	O
we	O
can	O
correct	O
using	O
any	O
code	O
?	O
</s>
<s>
A	O
linear	O
burst-error-correcting	O
code	O
achieving	O
the	O
above	O
Rieger	O
bound	O
is	O
called	O
an	O
optimal	O
burst-error-correcting	O
code	O
.	O
</s>
<s>
While	O
cyclic	O
codes	O
in	O
general	O
are	O
powerful	O
tools	O
for	O
detecting	O
burst	O
errors	O
,	O
we	O
now	O
consider	O
a	O
family	O
of	O
binary	O
cyclic	O
codes	O
named	O
Fire	O
Codes	O
,	O
which	O
possess	O
good	O
single	O
burst	O
error	B-Error_Name
correction	I-Error_Name
capabilities	O
.	O
</s>
<s>
If	O
we	O
can	O
show	O
that	O
all	O
bursts	O
of	O
length	O
or	O
less	O
occur	O
in	O
different	O
cosets	O
,	O
we	O
can	O
use	O
them	O
as	O
coset	B-Error_Name
leaders	I-Error_Name
that	O
form	O
correctable	O
error	O
patterns	O
.	O
</s>
<s>
The	O
reason	O
is	O
simple	O
:	O
we	O
know	O
that	O
each	O
coset	O
has	O
a	O
unique	O
syndrome	O
decoding	O
associated	O
with	O
it	O
,	O
and	O
if	O
all	O
bursts	O
of	O
different	O
lengths	O
occur	O
in	O
different	O
cosets	O
,	O
then	O
all	O
have	O
unique	O
syndromes	O
,	O
facilitating	O
error	B-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
Certain	O
families	O
of	O
codes	O
,	O
such	O
as	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
,	O
operate	O
on	O
alphabet	O
sizes	O
larger	O
than	O
binary	O
.	O
</s>
<s>
This	O
property	O
awards	O
such	O
codes	O
powerful	O
burst	O
error	B-Error_Name
correction	I-Error_Name
capabilities	O
.	O
</s>
<s>
If	O
is	O
an	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
over	O
,	O
we	O
can	O
think	O
of	O
as	O
an	O
code	O
over	O
.	O
</s>
<s>
The	O
reason	O
such	O
codes	O
are	O
powerful	O
for	O
burst	O
error	B-Error_Name
correction	I-Error_Name
is	O
that	O
each	O
symbol	O
is	O
represented	O
by	O
bits	O
,	O
and	O
in	O
general	O
,	O
it	O
is	O
irrelevant	O
how	O
many	O
of	O
those	O
bits	O
are	O
erroneous	O
;	O
whether	O
a	O
single	O
bit	O
,	O
or	O
all	O
of	O
the	O
bits	O
contain	O
errors	O
,	O
from	O
a	O
decoding	O
perspective	O
it	O
is	O
still	O
a	O
single	O
symbol	O
error	O
.	O
</s>
<s>
Let	O
be	O
a	O
RS	B-Error_Name
code	I-Error_Name
over	O
.	O
</s>
<s>
We	O
now	O
construct	O
a	O
Binary	O
RS	B-Error_Name
Code	I-Error_Name
from	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Binary	O
RS	B-Error_Name
code	I-Error_Name
will	O
have	O
as	O
its	O
parameters	O
.	O
</s>
<s>
By	O
the	O
theorem	O
above	O
for	O
error	B-Error_Name
correction	I-Error_Name
capacity	O
up	O
to	O
the	O
maximum	O
burst	O
length	O
allowed	O
is	O
For	O
burst	O
length	O
of	O
,	O
the	O
decoder	O
may	O
fail	O
.	O
</s>
<s>
Since	O
the	O
separation	O
between	O
consecutive	O
symbols	O
is	O
the	O
number	O
of	O
errors	O
that	O
the	O
deinterleaved	O
output	O
may	O
contain	O
is	O
By	O
the	O
theorem	O
above	O
,	O
for	O
error	B-Error_Name
correction	I-Error_Name
capacity	O
up	O
to	O
,	O
maximum	O
burst	O
length	O
allowed	O
is	O
For	O
burst	O
length	O
of	O
decoder	O
may	O
fail	O
.	O
</s>
<s>
The	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
codes	I-Error_Name
can	O
correct	O
a	O
corrupted	O
symbol	O
with	O
a	O
single	O
bit	O
error	O
just	O
as	O
easily	O
as	O
it	O
can	O
correct	O
a	O
symbol	O
with	O
all	O
bits	O
wrong	O
.	O
</s>
<s>
This	O
makes	O
the	O
RS	B-Error_Name
codes	I-Error_Name
particularly	O
suitable	O
for	O
correcting	O
burst	O
errors	O
.	O
</s>
<s>
By	O
far	O
,	O
the	O
most	O
common	O
application	O
of	O
RS	B-Error_Name
codes	I-Error_Name
is	O
in	O
compact	O
discs	O
.	O
</s>
<s>
In	O
addition	O
to	O
basic	O
error	B-Error_Name
correction	I-Error_Name
provided	O
by	O
RS	B-Error_Name
codes	I-Error_Name
,	O
protection	O
against	O
burst	O
errors	O
due	O
to	O
scratches	O
on	O
the	O
disc	O
is	O
provided	O
by	O
a	O
cross	O
interleaver	O
.	O
</s>
<s>
CIRC	O
(	O
Cross-Interleaved	B-Error_Name
Reed	I-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
)	O
is	O
the	O
basis	O
for	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
in	O
the	O
CD	O
process	O
.	O
</s>
<s>
The	O
1.41	O
Mbit/s	O
sampled	O
data	O
stream	O
passes	O
through	O
the	O
error	B-Error_Name
correction	I-Error_Name
system	O
eventually	O
getting	O
converted	O
to	O
a	O
stream	O
of	O
1.88	O
Mbit/s	O
.	O
</s>
<s>
Next	O
,	O
these	O
24	O
message	O
symbols	O
are	O
encoded	O
using	O
C2	O
(	O
28	O
,	O
24	O
,	O
5	O
)	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
which	O
is	O
a	O
shortened	O
RS	B-Error_Name
code	I-Error_Name
over	O
.	O
</s>
<s>
This	O
is	O
two-error-correcting	O
,	O
being	O
of	O
minimum	O
distance	O
5	O
.	O
</s>
<s>
These	O
are	O
then	O
passed	O
through	O
C1	O
(	O
32	O
,	O
28	O
,	O
5	O
)	O
RS	B-Error_Name
code	I-Error_Name
,	O
resulting	O
in	O
codewords	O
of	O
32	O
coded	O
output	O
symbols	O
.	O
</s>
