<s>
Coding	B-Error_Name
theory	I-Error_Name
is	O
the	O
study	O
of	O
the	O
properties	O
of	O
codes	O
and	O
their	O
respective	O
fitness	O
for	O
specific	O
applications	O
.	O
</s>
<s>
Codes	O
are	O
used	O
for	O
data	B-General_Concept
compression	I-General_Concept
,	O
cryptography	O
,	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
,	O
data	O
transmission	O
and	O
data	B-General_Concept
storage	I-General_Concept
.	O
</s>
<s>
Codes	O
are	O
studied	O
by	O
various	O
scientific	O
disciplines	O
—	O
such	O
as	O
information	O
theory	O
,	O
electrical	O
engineering	O
,	O
mathematics	O
,	O
linguistics	O
,	O
and	O
computer	B-General_Concept
science	I-General_Concept
—	O
for	O
the	O
purpose	O
of	O
designing	O
efficient	O
and	O
reliable	O
data	O
transmission	O
methods	O
.	O
</s>
<s>
This	O
typically	O
involves	O
the	O
removal	O
of	O
redundancy	B-General_Concept
and	O
the	O
correction	O
or	O
detection	O
of	O
errors	O
in	O
the	O
transmitted	O
data	O
.	O
</s>
<s>
Data	B-General_Concept
compression	I-General_Concept
attempts	O
to	O
remove	O
unwanted	O
redundancy	B-General_Concept
from	O
the	O
data	O
from	O
a	O
source	O
in	O
order	O
to	O
transmit	O
it	O
more	O
efficiently	O
.	O
</s>
<s>
For	O
example	O
,	O
ZIP	B-General_Concept
data	I-General_Concept
compression	I-General_Concept
makes	O
data	O
files	O
smaller	O
,	O
for	O
purposes	O
such	O
as	O
to	O
reduce	O
Internet	O
traffic	O
.	O
</s>
<s>
Data	B-General_Concept
compression	I-General_Concept
and	O
error	B-Error_Name
correction	I-Error_Name
may	O
be	O
studied	O
in	O
combination	O
.	O
</s>
<s>
Error	B-Error_Name
correction	I-Error_Name
adds	O
useful	O
redundancy	B-General_Concept
to	O
the	O
data	O
from	O
a	O
source	O
to	O
make	O
the	O
transmission	O
more	O
robust	O
to	O
disturbances	O
present	O
on	O
the	O
transmission	O
channel	O
.	O
</s>
<s>
The	O
ordinary	O
user	O
may	O
not	O
be	O
aware	O
of	O
many	O
applications	O
using	O
error	B-Error_Name
correction	I-Error_Name
.	O
</s>
<s>
A	O
typical	O
music	O
compact	O
disc	O
(	O
CD	O
)	O
uses	O
the	O
Reed	B-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
code	I-Error_Name
to	O
correct	O
for	O
scratches	O
and	O
dust	O
.	O
</s>
<s>
Cell	O
phones	O
also	O
use	O
coding	B-General_Concept
techniques	I-General_Concept
to	O
correct	O
for	O
the	O
fading	O
and	O
noise	O
of	O
high	O
frequency	O
radio	O
transmission	O
.	O
</s>
<s>
Data	O
modems	O
,	O
telephone	O
transmissions	O
,	O
and	O
the	O
NASA	O
Deep	O
Space	O
Network	O
all	O
employ	O
channel	O
coding	B-General_Concept
techniques	I-General_Concept
to	O
get	O
the	O
bits	O
through	O
,	O
for	O
example	O
the	O
turbo	B-Error_Name
code	I-Error_Name
and	O
LDPC	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
The	O
binary	B-Error_Name
Golay	I-Error_Name
code	I-Error_Name
was	O
developed	O
in	O
1949	O
.	O
</s>
<s>
It	O
is	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
capable	O
of	O
correcting	O
up	O
to	O
three	O
errors	O
in	O
each	O
24-bit	O
word	O
,	O
and	O
detecting	O
a	O
fourth	O
.	O
</s>
<s>
Richard	O
Hamming	O
won	O
the	O
Turing	O
Award	O
in	O
1968	O
for	O
his	O
work	O
at	O
Bell	O
Labs	O
in	O
numerical	O
methods	O
,	O
automatic	O
coding	O
systems	O
,	O
and	O
error-detecting	O
and	O
error-correcting	B-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
He	O
invented	O
the	O
concepts	O
known	O
as	O
Hamming	B-Error_Name
codes	I-Error_Name
,	O
Hamming	O
windows	O
,	O
Hamming	O
numbers	O
,	O
and	O
Hamming	O
distance	O
.	O
</s>
<s>
In	O
1972	O
,	O
Nasir	O
Ahmed	O
proposed	O
the	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
DCT	B-General_Concept
)	O
,	O
which	O
he	O
developed	O
with	O
T	O
.	O
Natarajan	O
and	O
K	O
.	O
R	O
.	O
Rao	O
in	O
1973	O
.	O
</s>
<s>
The	O
DCT	B-General_Concept
is	O
the	O
most	O
widely	O
used	O
lossy	B-Algorithm
compression	I-Algorithm
algorithm	O
,	O
the	O
basis	O
for	O
multimedia	O
formats	O
such	O
as	O
JPEG	O
,	O
MPEG	O
and	O
MP3	B-Application
.	O
</s>
<s>
The	O
aim	O
of	O
source	B-General_Concept
coding	I-General_Concept
is	O
to	O
take	O
the	O
source	O
data	O
and	O
make	O
it	O
smaller	O
.	O
</s>
<s>
Basically	O
,	O
source	O
codes	O
try	O
to	O
reduce	O
the	O
redundancy	B-General_Concept
present	O
in	O
the	O
source	O
,	O
and	O
represent	O
the	O
source	O
with	O
fewer	O
bits	O
that	O
carry	O
more	O
information	O
.	O
</s>
<s>
Data	B-General_Concept
compression	I-General_Concept
which	O
explicitly	O
tries	O
to	O
minimize	O
the	O
average	O
length	O
of	O
messages	O
according	O
to	O
a	O
particular	O
assumed	O
probability	O
model	O
is	O
called	O
entropy	B-Algorithm
encoding	I-Algorithm
.	O
</s>
<s>
Various	O
techniques	O
used	O
by	O
source	B-General_Concept
coding	I-General_Concept
schemes	O
try	O
to	O
achieve	O
the	O
limit	O
of	O
entropy	O
of	O
the	O
source	O
.	O
</s>
<s>
In	O
particular	O
,	O
no	O
source	B-General_Concept
coding	I-General_Concept
scheme	O
can	O
be	O
better	O
than	O
the	O
entropy	O
of	O
the	O
source	O
.	O
</s>
<s>
Facsimile	O
transmission	O
uses	O
a	O
simple	O
run	B-Algorithm
length	I-Algorithm
code	I-Algorithm
.	O
</s>
<s>
Source	B-General_Concept
coding	I-General_Concept
removes	O
all	O
data	O
superfluous	O
to	O
the	O
need	O
of	O
the	O
transmitter	O
,	O
decreasing	O
the	O
bandwidth	O
required	O
for	O
transmission	O
.	O
</s>
<s>
The	O
purpose	O
of	O
channel	O
coding	B-Error_Name
theory	I-Error_Name
is	O
to	O
find	O
codes	O
which	O
transmit	O
quickly	O
,	O
contain	O
many	O
valid	O
code	O
words	O
and	O
can	O
correct	O
or	O
at	O
least	O
detect	B-Error_Name
many	O
errors	O
.	O
</s>
<s>
CDs	O
use	O
cross-interleaved	B-Error_Name
Reed	I-Error_Name
–	I-Error_Name
Solomon	I-Error_Name
coding	I-Error_Name
to	O
spread	O
the	O
data	O
out	O
over	O
the	O
disk	O
.	O
</s>
<s>
The	O
term	O
algebraic	B-Error_Name
coding	I-Error_Name
theory	I-Error_Name
denotes	O
the	O
sub-field	O
of	O
coding	B-Error_Name
theory	I-Error_Name
where	O
the	O
properties	O
of	O
codes	O
are	O
expressed	O
in	O
algebraic	O
terms	O
and	O
then	O
further	O
researched	O
.	O
</s>
<s>
Algebraic	B-Error_Name
coding	I-Error_Name
theory	I-Error_Name
is	O
basically	O
divided	O
into	O
two	O
major	O
types	O
of	O
codes	O
:	O
</s>
<s>
The	O
powerful	O
(	O
24	O
,	O
12	O
)	O
Golay	B-Error_Name
code	I-Error_Name
used	O
in	O
deep	O
space	O
communications	O
uses	O
24	O
dimensions	O
.	O
</s>
<s>
The	O
only	O
nontrivial	O
and	O
useful	O
perfect	O
codes	O
are	O
the	O
distance-3	O
Hamming	B-Error_Name
codes	I-Error_Name
with	O
parameters	O
satisfying	O
(	O
2r	O
–	O
1	O
,	O
2r	O
–	O
1	O
–	O
r	O
,	O
3	O
)	O
,	O
and	O
the	O
 [ 23 , 12 , 7 ] 	O
binary	O
and	O
 [ 11 , 6 , 5 ] 	O
ternary	O
Golay	O
codes	O
.	O
</s>
<s>
This	O
is	O
like	O
convolution	B-Language
used	O
in	O
LTI	O
systems	O
to	O
find	O
the	O
output	O
of	O
a	O
system	O
,	O
when	O
you	O
know	O
the	O
input	O
and	O
impulse	O
response	O
.	O
</s>
<s>
So	O
we	O
generally	O
find	O
the	O
output	O
of	O
the	O
system	O
convolutional	O
encoder	O
,	O
which	O
is	O
the	O
convolution	B-Language
of	O
the	O
input	O
bit	O
,	O
against	O
the	O
states	O
of	O
the	O
convolution	B-Language
encoder	O
,	O
registers	O
.	O
</s>
<s>
The	O
Viterbi	B-Algorithm
algorithm	I-Algorithm
is	O
the	O
optimum	O
algorithm	O
used	O
to	O
decode	O
convolutional	O
codes	O
.	O
</s>
<s>
Cryptography	O
or	O
cryptographic	O
coding	O
is	O
the	O
practice	O
and	O
study	O
of	O
techniques	O
for	O
secure	O
communication	O
in	O
the	O
presence	O
of	O
third	O
parties	O
(	O
called	O
adversaries	B-General_Concept
)	O
.	O
</s>
<s>
More	O
generally	O
,	O
it	O
is	O
about	O
constructing	O
and	O
analyzing	O
protocols	O
that	O
block	O
adversaries	B-General_Concept
;	O
various	O
aspects	O
in	O
information	O
security	O
such	O
as	O
data	O
confidentiality	O
,	O
data	O
integrity	O
,	O
authentication	O
,	O
and	O
non-repudiation	O
are	O
central	O
to	O
modern	O
cryptography	O
.	O
</s>
<s>
Modern	O
cryptography	O
exists	O
at	O
the	O
intersection	O
of	O
the	O
disciplines	O
of	O
mathematics	O
,	O
computer	B-General_Concept
science	I-General_Concept
,	O
and	O
electrical	O
engineering	O
.	O
</s>
<s>
Modern	O
cryptography	O
is	O
heavily	O
based	O
on	O
mathematical	O
theory	O
and	O
computer	B-General_Concept
science	I-General_Concept
practice	O
;	O
cryptographic	O
algorithms	O
are	O
designed	O
around	O
computational	O
hardness	O
assumptions	O
,	O
making	O
such	O
algorithms	O
hard	O
to	O
break	O
in	O
practice	O
by	O
any	O
adversary	B-General_Concept
.	O
</s>
<s>
There	O
exist	O
information-theoretically	B-Algorithm
secure	I-Algorithm
schemes	O
that	O
cannot	O
be	O
broken	O
even	O
with	O
unlimited	O
computing	O
power	O
—	O
an	O
example	O
is	O
the	O
one-time	B-Algorithm
pad	I-Algorithm
—	O
but	O
these	O
schemes	O
are	O
more	O
difficult	O
to	O
implement	O
than	O
the	O
best	O
theoretically	O
breakable	O
but	O
computationally	O
secure	O
mechanisms	O
.	O
</s>
<s>
A	O
line	B-Protocol
code	I-Protocol
(	O
also	O
called	O
digital	B-Protocol
baseband	I-Protocol
modulation	I-Protocol
or	O
digital	B-General_Concept
baseband	B-Algorithm
transmission	I-Algorithm
method	O
)	O
is	O
a	O
code	O
chosen	O
for	O
use	O
within	O
a	O
communications	O
system	O
for	O
baseband	B-Algorithm
transmission	I-Algorithm
purposes	O
.	O
</s>
<s>
Line	B-Protocol
coding	I-Protocol
is	O
often	O
used	O
for	O
digital	B-General_Concept
data	I-General_Concept
transport	O
.	O
</s>
<s>
Line	B-Protocol
coding	I-Protocol
consists	O
of	O
representing	O
the	O
digital	B-General_Concept
signal	O
to	O
be	O
transported	O
by	O
an	O
amplitude	O
-	O
and	O
time-discrete	O
signal	O
that	O
is	O
optimally	O
tuned	O
for	O
the	O
specific	O
properties	O
of	O
the	O
physical	O
channel	O
(	O
and	O
of	O
the	O
receiving	O
equipment	O
)	O
.	O
</s>
<s>
The	O
waveform	O
pattern	O
of	O
voltage	O
or	O
current	O
used	O
to	O
represent	O
the	O
1s	O
and	O
0s	O
of	O
a	O
digital	B-General_Concept
data	I-General_Concept
on	O
a	O
transmission	O
link	O
is	O
called	O
line	B-Protocol
encoding	I-Protocol
.	O
</s>
<s>
The	O
common	O
types	O
of	O
line	B-Protocol
encoding	I-Protocol
are	O
unipolar	B-Protocol
,	O
polar	B-Protocol
,	O
bipolar	B-Protocol
,	O
and	O
Manchester	B-Protocol
encoding	I-Protocol
.	O
</s>
<s>
Another	O
concern	O
of	O
coding	B-Error_Name
theory	I-Error_Name
is	O
designing	O
codes	O
that	O
help	O
synchronization	O
.	O
</s>
<s>
Another	O
application	O
of	O
codes	O
,	O
used	O
in	O
some	O
mobile	O
phone	O
systems	O
,	O
is	O
code-division	B-Protocol
multiple	I-Protocol
access	I-Protocol
(	O
CDMA	B-Protocol
)	O
.	O
</s>
<s>
Another	O
general	O
class	O
of	O
codes	O
are	O
the	O
automatic	B-Error_Name
repeat-request	I-Error_Name
(	O
ARQ	O
)	O
codes	O
.	O
</s>
<s>
In	O
these	O
codes	O
the	O
sender	O
adds	O
redundancy	B-General_Concept
to	O
each	O
message	O
for	O
error	B-Error_Name
checking	I-Error_Name
,	O
usually	O
by	O
adding	O
check	B-Error_Name
bits	I-Error_Name
.	O
</s>
<s>
If	O
the	O
check	B-Error_Name
bits	I-Error_Name
are	O
not	O
consistent	O
with	O
the	O
rest	O
of	O
the	O
message	O
when	O
it	O
arrives	O
,	O
the	O
receiver	O
will	O
ask	O
the	O
sender	O
to	O
retransmit	O
the	O
message	O
.	O
</s>
<s>
Common	O
protocols	O
include	O
SDLC	B-Protocol
(	O
IBM	O
)	O
,	O
TCP	B-Protocol
(	O
Internet	O
)	O
,	O
X.25	B-Protocol
(	O
International	O
)	O
and	O
many	O
others	O
.	O
</s>
<s>
Typically	O
numbering	O
schemes	O
are	O
used	O
,	O
as	O
in	O
TCP	B-Protocol
.	O
</s>
<s>
Information	O
is	O
encoded	O
analogously	O
in	O
the	O
neural	B-Architecture
networks	I-Architecture
of	O
brains	O
,	O
in	O
analog	B-Algorithm
signal	I-Algorithm
processing	I-Algorithm
,	O
and	O
analog	O
electronics	O
.	O
</s>
<s>
Aspects	O
of	O
analog	B-Error_Name
coding	I-Error_Name
include	O
analog	O
error	B-Error_Name
correction	I-Error_Name
,	O
</s>
<s>
analog	O
data	B-General_Concept
compression	I-General_Concept
and	O
analog	O
encryption	O
.	O
</s>
<s>
Neural	O
coding	O
is	O
a	O
neuroscience-related	O
field	O
concerned	O
with	O
how	O
sensory	O
and	O
other	O
information	O
is	O
represented	O
in	O
the	O
brain	O
by	O
networks	B-Architecture
of	O
neurons	O
.	O
</s>
