<s>
In	O
computer	B-General_Concept
science	I-General_Concept
and	O
telecommunication	O
,	O
Hamming	B-Error_Name
codes	I-Error_Name
are	O
a	O
family	O
of	O
linear	O
error-correcting	O
codes	O
.	O
</s>
<s>
Hamming	B-Error_Name
codes	I-Error_Name
can	O
detect	O
one-bit	O
and	O
two-bit	O
errors	O
,	O
or	O
correct	O
one-bit	O
errors	O
without	O
detection	O
of	O
uncorrected	O
errors	O
.	O
</s>
<s>
By	O
contrast	O
,	O
the	O
simple	O
parity	B-Error_Name
code	I-Error_Name
cannot	O
correct	O
errors	O
,	O
and	O
can	O
detect	O
only	O
an	O
odd	O
number	O
of	O
bits	O
in	O
error	O
.	O
</s>
<s>
Hamming	B-Error_Name
codes	I-Error_Name
are	O
perfect	O
codes	O
,	O
that	O
is	O
,	O
they	O
achieve	O
the	O
highest	O
possible	O
rate	O
for	O
codes	O
with	O
their	O
block	O
length	O
and	O
minimum	O
distance	O
of	O
three	O
.	O
</s>
<s>
Richard	O
W	O
.	O
Hamming	B-Error_Name
invented	O
Hamming	B-Error_Name
codes	I-Error_Name
in	O
1950	O
as	O
a	O
way	O
of	O
automatically	O
correcting	O
errors	O
introduced	O
by	O
punched	B-Architecture
card	I-Architecture
readers	O
.	O
</s>
<s>
In	O
his	O
original	O
paper	O
,	O
Hamming	B-Error_Name
elaborated	O
his	O
general	O
idea	O
,	O
but	O
specifically	O
focused	O
on	O
the	O
Hamming(7,4 )	O
code	O
which	O
adds	O
three	O
parity	B-Error_Name
bits	I-Error_Name
to	O
four	O
bits	O
of	O
data	O
.	O
</s>
<s>
In	O
mathematical	O
terms	O
,	O
Hamming	B-Error_Name
codes	I-Error_Name
are	O
a	O
class	O
of	O
binary	O
linear	O
code	O
.	O
</s>
<s>
Hence	O
the	O
rate	O
of	O
Hamming	B-Error_Name
codes	I-Error_Name
is	O
,	O
which	O
is	O
the	O
highest	O
possible	O
for	O
codes	O
with	O
minimum	O
distance	O
of	O
three	O
(	O
i.e.	O
,	O
the	O
minimal	O
number	O
of	O
bit	O
changes	O
needed	O
to	O
go	O
from	O
any	O
code	O
word	O
to	O
any	O
other	O
code	O
word	O
is	O
three	O
)	O
and	O
block	O
length	O
.	O
</s>
<s>
The	O
parity-check	O
matrix	O
of	O
a	O
Hamming	B-Error_Name
code	I-Error_Name
is	O
constructed	O
by	O
listing	O
all	O
columns	O
of	O
length	O
that	O
are	O
non-zero	O
,	O
which	O
means	O
that	O
the	O
dual	O
code	O
of	O
the	O
Hamming	B-Error_Name
code	I-Error_Name
is	O
the	O
shortened	B-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
,	O
also	O
known	O
as	O
a	O
Simplex	O
code	O
.	O
</s>
<s>
The	O
parity-check	O
matrix	O
has	O
the	O
property	O
that	O
any	O
two	O
columns	O
are	O
pairwise	O
linearly	O
independent	O
.	O
</s>
<s>
Due	O
to	O
the	O
limited	O
redundancy	O
that	O
Hamming	B-Error_Name
codes	I-Error_Name
add	O
to	O
the	O
data	O
,	O
they	O
can	O
only	O
detect	O
and	O
correct	O
errors	O
when	O
the	O
error	O
rate	O
is	O
low	O
.	O
</s>
<s>
This	O
is	O
the	O
case	O
in	O
computer	O
memory	O
(	O
usually	O
RAM	O
)	O
,	O
where	O
bit	O
errors	O
are	O
extremely	O
rare	O
and	O
Hamming	B-Error_Name
codes	I-Error_Name
are	O
widely	O
used	O
,	O
and	O
a	O
RAM	O
with	O
this	O
correction	O
system	O
is	O
a	O
ECC	B-General_Concept
RAM	I-General_Concept
(	O
ECC	B-General_Concept
memory	I-General_Concept
)	O
.	O
</s>
<s>
In	O
this	O
context	O
,	O
an	O
extended	O
Hamming	B-Error_Name
code	I-Error_Name
having	O
one	O
extra	O
parity	B-Error_Name
bit	I-Error_Name
is	O
often	O
used	O
.	O
</s>
<s>
Extended	O
Hamming	B-Error_Name
codes	I-Error_Name
achieve	O
a	O
Hamming	B-Error_Name
distance	O
of	O
four	O
,	O
which	O
allows	O
the	O
decoder	O
to	O
distinguish	O
between	O
when	O
at	O
most	O
one	O
one-bit	O
error	O
occurs	O
and	O
when	O
any	O
two-bit	O
errors	O
occur	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
extended	O
Hamming	B-Error_Name
codes	I-Error_Name
are	O
single-error	O
correcting	O
and	O
double-error	O
detecting	O
,	O
abbreviated	O
as	O
SECDED	O
.	O
</s>
<s>
Richard	O
Hamming	B-Error_Name
,	O
the	O
inventor	O
of	O
Hamming	B-Error_Name
codes	I-Error_Name
,	O
worked	O
at	O
Bell	O
Labs	O
in	O
the	O
late	O
1940s	O
on	O
the	O
Bell	O
Model	B-Device
V	I-Device
computer	O
,	O
an	O
electromechanical	O
relay-based	O
machine	O
with	O
cycle	O
times	O
in	O
seconds	O
.	O
</s>
<s>
Hamming	B-Error_Name
worked	O
on	O
weekends	O
,	O
and	O
grew	O
increasingly	O
frustrated	O
with	O
having	O
to	O
restart	O
his	O
programs	O
from	O
scratch	O
due	O
to	O
detected	O
errors	O
.	O
</s>
<s>
In	O
a	O
taped	O
interview	O
,	O
Hamming	B-Error_Name
said	O
,	O
"	O
And	O
so	O
I	O
said	O
,	O
'	O
Damn	O
it	O
,	O
if	O
the	O
machine	O
can	O
detect	O
an	O
error	O
,	O
why	O
ca	O
n't	O
it	O
locate	O
the	O
position	O
of	O
the	O
error	O
and	O
correct	O
it	O
?	O
'	O
 "	O
.	O
</s>
<s>
In	O
1950	O
,	O
he	O
published	O
what	O
is	O
now	O
known	O
as	O
Hamming	B-Error_Name
code	I-Error_Name
,	O
which	O
remains	O
in	O
use	O
today	O
in	O
applications	O
such	O
as	O
ECC	B-General_Concept
memory	I-General_Concept
.	O
</s>
<s>
A	O
number	O
of	O
simple	O
error-detecting	O
codes	O
were	O
used	O
before	O
Hamming	B-Error_Name
codes	I-Error_Name
,	O
but	O
none	O
were	O
as	O
effective	O
as	O
Hamming	B-Error_Name
codes	I-Error_Name
in	O
the	O
same	O
overhead	O
of	O
space	O
.	O
</s>
<s>
Parity	B-Error_Name
adds	O
a	O
single	O
bit	O
that	O
indicates	O
whether	O
the	O
number	O
of	O
ones	O
(	O
bit-positions	O
with	O
values	O
of	O
one	O
)	O
in	O
the	O
preceding	O
data	O
was	O
even	O
or	O
odd	O
.	O
</s>
<s>
If	O
an	O
odd	O
number	O
of	O
bits	O
is	O
changed	O
in	O
transmission	O
,	O
the	O
message	O
will	O
change	O
parity	B-Error_Name
and	O
the	O
error	O
can	O
be	O
detected	O
at	O
this	O
point	O
;	O
however	O
,	O
the	O
bit	O
that	O
changed	O
may	O
have	O
been	O
the	O
parity	B-Error_Name
bit	I-Error_Name
itself	O
.	O
</s>
<s>
The	O
most	O
common	O
convention	O
is	O
that	O
a	O
parity	B-Error_Name
value	O
of	O
one	O
indicates	O
that	O
there	O
is	O
an	O
odd	O
number	O
of	O
ones	O
in	O
the	O
data	O
,	O
and	O
a	O
parity	B-Error_Name
value	O
of	O
zero	O
indicates	O
that	O
there	O
is	O
an	O
even	O
number	O
of	O
ones	O
.	O
</s>
<s>
If	O
the	O
number	O
of	O
bits	O
changed	O
is	O
even	O
,	O
the	O
check	B-Error_Name
bit	I-Error_Name
will	O
be	O
valid	O
and	O
the	O
error	O
will	O
not	O
be	O
detected	O
.	O
</s>
<s>
Moreover	O
,	O
parity	B-Error_Name
does	O
not	O
indicate	O
which	O
bit	O
contained	O
the	O
error	O
,	O
even	O
when	O
it	O
can	O
detect	O
it	O
.	O
</s>
<s>
However	O
,	O
while	O
the	O
quality	O
of	O
parity	B-Error_Name
checking	I-Error_Name
is	O
poor	O
,	O
since	O
it	O
uses	O
only	O
a	O
single	O
bit	O
,	O
this	O
method	O
results	O
in	O
the	O
least	O
overhead	O
.	O
</s>
<s>
For	O
instance	O
,	O
if	O
the	O
data	O
bit	O
to	O
be	O
sent	O
is	O
a	O
1	O
,	O
an	O
repetition	B-Error_Name
code	I-Error_Name
will	O
send	O
111	O
.	O
</s>
<s>
This	O
triple	O
repetition	B-Error_Name
code	I-Error_Name
is	O
a	O
Hamming	B-Error_Name
code	I-Error_Name
with	O
since	O
there	O
are	O
two	O
parity	B-Error_Name
bits	I-Error_Name
,	O
and	O
data	O
bit	O
.	O
</s>
<s>
If	O
we	O
increase	O
the	O
size	O
of	O
the	O
bit	O
string	O
to	O
four	O
,	O
we	O
can	O
detect	O
all	O
two-bit	O
errors	O
but	O
cannot	O
correct	O
them	O
(	O
the	O
quantity	O
of	O
parity	B-Error_Name
bits	I-Error_Name
is	O
even	O
)	O
;	O
at	O
five	O
bits	O
,	O
we	O
can	O
both	O
detect	O
and	O
correct	O
all	O
two-bit	O
errors	O
,	O
but	O
not	O
all	O
three-bit	O
errors	O
.	O
</s>
<s>
Moreover	O
,	O
increasing	O
the	O
size	O
of	O
the	O
parity	B-Error_Name
bit	I-Error_Name
string	O
is	O
inefficient	O
,	O
reducing	O
throughput	O
by	O
three	O
times	O
in	O
our	O
original	O
case	O
,	O
and	O
the	O
efficiency	O
drops	O
drastically	O
as	O
we	O
increase	O
the	O
number	O
of	O
times	O
each	O
bit	O
is	O
duplicated	O
in	O
order	O
to	O
detect	O
and	O
correct	O
more	O
errors	O
.	O
</s>
<s>
Hamming	B-Error_Name
studied	O
the	O
existing	O
coding	O
schemes	O
,	O
including	O
two-of-five	O
,	O
and	O
generalized	O
their	O
concepts	O
.	O
</s>
<s>
For	O
instance	O
,	O
parity	B-Error_Name
includes	O
a	O
single	O
bit	O
for	O
any	O
data	O
word	O
,	O
so	O
assuming	O
ASCII	B-Protocol
words	O
with	O
seven	O
bits	O
,	O
Hamming	B-Error_Name
described	O
this	O
as	O
an	O
(	O
8	O
,	O
7	O
)	O
code	O
,	O
with	O
eight	O
bits	O
in	O
total	O
,	O
of	O
which	O
seven	O
are	O
data	O
.	O
</s>
<s>
Hamming	B-Error_Name
also	O
noticed	O
the	O
problems	O
with	O
flipping	O
two	O
or	O
more	O
bits	O
,	O
and	O
described	O
this	O
as	O
the	O
"	O
distance	O
"	O
(	O
it	O
is	O
now	O
called	O
the	O
Hamming	B-Error_Name
distance	O
,	O
after	O
him	O
)	O
.	O
</s>
<s>
Parity	B-Error_Name
has	O
a	O
distance	O
of	O
2	O
,	O
so	O
one	O
bit	O
flip	O
can	O
be	O
detected	O
but	O
not	O
corrected	O
,	O
and	O
any	O
two	O
bit	O
flips	O
will	O
be	O
invisible	O
.	O
</s>
<s>
Hamming	B-Error_Name
was	O
interested	O
in	O
two	O
problems	O
at	O
once	O
:	O
increasing	O
the	O
distance	O
as	O
much	O
as	O
possible	O
,	O
while	O
at	O
the	O
same	O
time	O
increasing	O
the	O
code	O
rate	O
as	O
much	O
as	O
possible	O
.	O
</s>
<s>
The	O
key	O
to	O
all	O
of	O
his	O
systems	O
was	O
to	O
have	O
the	O
parity	B-Error_Name
bits	I-Error_Name
overlap	O
,	O
such	O
that	O
they	O
managed	O
to	O
check	O
each	O
other	O
as	O
well	O
as	O
the	O
data	O
.	O
</s>
<s>
All	O
bit	O
positions	O
that	O
are	O
powers	O
of	O
two	O
(	O
have	O
a	O
single	O
1	O
bit	O
in	O
the	O
binary	O
form	O
of	O
their	O
position	O
)	O
are	O
parity	B-Error_Name
bits	I-Error_Name
:	O
1	O
,	O
2	O
,	O
4	O
,	O
8	O
,	O
etc	O
.	O
</s>
<s>
Each	O
data	O
bit	O
is	O
included	O
in	O
a	O
unique	O
set	O
of	O
2	O
or	O
more	O
parity	B-Error_Name
bits	I-Error_Name
,	O
as	O
determined	O
by	O
the	O
binary	O
form	O
of	O
its	O
bit	O
position	O
.	O
</s>
<s>
Parity	B-Error_Name
bit	I-Error_Name
1	O
covers	O
all	O
bit	O
positions	O
which	O
have	O
the	O
least	O
significant	O
bit	O
set	O
:	O
bit	O
1	O
(	O
the	O
parity	B-Error_Name
bit	I-Error_Name
itself	O
)	O
,	O
3	O
,	O
5	O
,	O
7	O
,	O
9	O
,	O
etc	O
.	O
</s>
<s>
Parity	B-Error_Name
bit	I-Error_Name
2	O
covers	O
all	O
bit	O
positions	O
which	O
have	O
the	O
second	O
least	O
significant	O
bit	O
set	O
:	O
bits	O
2-3	O
,	O
6-7	O
,	O
10-11	O
,	O
etc	O
.	O
</s>
<s>
Parity	B-Error_Name
bit	I-Error_Name
4	O
covers	O
all	O
bit	O
positions	O
which	O
have	O
the	O
third	O
least	O
significant	O
bit	O
set	O
:	O
bits	O
4	O
–	O
7	O
,	O
12	O
–	O
15	O
,	O
20	O
–	O
23	O
,	O
etc	O
.	O
</s>
<s>
Parity	B-Error_Name
bit	I-Error_Name
8	O
covers	O
all	O
bit	O
positions	O
which	O
have	O
the	O
fourth	O
least	O
significant	O
bit	O
set	O
:	O
bits	O
8	O
–	O
15	O
,	O
24	O
–	O
31	O
,	O
40	O
–	O
47	O
,	O
etc	O
.	O
</s>
<s>
In	O
general	O
each	O
parity	B-Error_Name
bit	I-Error_Name
covers	O
all	O
bits	O
where	O
the	O
bitwise	O
AND	O
of	O
the	O
parity	B-Error_Name
position	O
and	O
the	O
bit	O
position	O
is	O
non-zero	O
.	O
</s>
<s>
If	O
a	O
byte	O
of	O
data	O
to	O
be	O
encoded	O
is	O
10011010	O
,	O
then	O
the	O
data	O
word	O
(	O
using	O
_	O
to	O
represent	O
the	O
parity	B-Error_Name
bits	I-Error_Name
)	O
would	O
be	O
__1_001_1010	O
,	O
and	O
the	O
code	O
word	O
is	O
011100101010	O
.	O
</s>
<s>
The	O
choice	O
of	O
the	O
parity	B-Error_Name
,	O
even	O
or	O
odd	O
,	O
is	O
irrelevant	O
but	O
the	O
same	O
choice	O
must	O
be	O
used	O
for	O
both	O
encoding	O
and	O
decoding	O
.	O
</s>
<s>
Shown	O
are	O
only	O
20	O
encoded	O
bits	O
(	O
5	O
parity	B-Error_Name
,	O
15	O
data	O
)	O
but	O
the	O
pattern	O
continues	O
indefinitely	O
.	O
</s>
<s>
The	O
key	O
thing	O
about	O
Hamming	B-Error_Name
Codes	I-Error_Name
that	O
can	O
be	O
seen	O
from	O
visual	O
inspection	O
is	O
that	O
any	O
given	O
bit	O
is	O
included	O
in	O
a	O
unique	O
set	O
of	O
parity	B-Error_Name
bits	I-Error_Name
.	O
</s>
<s>
To	O
check	O
for	O
errors	O
,	O
check	O
all	O
of	O
the	O
parity	B-Error_Name
bits	I-Error_Name
.	O
</s>
<s>
If	O
all	O
parity	B-Error_Name
bits	I-Error_Name
are	O
correct	O
,	O
there	O
is	O
no	O
error	O
.	O
</s>
<s>
Otherwise	O
,	O
the	O
sum	O
of	O
the	O
positions	O
of	O
the	O
erroneous	O
parity	B-Error_Name
bits	I-Error_Name
identifies	O
the	O
erroneous	O
bit	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
the	O
parity	B-Error_Name
bits	I-Error_Name
in	O
positions	O
1	O
,	O
2	O
and	O
8	O
indicate	O
an	O
error	O
,	O
then	O
bit	O
1+2+8	O
=	O
11	O
is	O
in	O
error	O
.	O
</s>
<s>
If	O
only	O
one	O
parity	B-Error_Name
bit	I-Error_Name
indicates	O
an	O
error	O
,	O
the	O
parity	B-Error_Name
bit	I-Error_Name
itself	O
is	O
in	O
error	O
.	O
</s>
<s>
With	O
parity	B-Error_Name
bits	I-Error_Name
,	O
bits	O
from	O
1	O
up	O
to	O
can	O
be	O
covered	O
.	O
</s>
<s>
After	O
discounting	O
the	O
parity	B-Error_Name
bits	I-Error_Name
,	O
bits	O
remain	O
for	O
use	O
as	O
data	O
.	O
</s>
<s>
As	O
varies	O
,	O
we	O
get	O
all	O
the	O
possible	O
Hamming	B-Error_Name
codes	I-Error_Name
:	O
</s>
<s>
Hamming	B-Error_Name
codes	I-Error_Name
have	O
a	O
minimum	O
distance	O
of	O
3	O
,	O
which	O
means	O
that	O
the	O
decoder	O
can	O
detect	O
and	O
correct	O
a	O
single	O
error	O
,	O
but	O
it	O
cannot	O
distinguish	O
a	O
double	O
bit	O
error	O
of	O
some	O
codeword	O
from	O
a	O
single	O
bit	O
error	O
of	O
a	O
different	O
codeword	O
.	O
</s>
<s>
To	O
remedy	O
this	O
shortcoming	O
,	O
Hamming	B-Error_Name
codes	I-Error_Name
can	O
be	O
extended	O
by	O
an	O
extra	O
parity	B-Error_Name
bit	I-Error_Name
.	O
</s>
<s>
This	O
way	O
,	O
it	O
is	O
possible	O
to	O
increase	O
the	O
minimum	O
distance	O
of	O
the	O
Hamming	B-Error_Name
code	I-Error_Name
to	O
4	O
,	O
which	O
allows	O
the	O
decoder	O
to	O
distinguish	O
between	O
single	O
bit	O
errors	O
and	O
two-bit	O
errors	O
.	O
</s>
<s>
This	O
extended	O
Hamming	B-Error_Name
code	I-Error_Name
was	O
popular	O
in	O
computer	O
memory	O
systems	O
,	O
starting	O
with	O
IBM	B-Device
7030	I-Device
Stretch	I-Device
in	O
1961	O
,	O
where	O
it	O
is	O
known	O
as	O
SECDED	O
(	O
or	O
SEC-DED	B-Error_Name
,	O
abbreviated	O
from	O
single	O
error	O
correction	O
,	O
double	O
error	O
detection	O
)	O
.	O
</s>
<s>
Server	O
computers	O
in	O
21st	O
century	O
,	O
while	O
typically	O
keeping	O
the	O
SECDED	O
level	O
of	O
protection	O
,	O
no	O
longer	O
use	O
the	O
Hamming	B-Error_Name
's	O
method	O
,	O
relying	O
instead	O
on	O
the	O
designs	O
with	O
longer	O
codewords	O
(	O
128	O
to	O
256	O
bits	O
of	O
data	O
)	O
and	O
modified	O
balanced	O
parity-check	O
trees	O
.	O
</s>
<s>
The	O
(	O
72	O
,	O
64	O
)	O
Hamming	B-Error_Name
code	I-Error_Name
is	O
still	O
popular	O
in	O
some	O
hardware	O
designs	O
,	O
including	O
Xilinx	O
FPGA	B-Architecture
families	O
.	O
</s>
<s>
In	O
1950	O
,	O
Hamming	B-Error_Name
introduced	O
the	O
 [ 7 , 4 ] 	O
Hamming	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
It	O
encodes	O
four	O
data	O
bits	O
into	O
seven	O
bits	O
by	O
adding	O
three	O
parity	B-Error_Name
bits	I-Error_Name
.	O
</s>
<s>
With	O
the	O
addition	O
of	O
an	O
overall	O
parity	B-Error_Name
bit	I-Error_Name
,	O
it	O
becomes	O
the	O
 [ 8 , 4 ] 	O
extended	O
Hamming	B-Error_Name
code	I-Error_Name
which	O
is	O
SECDED	O
and	O
can	O
both	O
detect	O
and	O
correct	O
single-bit	O
errors	O
and	O
detect	O
(	O
but	O
not	O
correct	O
)	O
double-bit	O
errors	O
.	O
</s>
<s>
and	O
is	O
called	O
a	O
parity-check	O
matrix	O
.	O
</s>
<s>
The	O
parity-check	O
matrix	O
H	O
of	O
a	O
Hamming	B-Error_Name
code	I-Error_Name
is	O
constructed	O
by	O
listing	O
all	O
columns	O
of	O
length	O
m	O
that	O
are	O
pair-wise	O
independent	O
.	O
</s>
<s>
The	O
code	O
generator	O
matrix	O
and	O
the	O
parity-check	O
matrix	O
are	O
:	O
</s>
<s>
The	O
 [ 7 , 4 ] 	O
Hamming	B-Error_Name
code	I-Error_Name
can	O
easily	O
be	O
extended	O
to	O
an	O
 [ 8 , 4 ] 	O
code	O
by	O
adding	O
an	O
extra	O
parity	B-Error_Name
bit	I-Error_Name
on	O
top	O
of	O
the	O
(	O
7	O
,	O
4	O
)	O
encoded	O
word	O
(	O
see	O
Hamming(7,4 )	O
)	O
.	O
</s>
<s>
The	O
addition	O
of	O
the	O
fourth	O
row	O
effectively	O
computes	O
the	O
sum	O
of	O
all	O
the	O
codeword	O
bits	O
(	O
data	O
and	O
parity	B-Error_Name
)	O
as	O
the	O
fourth	O
parity	B-Error_Name
bit	I-Error_Name
.	O
</s>
<s>
For	O
example	O
,	O
1011	O
is	O
encoded	O
(	O
using	O
the	O
non-systematic	O
form	O
of	O
G	O
at	O
the	O
start	O
of	O
this	O
section	O
)	O
into	O
01100110	O
where	O
blue	O
digits	O
are	O
data	O
;	O
red	O
digits	O
are	O
parity	B-Error_Name
bits	I-Error_Name
from	O
the	O
 [ 7 , 4 ] 	O
Hamming	B-Error_Name
code	I-Error_Name
;	O
and	O
the	O
green	O
digit	O
is	O
the	O
parity	B-Error_Name
bit	I-Error_Name
added	O
by	O
the	O
 [ 8 , 4 ] 	O
code	O
.	O
</s>
<s>
The	O
green	O
digit	O
makes	O
the	O
parity	B-Error_Name
of	O
the	O
 [ 7 , 4 ] 	O
codewords	O
even	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
code	O
can	O
be	O
defined	O
as	O
 [ 8 , 4 ] 	O
Hamming	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
To	O
decode	O
the	O
 [ 8 , 4 ] 	O
Hamming	B-Error_Name
code	I-Error_Name
,	O
first	O
check	O
the	O
parity	B-Error_Name
bit	I-Error_Name
.	O
</s>
<s>
If	O
the	O
parity	B-Error_Name
bit	I-Error_Name
indicates	O
an	O
error	O
,	O
single	O
error	O
correction	O
(	O
the	O
 [ 7 , 4 ] 	O
Hamming	B-Error_Name
code	I-Error_Name
)	O
will	O
indicate	O
the	O
error	O
location	O
,	O
with	O
"	O
no	O
error	O
"	O
indicating	O
the	O
parity	B-Error_Name
bit	I-Error_Name
.	O
</s>
<s>
If	O
the	O
parity	B-Error_Name
bit	I-Error_Name
is	O
correct	O
,	O
then	O
single	O
error	O
correction	O
will	O
indicate	O
the	O
(	O
bitwise	O
)	O
exclusive-or	O
of	O
two	O
error	O
locations	O
.	O
</s>
