<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
an	O
error-correcting	B-Error_Name
code	I-Error_Name
named	O
after	O
Jacques	O
Hadamard	O
that	O
is	O
used	O
for	O
error	B-Error_Name
detection	I-Error_Name
and	I-Error_Name
correction	I-Error_Name
when	O
transmitting	O
messages	O
over	O
very	O
noisy	O
or	O
unreliable	O
channels	O
.	O
</s>
<s>
Because	O
of	O
its	O
unique	O
mathematical	O
properties	O
,	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
not	O
only	O
used	O
by	O
engineers	O
,	O
but	O
also	O
intensely	O
studied	O
in	O
coding	B-Error_Name
theory	I-Error_Name
,	O
mathematics	O
,	O
and	O
theoretical	O
computer	O
science	O
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
also	O
known	O
under	O
the	O
names	O
Walsh	B-Error_Name
code	I-Error_Name
,	O
Walsh	O
family	O
,	O
and	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
in	O
recognition	O
of	O
the	O
American	O
mathematician	O
Joseph	O
Leonard	O
Walsh	O
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
an	O
example	O
of	O
a	O
linear	O
code	O
of	O
length	O
over	O
a	O
binary	O
alphabet	O
.	O
</s>
<s>
In	O
this	O
article	O
,	O
the	O
first	O
case	O
is	O
called	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
while	O
the	O
second	O
is	O
called	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
unique	O
in	O
that	O
each	O
non-zero	O
codeword	O
has	O
a	O
Hamming	O
weight	O
of	O
exactly	O
,	O
which	O
implies	O
that	O
the	O
distance	O
of	O
the	O
code	O
is	O
also	O
.	O
</s>
<s>
In	O
standard	O
coding	B-Error_Name
theory	I-Error_Name
notation	O
for	O
block	O
codes	O
,	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
a	O
-code	O
,	O
that	O
is	O
,	O
it	O
is	O
a	O
linear	O
code	O
over	O
a	O
binary	O
alphabet	O
,	O
has	O
block	O
length	O
,	O
message	O
length	O
(	O
or	O
dimension	O
)	O
,	O
and	O
minimum	O
distance	O
.	O
</s>
<s>
The	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
a	O
slightly	O
improved	O
version	O
of	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
;	O
it	O
is	O
a	O
-code	O
and	O
thus	O
has	O
a	O
slightly	O
better	O
rate	O
while	O
maintaining	O
the	O
relative	O
distance	O
of	O
,	O
and	O
is	O
thus	O
preferred	O
in	O
practical	O
applications	O
.	O
</s>
<s>
In	O
communication	O
theory	O
,	O
this	O
is	O
simply	O
called	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
and	O
it	O
is	O
the	O
same	O
as	O
the	O
first	O
order	O
Reed	O
–	O
Muller	O
code	O
over	O
the	O
binary	O
alphabet	O
.	O
</s>
<s>
Normally	O
,	O
Hadamard	B-Error_Name
codes	I-Error_Name
are	O
based	O
on	O
Sylvester	O
's	O
construction	O
of	O
Hadamard	O
matrices	O
,	O
but	O
the	O
term	O
“	O
Hadamard	B-Error_Name
code	I-Error_Name
”	O
is	O
also	O
used	O
to	O
refer	O
to	O
codes	O
constructed	O
from	O
arbitrary	O
Hadamard	O
matrices	O
,	O
which	O
are	O
not	O
necessarily	O
of	O
Sylvester	O
type	O
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
a	O
locally	B-Error_Name
decodable	I-Error_Name
code	I-Error_Name
,	O
which	O
provides	O
a	O
way	O
to	O
recover	O
parts	O
of	O
the	O
original	O
message	O
with	O
high	O
probability	O
,	O
while	O
only	O
looking	O
at	O
a	O
small	O
fraction	O
of	O
the	O
received	O
word	O
.	O
</s>
<s>
Since	O
the	O
relative	O
distance	O
of	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
1/2	O
,	O
normally	O
one	O
can	O
only	O
hope	O
to	O
recover	O
from	O
at	O
most	O
a	O
1/4	O
fraction	O
of	O
error	O
.	O
</s>
<s>
In	O
code-division	B-Protocol
multiple	I-Protocol
access	I-Protocol
(	O
CDMA	B-Protocol
)	O
communication	O
,	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
referred	O
to	O
as	O
Walsh	B-Error_Name
Code	I-Error_Name
,	O
and	O
is	O
used	O
to	O
define	O
individual	O
communication	O
channels	O
.	O
</s>
<s>
It	O
is	O
usual	O
in	O
the	O
CDMA	B-Protocol
literature	O
to	O
refer	O
to	O
codewords	O
as	O
“	O
codes	O
”	O
.	O
</s>
<s>
Because	O
Walsh	O
codewords	O
are	O
mathematically	O
orthogonal	O
,	O
a	O
Walsh-encoded	O
signal	O
appears	O
as	O
random	O
noise	O
to	O
a	O
CDMA	B-Protocol
capable	O
mobile	O
terminal	O
,	O
unless	O
that	O
terminal	O
uses	O
the	O
same	O
codeword	O
as	O
the	O
one	O
used	O
to	O
encode	O
the	O
incoming	O
signal	O
.	O
</s>
<s>
Hadamard	B-Error_Name
code	I-Error_Name
is	O
the	O
name	O
that	O
is	O
most	O
commonly	O
used	O
for	O
this	O
code	O
in	O
the	O
literature	O
.	O
</s>
<s>
However	O
,	O
in	O
modern	O
use	O
these	O
error	B-Error_Name
correcting	I-Error_Name
codes	I-Error_Name
are	O
referred	O
to	O
as	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
codes	I-Error_Name
.	O
</s>
<s>
Jacques	O
Hadamard	O
did	O
not	O
invent	O
the	O
code	O
himself	O
,	O
but	O
he	O
defined	O
Hadamard	O
matrices	O
around	O
1893	O
,	O
long	O
before	O
the	O
first	O
error-correcting	B-Error_Name
code	I-Error_Name
,	O
the	O
Hamming	B-Error_Name
code	I-Error_Name
,	O
was	O
developed	O
in	O
the	O
1940s	O
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
based	O
on	O
Hadamard	O
matrices	O
,	O
and	O
while	O
there	O
are	O
many	O
different	O
Hadamard	O
matrices	O
that	O
could	O
be	O
used	O
here	O
,	O
normally	O
only	O
Sylvester	O
's	O
construction	O
of	O
Hadamard	O
matrices	O
is	O
used	O
to	O
obtain	O
the	O
codewords	O
of	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Hence	O
the	O
name	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
disputed	O
and	O
sometimes	O
the	O
code	O
is	O
called	O
Walsh	B-Error_Name
code	I-Error_Name
,	O
honoring	O
the	O
American	O
mathematician	O
Joseph	O
Leonard	O
Walsh	O
.	O
</s>
<s>
An	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
was	O
used	O
during	O
the	O
1971	O
Mariner	O
9	O
mission	O
to	O
correct	O
for	O
picture	O
transmission	O
errors	O
.	O
</s>
<s>
Instead	O
of	O
using	O
a	O
repetition	B-Error_Name
code	I-Error_Name
,	O
a	O
[	O
32	O
,	O
6	O
,	O
16 ]	O
Hadamard	B-Error_Name
code	I-Error_Name
was	O
used	O
.	O
</s>
<s>
Compared	O
to	O
a	O
5-repetition	O
code	O
,	O
the	O
error	O
correcting	O
properties	O
of	O
this	O
Hadamard	B-Error_Name
code	I-Error_Name
are	O
much	O
better	O
,	O
yet	O
its	O
rate	O
is	O
comparable	O
.	O
</s>
<s>
Since	O
the	O
1990s	O
use	O
of	O
this	O
code	O
by	O
space	O
programs	O
has	O
more	O
or	O
less	O
ceased	O
,	O
and	O
the	O
NASA	O
Deep	O
Space	O
Network	O
does	O
not	O
support	O
this	O
error	B-Error_Name
correction	I-Error_Name
scheme	O
for	O
its	O
dishes	O
that	O
are	O
greater	O
than	O
26m	O
.	O
</s>
<s>
While	O
all	O
Hadamard	B-Error_Name
codes	I-Error_Name
are	O
based	O
on	O
Hadamard	O
matrices	O
,	O
the	O
constructions	O
differ	O
in	O
subtle	O
ways	O
for	O
different	O
scientific	O
fields	O
,	O
authors	O
,	O
and	O
uses	O
.	O
</s>
<s>
Engineers	O
,	O
who	O
use	O
the	O
codes	O
for	O
data	O
transmission	O
,	O
and	O
coding	B-Error_Name
theorists	I-Error_Name
,	O
who	O
analyse	O
extremal	O
properties	O
of	O
codes	O
,	O
typically	O
want	O
the	O
rate	O
of	O
the	O
code	O
to	O
be	O
as	O
high	O
as	O
possible	O
,	O
even	O
if	O
this	O
means	O
that	O
the	O
construction	O
becomes	O
mathematically	O
slightly	O
less	O
elegant	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
for	O
many	O
applications	O
of	O
Hadamard	B-Error_Name
codes	I-Error_Name
in	O
theoretical	O
computer	O
science	O
it	O
is	O
not	O
so	O
important	O
to	O
achieve	O
the	O
optimal	O
rate	O
,	O
and	O
hence	O
simpler	O
constructions	O
of	O
Hadamard	B-Error_Name
codes	I-Error_Name
are	O
preferred	O
since	O
they	O
can	O
be	O
analyzed	O
more	O
elegantly	O
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
used	O
in	O
practice	O
since	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
itself	O
is	O
somewhat	O
wasteful	O
.	O
</s>
<s>
Hence	O
it	O
makes	O
sense	O
to	O
restrict	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
to	O
these	O
positions	O
,	O
which	O
gives	O
rise	O
to	O
the	O
augmented	O
Hadamard	O
encoding	O
of	O
;	O
that	O
is	O
,	O
.	O
</s>
<s>
The	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
a	O
linear	O
code	O
,	O
and	O
all	O
linear	O
codes	O
can	O
be	O
generated	O
by	O
a	O
generator	O
matrix	O
.	O
</s>
<s>
In	O
particular	O
,	O
an	O
equivalent	O
way	O
to	O
write	O
the	O
inner	O
product	O
definition	O
for	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
arises	O
by	O
using	O
the	O
generator	O
matrix	O
whose	O
columns	O
consist	O
of	O
all	O
strings	O
of	O
length	O
,	O
that	O
is	O
,	O
</s>
<s>
For	O
example	O
,	O
the	O
generator	O
matrix	O
for	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
of	O
dimension	O
is	O
:	O
</s>
<s>
The	O
matrix	O
is	O
a	O
-matrix	O
and	O
gives	O
rise	O
to	O
the	O
linear	B-Architecture
operator	I-Architecture
.	O
</s>
<s>
The	O
generator	O
matrix	O
of	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
obtained	O
by	O
restricting	O
the	O
matrix	O
to	O
the	O
columns	O
whose	O
first	O
entry	O
is	O
one	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
generator	O
matrix	O
for	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
of	O
dimension	O
is	O
:	O
</s>
<s>
Then	O
is	O
a	O
linear	B-Architecture
mapping	I-Architecture
with	O
.	O
</s>
<s>
For	O
general	O
,	O
the	O
generator	O
matrix	O
of	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
a	O
parity-check	O
matrix	O
for	O
the	O
extended	O
Hamming	B-Error_Name
code	I-Error_Name
of	O
length	O
and	O
dimension	O
,	O
which	O
makes	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
the	O
dual	O
code	O
of	O
the	O
extended	O
Hamming	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Hence	O
an	O
alternative	O
way	O
to	O
define	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
in	O
terms	O
of	O
its	O
parity-check	O
matrix	O
:	O
the	O
parity-check	O
matrix	O
of	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
equal	O
to	O
the	O
generator	O
matrix	O
of	O
the	O
Hamming	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
Hadamard	B-Error_Name
codes	I-Error_Name
are	O
obtained	O
from	O
an	O
n-by-n	O
Hadamard	O
matrix	O
H	O
.	O
In	O
particular	O
,	O
the	O
2n	O
codewords	O
of	O
the	O
code	O
are	O
the	O
rows	O
of	O
H	O
and	O
the	O
rows	O
of	O
−H	O
.	O
</s>
<s>
To	O
get	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
above	O
with	O
,	O
the	O
chosen	O
Hadamard	O
matrix	O
H	O
has	O
to	O
be	O
of	O
Sylvester	O
type	O
,	O
which	O
gives	O
rise	O
to	O
a	O
message	O
length	O
of	O
.	O
</s>
<s>
Since	O
the	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
is	O
a	O
linear	O
code	O
,	O
the	O
distance	O
is	O
equal	O
to	O
the	O
minimum	O
Hamming	O
weight	O
among	O
all	O
of	O
its	O
non-zero	O
codewords	O
.	O
</s>
<s>
All	O
non-zero	O
codewords	O
of	O
the	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
have	O
a	O
Hamming	O
weight	O
of	O
exactly	O
by	O
the	O
following	O
argument	O
.	O
</s>
<s>
Thus	O
,	O
in	O
fact	O
,	O
all	O
non-zero	O
codewords	O
of	O
the	O
Hadamard	B-Error_Name
code	I-Error_Name
have	O
relative	O
Hamming	O
weight	O
,	O
and	O
thus	O
,	O
its	O
relative	O
distance	O
is	O
.	O
</s>
<s>
The	O
relative	O
distance	O
of	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
is	O
as	O
well	O
,	O
but	O
it	O
no	O
longer	O
has	O
the	O
property	O
that	O
every	O
non-zero	O
codeword	O
has	O
weight	O
exactly	O
since	O
the	O
all	O
s	O
vector	O
is	O
a	O
codeword	O
of	O
the	O
augmented	O
Hadamard	B-Error_Name
code	I-Error_Name
.	O
</s>
<s>
A	O
locally	B-Error_Name
decodable	I-Error_Name
code	I-Error_Name
is	O
a	O
code	O
that	O
allows	O
a	O
single	O
bit	O
of	O
the	O
original	O
message	O
to	O
be	O
recovered	O
with	O
high	O
probability	O
by	O
only	O
looking	O
at	O
a	O
small	O
portion	O
of	O
the	O
received	O
word	O
.	O
</s>
<s>
A	O
code	O
is	O
-query	O
locally	B-Error_Name
decodable	I-Error_Name
if	O
a	O
message	O
bit	O
,	O
,	O
can	O
be	O
recovered	O
by	O
checking	O
bits	O
of	O
the	O
received	O
word	O
.	O
</s>
<s>
Theorem	O
1	O
:	O
The	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
is	O
-locally	O
decodable	O
for	O
all	O
.	O
</s>
<s>
Lemma	O
1	O
:	O
For	O
all	O
codewords	O
,	O
in	O
a	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
,	O
,	O
,	O
where	O
represent	O
the	O
bits	O
in	O
in	O
positions	O
and	O
respectively	O
,	O
and	O
represents	O
the	O
bit	O
at	O
position	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
Walsh	B-Error_Name
–	I-Error_Name
Hadamard	I-Error_Name
code	I-Error_Name
is	O
locally	B-Error_Name
decodable	I-Error_Name
for	O
.	O
</s>
<s>
For	O
k≤7	O
the	O
linear	O
Hadamard	B-Error_Name
codes	I-Error_Name
have	O
been	O
proven	O
optimal	O
in	O
the	O
sense	O
of	O
minimum	O
distance	O
.	O
</s>
