<s>
In	O
mathematics	O
,	O
the	O
Paley	B-Algorithm
construction	I-Algorithm
is	O
a	O
method	O
for	O
constructing	O
Hadamard	O
matrices	O
using	O
finite	O
fields	O
.	O
</s>
<s>
The	O
Paley	B-Algorithm
construction	I-Algorithm
uses	O
quadratic	O
residues	O
in	O
a	O
finite	O
field	O
GF(q )	O
where	O
q	O
is	O
a	O
power	O
of	O
an	O
odd	O
prime	O
number	O
.	O
</s>
<s>
The	O
Jacobsthal	B-Algorithm
matrix	I-Algorithm
Q	O
for	O
GF(q )	O
is	O
the	O
q×q	O
matrix	O
with	O
rows	O
and	O
columns	O
indexed	O
by	O
finite	O
field	O
elements	O
such	O
that	O
the	O
entry	O
in	O
row	O
a	O
and	O
column	O
b	O
is	O
χ( a−b	O
)	O
.	O
</s>
<s>
The	O
Jacobsthal	B-Algorithm
matrix	I-Algorithm
has	O
the	O
properties	O
Q	O
QT	O
=	O
q	O
I−J	O
and	O
Q	O
J	O
=	O
J	O
Q	O
=	O
0	O
where	O
I	O
is	O
the	O
q×q	O
identity	O
matrix	O
and	O
J	O
is	O
the	O
q×q	O
all	O
1	O
matrix	O
.	O
</s>
<s>
which	O
implies	O
that	O
Q	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
skew-symmetric	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
It	O
is	O
a	O
symmetric	B-Algorithm
Hadamard	O
matrix	O
.	O
</s>
<s>
Applying	O
Paley	B-Algorithm
Construction	I-Algorithm
I	O
to	O
the	O
Jacobsthal	B-Algorithm
matrix	I-Algorithm
for	O
GF(7 )	O
,	O
one	O
produces	O
the	O
8×8	O
Hadamard	O
matrix	O
,	O
</s>
<s>
It	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
consisting	O
of	O
nine	O
3×3	O
circulant	O
blocks	O
.	O
</s>
<s>
Paley	B-Algorithm
Construction	I-Algorithm
II	O
produces	O
the	O
symmetric	B-Algorithm
20×20	O
Hadamard	O
matrix	O
,	O
</s>
<s>
By	O
forming	O
Kronecker	O
products	O
of	O
matrices	O
from	O
the	O
Paley	B-Algorithm
construction	I-Algorithm
and	O
the	O
2×2	O
matrix	O
,	O
</s>
<s>
In	O
his	O
1933	O
paper	O
,	O
Paley	O
says	O
“	O
It	O
seems	O
probable	O
that	O
,	O
whenever	O
m	O
is	O
divisible	O
by	O
4	O
,	O
it	O
is	O
possible	O
to	O
construct	O
an	O
orthogonal	B-Algorithm
matrix	I-Algorithm
of	O
order	O
m	O
composed	O
of	O
±1	O
,	O
but	O
the	O
general	O
theorem	O
has	O
every	O
appearance	O
of	O
difficulty.	O
”	O
This	O
appears	O
to	O
be	O
the	O
first	O
published	O
statement	O
of	O
the	O
Hadamard	O
conjecture	O
.	O
</s>
