<s>
In	O
mathematics	O
,	O
a	O
Costas	B-Algorithm
array	I-Algorithm
can	O
be	O
regarded	O
geometrically	O
as	O
a	O
set	O
of	O
n	O
points	O
,	O
each	O
at	O
the	O
center	O
of	O
a	O
square	O
in	O
an	O
nn	O
square	O
tiling	O
such	O
that	O
each	O
row	O
or	O
column	O
contains	O
only	O
one	O
point	O
,	O
and	O
all	O
of	O
the	O
n(n1 )	O
/2	O
displacement	O
vectors	O
between	O
each	O
pair	O
of	O
dots	O
are	O
distinct	O
.	O
</s>
<s>
This	O
results	O
in	O
an	O
ideal	O
"	O
thumbtack	O
"	O
auto-ambiguity	O
function	O
,	O
making	O
the	O
arrays	O
useful	O
in	O
applications	O
such	O
as	O
sonar	B-Application
and	O
radar	B-Application
.	O
</s>
<s>
Costas	B-Algorithm
arrays	I-Algorithm
can	O
be	O
regarded	O
as	O
two-dimensional	O
cousins	O
of	O
the	O
one-dimensional	O
Golomb	O
ruler	O
construction	O
,	O
and	O
,	O
as	O
well	O
as	O
being	O
of	O
mathematical	O
interest	O
,	O
have	O
similar	O
applications	O
in	O
experimental	O
design	O
and	O
phased	O
array	O
radar	B-Application
engineering	O
.	O
</s>
<s>
Costas	B-Algorithm
arrays	I-Algorithm
are	O
named	O
after	O
John	O
P	O
.	O
Costas	O
,	O
who	O
first	O
wrote	O
about	O
them	O
in	O
a	O
1965	O
technical	O
report	O
.	O
</s>
<s>
Independently	O
,	O
Edgar	O
Gilbert	O
also	O
wrote	O
about	O
them	O
in	O
the	O
same	O
year	O
,	O
publishing	O
what	O
is	O
now	O
known	O
as	O
the	O
logarithmic	O
Welch	O
method	O
of	O
constructing	O
Costas	B-Algorithm
arrays	I-Algorithm
.	O
</s>
<s>
A	O
Costas	B-Algorithm
array	I-Algorithm
may	O
be	O
represented	O
numerically	O
as	O
an	O
nn	O
array	O
of	O
numbers	O
,	O
where	O
each	O
entry	O
is	O
either	O
1	O
,	O
for	O
a	O
point	O
,	O
or	O
0	O
,	O
for	O
the	O
absence	O
of	O
a	O
point	O
.	O
</s>
<s>
When	O
interpreted	O
as	O
binary	B-Algorithm
matrices	I-Algorithm
,	O
these	O
arrays	O
of	O
numbers	O
have	O
the	O
property	O
that	O
,	O
since	O
each	O
row	O
and	O
column	O
has	O
the	O
constraint	O
that	O
it	O
only	O
has	O
one	O
point	O
on	O
it	O
,	O
they	O
are	O
therefore	O
also	O
permutation	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
Thus	O
,	O
the	O
Costas	B-Algorithm
arrays	I-Algorithm
for	O
any	O
given	O
n	O
are	O
a	O
subset	O
of	O
the	O
permutation	B-Algorithm
matrices	I-Algorithm
of	O
order	O
n	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
following	O
is	O
a	O
valid	O
Costas	B-Algorithm
array	I-Algorithm
of	O
order	O
N	O
=	O
4	O
:	O
</s>
<s>
This	O
defines	O
a	O
permutation	B-Algorithm
.	O
</s>
<s>
Costas	B-Algorithm
array	I-Algorithm
counts	O
are	O
known	O
for	O
orders	O
1	O
through	O
29	O
:	O
</s>
<s>
Enumeration	O
of	O
known	O
Costas	B-Algorithm
arrays	I-Algorithm
to	O
order	O
200	O
,	O
order	O
500	O
and	O
to	O
order	O
1030	O
are	O
available	O
.	O
</s>
<s>
Although	O
these	O
lists	O
and	O
databases	O
of	O
these	O
Costas	B-Algorithm
arrays	I-Algorithm
are	O
likely	O
near	O
complete	O
,	O
other	O
Costas	B-Algorithm
arrays	I-Algorithm
with	O
orders	O
above	O
29	O
that	O
are	O
not	O
in	O
these	O
lists	O
may	O
exist	O
.	O
</s>
<s>
A	O
Welch	O
–	O
Costas	B-Algorithm
array	I-Algorithm
,	O
or	O
just	O
Welch	O
array	O
,	O
is	O
a	O
Costas	B-Algorithm
array	I-Algorithm
generated	O
using	O
the	O
following	O
method	O
,	O
first	O
discovered	O
by	O
Edgar	O
Gilbert	O
in	O
1965	O
and	O
rediscovered	O
in	O
1982	O
by	O
Lloyd	O
R	O
.	O
Welch	O
.	O
</s>
<s>
The	O
Welch	O
–	O
Costas	B-Algorithm
array	I-Algorithm
is	O
constructed	O
by	O
taking	O
a	O
primitive	O
root	O
g	O
of	O
a	O
prime	O
number	O
p	O
and	O
defining	O
the	O
array	O
A	O
by	O
if	O
,	O
otherwise	O
0	O
.	O
</s>
<s>
The	O
result	O
is	O
a	O
Costas	B-Algorithm
array	I-Algorithm
of	O
size	O
p1	O
.	O
</s>
<s>
Therefore	O
,	O
[	O
3	O
4	O
2	O
1 ]	O
is	O
a	O
Costas	O
permutation	B-Algorithm
.	O
</s>
<s>
The	O
number	O
of	O
Welch	O
–	O
Costas	B-Algorithm
arrays	I-Algorithm
which	O
exist	O
for	O
a	O
given	O
size	O
depends	O
on	O
the	O
totient	O
function	O
.	O
</s>
<s>
The	O
result	O
is	O
a	O
Costas	B-Algorithm
array	I-Algorithm
of	O
size	O
q2	O
.	O
</s>
<s>
If	O
α+β	O
=	O
1	O
then	O
the	O
first	O
row	O
and	O
column	O
may	O
be	O
deleted	O
to	O
form	O
another	O
Costas	B-Algorithm
array	I-Algorithm
of	O
size	O
q3	O
:	O
such	O
a	O
pair	O
of	O
primitive	O
elements	O
exists	O
for	O
every	O
prime	O
power	O
q>2	O
.	O
</s>
<s>
Generation	O
of	O
new	O
Costas	B-Algorithm
arrays	I-Algorithm
by	O
adding	O
or	O
subtracting	O
a	O
row/column	O
or	O
two	O
with	O
a	O
1	O
or	O
a	O
pair	O
of	O
1	O
's	O
in	O
a	O
corner	O
were	O
published	O
in	O
a	O
paper	O
focused	O
on	O
generation	O
methods	O
and	O
in	O
Golomb	O
and	O
Taylor	O
's	O
landmark	O
1984	O
paper	O
.	O
</s>
<s>
More	O
sophisticated	O
methods	O
of	O
generating	O
new	O
Costas	B-Algorithm
arrays	I-Algorithm
by	O
deleting	O
rows	O
and	O
columns	O
of	O
existing	O
Costas	B-Algorithm
arrays	I-Algorithm
that	O
were	O
generated	O
by	O
the	O
Welch	O
,	O
Lempel	O
or	O
Golomb	O
generators	O
were	O
published	O
in	O
1992	O
.	O
</s>
<s>
There	O
is	O
no	O
upper	O
limit	O
on	O
the	O
order	O
for	O
which	O
these	O
generators	O
will	O
produce	O
Costas	B-Algorithm
arrays	I-Algorithm
.	O
</s>
<s>
Two	O
methods	O
that	O
found	O
Costas	B-Algorithm
arrays	I-Algorithm
up	O
to	O
order	O
52	O
using	O
more	O
complicated	O
methods	O
of	O
adding	O
or	O
deleting	O
rows	O
and	O
columns	O
were	O
published	O
in	O
2004	O
and	O
2007	O
.	O
</s>
<s>
Costas	B-Algorithm
arrays	I-Algorithm
on	O
a	O
hexagonal	O
lattice	O
are	O
known	O
as	O
honeycomb	O
arrays	O
.	O
</s>
