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Corollary 1.
Let p be a prime number, k be any positive integer such that
1
1,k
)
=1, then the kth power residue FH sequence set
C
(
p, k
)
has the following characteristics.
1) The Hamming autocorrelation functions of C
(
p, k
)
are given by
H
(
e
(
i
)
,e
(
i
)
;
τ
)=
p, τ
=0
0
,
1
≤
k
≤
p-2 and gcd(p
−
≤
τ
≤
p
−
1
The average Hamming autocorrelation of C
(
p, k
)
is
A
a
=0
.
2) The Hamming crosscorrelation functions of C
(
p, k
)
are given by
H
(
e
(
i
)
,e
(
j
)
;
τ
)=1
.
(7)
The average Hamming crosscorrelation of C
(
p, k
)
is
A
c
=1
.
(8)
3) C
(
p, k
)
is an optimal maximum Hamming correlation family.
Proof.
The proof of 1) is straightforward.
We prove now 2). For any fixed 1
j
−
1
≤
j
≤
p
−
1, it is noted that
i
·
runs
independently over all elements in
{
1,2,. . . ,
p
−
1
}
as
i
takes independently all
j
−
1
takes
k
th power residues
elements in
{
1,2,. . . ,
p
−
1
}
.Sincegcd(
p
−
1,
k
)=1,
i
·
modulo
p
altogether (
p
−
1)/gcd(
p
−
1,
k
)=
p
−
1 times from number theory [14].
j
−
1
are
k
th power residues
modulo
p
. Hence, (7) follows immediately from (5), and (8) follows immediately
from (7).
It is easy to see that the FH sequence set
C
(
p
,
k
) has the following parameters:
q
=
p
,
L
=
p
,
M
=
p
That is, for all
i
and
j
with 1
≤
i
,
j
≤
p
−
1,
i
·
−
1,
H
a
=0 and
H
c
=1. By applying these parameters to (1),
it follows that
2)
p
2
2)
p
2
,
(
L
−
1)
qH
a
+(
M
−
1)
LqH
c
=(
p
−
≥
(
LM
−
q
)
L
=(
p
−
hence, the FH sequence set
C
(
p
,
k
) is an optimal maximum Hamming correlation
family
.
This completes the proof.
4
Illustrative Examples
For
p
=7,
k
=3, one can design 3rd power residue FH sequence set
C
(7, 3) as
shown below:
C
(7, 3)=
e
(
1
)
=(0,1,1,6,1,6,6),
e
(
2
)
=(0,2,2,5,2,5,5),
e
(
3
)
=(0,3,3,4,3,4,4),
e
(
4
)
=(0,4,4,3,4,3,3),
e
(
5
)
=(0,5,5,2,5,2,2),
e
(
6
)
=(0,6,6,1,6,1,1)
{
}
.Theperiodic
Hamming correlations of
C
(7, 3) are given by
⎧
⎨
(7
,
2
,
2
,
2
,
2
,
2
,
2)
,i
=
j,
(1
,
3
,
3
,
3
,
3
,
3
,
3)
,
(
i, j
)=(1
,
6)
,
(2
,
5)
,
(3
,
4)
,
(1
,
0
,
0
,
0
,
0
,
0
,
0)
,
otherwise
.
H
(
e
(
i
)
,e
(
j
)
;
τ
)=
⎩
