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For any 1
≤
1, we have from number theory [14]
H
(
e
(
i
)
,e
(
i
)
;
τ
)=
N
((1 +
τ
τ
≤
p
−
n
−
1
)
k
=1)=gcd(
p
·
−
1
,k
)
−
1
,
hence (3) follows. Then (4) follows immediately from (3).
We prove now part 3). For any two
k
th power residue FH sequences
e
(
i
)
,
e
(
j
)
∈
C
(
p
,
k
), their Hamming crosscorrelation functions can be written as
n
=0
h
(
c
(
i
n
,e
(
j
)
n
=0
h
(
e
(
j
)
p−
1
p−
1
e
(
i
n
,
0)
H
(
e
(
i
)
,e
(
j
)
;
τ
)=
n
+
τ
)=
−
n
+
τ
=
N
(
j
(
n
+
τ
)
k
n
k
=0)=
N
(
j
(
n
+
τ
)
k
=
i
n
k
)
.
−
i
·
·
For
τ
=0, since
i
=
j
,weobtain
H
(
e
(
i
)
,e
(
j
)
;0)=
N
(
j
n
k
=
i
n
k
)=1
.
·
·
For any 1
≤
τ
≤
p
−
1, we have from number theory
H
(
e
(
i
)
,e
(
j
)
;
τ
)=
N
((1 +
τ
n
−
1
)
k
=
i
j
−
1
)
·
·
=
gcd(
p
j
−
1
is a
k
th power residue modulo
p
−
1
,k
)
,i
·
j
−
1
is a
k
th power nonresidue modulo
p
0
,i
·
Hence (5) follows.
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 elements in
{
1,2,. . . ,
j
−
1
p
−
1
}
. Therefore,
i
·
takes
k
th power residues modulo
p
(
p
−
1)/gcd(
p
−
1,
k
)times,
k
th power nonresidues modulo
pp
−
1-(
p
−
1)/gcd(
p
−
1,
k
)timesfrom
number theory. Therefore
1
2
H
(
e
(
i
)
,e
(
j
)
;
τ
)
S
c
=
1
≤i,j≤p−
1
,
0
≤τ≤p−
1
,i
=
j
(
p
−
1)(
p
−
2)
+
2
=
H
(
e
(
i
)
,e
(
j
)
;
τ
)
2
1
≤
i,j
≤
p
−
1
,
1
≤
τ
≤
p
−
1
,i
p−
1
gcd(
p
=
j
1
(
p−
1)(
p−
2)
2
+
2
=
1
,k
)
−
·
gcd(
p
−
1
,k
)
−
p
2
1
≤j≤p−
1
,
1
≤τ≤p−
1
1
,k
)
.
p−
1
2
=
−
p
−
1
−
(
p
−
1)gcd(
p
−
The average Hamming crosscorrelation of
C
(
p
,
k
)is
2
S
c
ML
(
M
A
c
=
−
1)
p
2
1
,k
)
(
p−
1)
p
(
p−
2)
.
p−
1
2
=
−
p
−
1
−
(
p
−
1)gcd(
p
−
p
(
p−
2)
p
2
2
1
,k
)
.
1
=
−
p
−
1
−
(
p
−
1)gcd(
p
−
Thus (6) is true. By applying 1), (4) and (5) to (2), it follows that
A
a
L
(
M
A
c
(
L
1)
+
−
−
1)
p
(
p−
2)
p
2
1
,k
)
1
1
(
p−
1)
1
=
p
(
p−
2)
[gcd(
p
−
1
,k
)
−
1] +
−
p
−
1
−
(
p
−
1)gcd(
p
−
LM−q
p
(
p−
1)
−p
p
(
p
1
1
=
≥
=
=
1)
.
(
p
−
1)
q
(
L
−
1)(
M
−
1)
−
1)(
p
−
2)
(
p
−
Thus, the FH sequence set
C
(
p
,
k
) is an optimal average Hamming correlation
set
.
This completes the proof.