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In-Depth Information
3
Construction and Analysis of the Power Residue
Frequency Hopping Sequences
This section gives the definition of the new family of FH sequences and then
establishes the relevant properties.
Definition 2.
Let p be a prime number, k be any positive integer such that
1
≤
k
≤
p-2. A kth power residue FH sequence set C
(
p, k
)
is defined as follows:
⎧
⎨
C
(
p, k
)=
{
c
(
i
)
|
≤
i
≤
p
−
}
,
1
1
c
(
i
)
=(
c
(
i
0
,c
(
i
1
,...,c
(
i
)
p−
1
)
,
⎩
c
(
i
n
=
i
n
k
mod
p
(0
·
≤
n
≤
p
−
1)
.
As for their Hamming correlations, we have the following results.
Theorem 2.
Let p be a prime number, k be any positive integer such that 1
≤
k
p-2, then the kth power residue FH sequence set C
(
p, k
)
has the following
characteristics.
1) M
=
p−1, L
=
q
=
p.
2) The Hamming autocorrelation functions of C
(
p, k
)
are given by
H
(
e
(
i
)
,e
(
i
)
;
τ
)=
p, τ
=0
gcd(
p
≤
(3)
−
1
,k
)
−
1
,
1
≤
τ
≤
p
−
1
The average Hamming autocorrelation of C
(
p, k
)
is
A
a
=gcd(
p
−
1
,k
)
−
1
.
(4)
3) The Hamming crosscorrelation functions of C
(
p, k
)
are given by
H
(
e
(
i
)
,e
(
j
)
;
τ
)
⎧
⎨
1
,τ
=0
0
,τ >
0
and i
(5)
j
−
1
is a kth power nonresidue modulo p
=
·
⎩
j
−
1
is a kth power residue modulo p
gcd(
p
−
1
,k
)
,τ >
0
and i
·
where 1
=
j.
The average Hamming crosscorrelation of C
(
p, k
)
is
≤
i, j
≤
p
−
1, i
2)
p
2
1
,k
)
.
1
A
c
=
−
p
−
1
−
(
p
−
1)gcd(
p
−
(6)
p
(
p
−
4) The C
(
p, k
)
is an optimal average Hamming correlation family.
Proof.
The proof of 1) is straightforward. We prove now 2). For any FH sequence
e
(
i
)
∈
C
(
p
,
k
), its Hamming autocorrelation functions can be written as
n
=0
h
(
e
(
i
n
,e
(
i
)
p−
1
p−
1
n
=0
h
(
e
(
i
)
e
(
i
n
,
0)
H
(
e
(
i
)
,e
(
i
)
;
τ
)=
n
+
τ
)=
−
n
+
τ
=
N
(
i
(
n
+
τ
)
k
in
k
=0)=
N
((
n
+
τ
)
k
n
k
=0)
.
−
−
