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The 3rd power residue FH sequence set
C
(7, 3) is an optimal average Hamming
correlation family by Theorem 2. It is easy to see that the FH sequence set
C
(7, 3)
has the following parameters:
q
=7,
L
=7,
M
=6,
H
a
=2 and
H
c
=3. By applying
these parameters to (1), it follows that
(
L
−
1)
qH
a
+(
M
−
1)
LqH
c
= 624
>
(
LM
−
q
)
L
= 245
,
hence, the FH sequence set
C
(7, 3) is not an optimal maximum Hamming cor-
relation family
.
For
p
=7,
k
=4, one can design 4th power residue FH sequence set
C
(7, 4) as
shown below:
C
(7, 4)=
e
(
1
)
=(0,1,2,4,4,2,1),
e
(
2
)
=(0,2,4,1,1,4,2),
e
(
3
)
=(0,3,6,5,5,6,3),
e
(
4
)
=(0,4,1,2,2,1,4),
e
(
5
)
=(0,5,3,6,6,3,5),
e
(
6
)
=(0,6,5,3,3,5,6)
{
}
.Theperiodic
Hamming correlations of
C
(7, 4) are given by
⎧
⎨
(7
,
1
,
1
,
1
,
1
,
1
,
1)
,i
=
j,
(1
,
2
,
2
,
2
,
2
,
2
,
2)
,
(
i, j
)=(1
,
2)
,
(1
,
4)
,
(2
,
4)
,
(3
,
5)
,
(3
,
6)
,
(5
,
6)
,
(1
,
0
,
0
,
0
,
0
,
0
,
0)
,
otherwise
.
H
(
e
(
i
)
,e
(
j
)
;
τ
)=
⎩
The 4th power residue FH sequence set
C
(7, 4) is an optimal average Hamming
correlation family by Theorem 2. It is easy to see that the FH sequence set
C
(7,
4) has the following parameters:
q
=7,
L
=7,
M
=6,
H
a
=1 and
H
c
=2. By applying
these parameters to (1), it follows that
(
L
−
1)
qH
a
+(
M
−
1)
LqH
c
= 532
>
(
LM
−
q
)
L
= 245
,
whence, the FH sequence set
C
(7, 4) is not an optimal maximum Hamming
correlation family
.
For
p
=7,
k
=5, one can design 5th power residue FH sequence set
C
(7, 5) as
shown below:
C
(7, 5)=
e
(
1
)
=(0,1,4,5,2,3,6),
e
(
2
)
=(0,2,1,3,4,6,5),
e
(
3
)
=(0,3,5,1,6,2,4),
e
(
4
)
=(0,4,2,6,1,5,3),
e
(
5
)
=(0,5,6,4,3,1,2),
e
(
6
)
=(0,6,3,2,5,4,1)
{
}
.Theperiodic
Hamming correlations of
C
(7, 5) are given by
H
(
e
(
i
)
,e
(
j
)
;
τ
)=
(7
,
0
,
0
,
0
,
0
,
0
,
0)
,i
=
j,
(1
,
1
,
1
,
1
,
1
,
1
,
1)
,
otherwise
.
Since gcd(5,7)=1, the 5th power residue FH sequence set
C
(7, 5) is an optimal
average Hamming correlation family by Theorem 2, and is an optimal maximum
Hamming correlation family by Corollary 1.
5
Concluding Remarks
In this paper, the lower bound on the average Hamming correlation for FH se-
quences with respect to the size
p
of the frequency slot set, the sequence length
L
, the family size
M
, the average Hamming autocorrelation
A
a
and the aver-
age Hamming crosscorrelation
A
c
, is established, and a class of power residue
