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have simultaneously good Hamming autocorrelation functions, small Hamming
crosscorrelation functions and large family size is therefore well motivated.
Frequency hopping sequence design normally involves the following parame-
ters: the size
q
of the frequency slot set
F
, the sequence length
L
, the family
size
M
, the maximum Hamming autocorrelation sidelobe
H
a
,themaximum
Hamming crosscorrelation
H
c
, the average Hamming autocorrelation
A
a
,and
the average Hamming crosscorrelation
A
c
. In applications, it is generally desired
that the family of FH sequences has the following properties [1,3,4].
1)
.
The maximum Hamming autocorrelation
H
a
should be as small as
possible;
2)
.
The maximum Hamming crosscorrelation
H
c
should be as small
as possible;
3)
.
The average Hamming autocorrelation
A
a
should be as small as possible;
4)
.
The average Hamming crosscorrelation
A
c
should be as small as possible;
5)
.
The family size
M
for given
H
a
,
H
c
,
A
a
,
A
c
,
q
and
L
should be as large
as possible.
However, these parameters
q
,
L
,
M
,
H
a
,
H
c
,
A
a
and
A
c
are not independent, and
are bounded by certain theoretical limits. In order to evaluate the theoretical
performance of the FH sequences, it is important to find the tight theoretical lim-
its which set bounded relation among these parameters. Early in 1974, Lempel
and Greenberger [4] established some bounds on the periodic Hamming corre-
lation of FH sequences for
M
=1 or 2. In 2004, Peng and Fan [5] obtained the
following theoretical limit which sets bounded relation among the parameters
q
,
L
,
M
,
H
a
and
H
c
.
(
L
−
1)
qH
a
+(
M
−
1)
LqH
c
≥
(
LM
−
q
)
L.
(1)
If
H
a
and
H
c
are a pair of the minimum integer solutions of inequality (1),
then the corresponding FH sequence set
S
is called optimal maximum Hamming
correlation family.
As far as the authors aware, the theoretical limits haven't been obtained yet
which set bounded relation among the parameters
q
,
L
,
M
,
A
a
and
A
c
.
There are a number of hopping sequences derived from the polynomials over fi-
nite fields, such as linear hopping sequences [6], general linear hopping sequences
[7], quadratic hopping sequences [8], extended quadratic hopping sequences [9],
cubic sequences [10], cubic + linear congruence sequences [11], extended cubic
hopping sequences [12], general cubic hopping sequences[3], polynomial hopping
sequences [12], etc.
In this paper, we will pay particular attention to the bounds on the average
Hamming correlation for FH sequences, and the construction and the character-
istics on the Hamming correlation for the power residue FH sequences.
The rest of the paper is organized as follows. Section 2 gives the bounds
on the average Hamming correlation for FH sequences; Section 3 presents the
construction of the family of power residue FH sequences and establishes the