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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
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