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as the overall number of Hamming autocorrelation (auto-hits) and Hamming
crosscorrelation (cross-hits) of S respectively, and call
S
a
(
S
)
M
(
L
A
a
(
S
)=
1)
,
−
2
S
c
(
S
)
LM
(
M
A
c
(
S
)=
1)
as the average Hamming autocorrelation (average of auto-hits) and the average
Hamming crosscorrelation (average of cross-hits) of S respectively.
−
Obviously, the average Hamming correlation (average of hits) indicates the aver-
age error (or interference) performance of the hopping systems. However, gener-
ally speaking, it is very dicult to derive the average Hamming correlation for a
given hopping sequence. For simplicity, we will denote
S
a
=
S
a
(
S
),
S
c
=
S
c
(
S
),
A
a
=
A
a
(
S
)and
A
c
=
A
c
(
S
).
We now give the theoretical limits which set a bounded relation among the
parameters
q
,
L
,
M
,
A
a
and
A
c
.
Theorem 1.
Let S be a set of M hopping sequences of length L over a given fre-
quency slot set F with size q, A
a
and A
c
be the average Hamming autocorrelation
and the average Hamming crosscorrelation of S respectively, then
A
a
L
(
M
A
c
(
L
LM
−
q
1)
+
1)
≥
1)
.
(2)
−
−
q
(
L
−
1)(
M
−
Proof.
We first have
H
(
x,y
;
τ
)=
x∈S
H
(
x,x
;0) +
H
(
x,y
;
τ
)+
H
(
x,y
;
τ
)
x,y∈S,τ
x∈S,τ
=0
x
=
y∈S,τ
=
LM
+
S
a
+2
S
c
.
In [5], a lower bound
L
2
M
2
/
q
on
x,y∈S,
0
≤τ≤L−
1
H
(
x,y
;
τ
) was given.
Therefore,
L
2
M
2
q
LM
+
S
a
+2
S
c
≥
,
and
1
S
a
2
S
c
LM
1)
≥
1)
,
1)
+
1)
+
(
L
−
1)(
M
−
LM
(
L
−
1)(
M
−
LM
(
L
−
1)(
M
−
q
(
L
−
1)(
M
−
1
A
a
L
(
M
A
c
(
L
LM
1)
+
1)
+
1)
≥
1)
,
(
L
−
1)(
M
−
−
−
q
(
L
−
1)(
M
−
and (2) follows immediately. This completes the proof.
If the parameters
q
,
L
,
M
,
A
a
,and
A
c
of the FH sequence set
S
satisfy inequality
(2) with equality, then it is said that sequence set
S
is an optimal average
Hamming correlation family.