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By Lemma 1,
f
(
β
))
η
1
p
n
ψ
(
f
(
ξ
+
ξ
τ
+
ω
)
−
f
(
ξ
)) =
ψ
(
f
(
β
+
ξ
τ
+
ω
)
−
ψ
(
η
(
ξ
−
β
))
.
β
∈
F
p
n
∈
F
p
n
We get
ψ
(
f
(
ξ
+
ξ
τ
+
ω
)
−
f
(
ξ
))
ξ∈P
ω
f
(
β
))
η∈
F
p
n
1
p
n
=
ψ
(
f
(
β
+
ξ
τ
+
ω
)
−
ψ
(
η
(
ξ
−
β
))
ξ∈P
ω
β∈
F
p
n
1
p
n
≤
ψ
(
f
(
β
+
ξ
τ
+
ω
)
−
f
(
β
)
−
ηβ
)
ψ
(
ηξ
)
.
η∈
F
p
n
β∈
F
p
n
ξ∈P
ω
By Lemmas 2 and 5,
1)
p
n/
2
/p
n
ψ
(
f
(
ξ
+
ξ
τ
+
ω
)
−
f
(
ξ
))
≤
max
ξ
=0
(deg(
g
ξ
)
−
ψ
(
ηξ
)
ξ∈P
ω
η∈
F
p
n
ξ∈P
ω
1)
p
n/
2
(1 + ln
p
)
n−
1
.
≤
max
ξ
=0
(deg(
g
ξ
)
−
Similarly, we have
1)
p
n/
2
(1 + ln
p
)
n−
1
.
ψ
(
f
(
ξ
+
ξ
τ
)
−
f
(
ξ
))
≤
max
ξ
=0
(deg(
g
ξ
)
−
ξ∈P
ω
Finally, by (2),
2(2
n−
1
1)
p
n/
2
(1 + ln
p
)
n−
1
+max
1)
p
n/
2
.
|
C
S
(
τ
)
|≤
−
ξ
=0
(deg(
g
ξ
)
−
ξ
=0
(deg(
g
ξ
)
−
1) max
Remark 1.
The bound in Theorem 2 is applicable only when
p
is large. The basic
idea in the proof above is from [7]. Note that the approach for establishing The-
orem 3 in [1] is also from [7]. Unfortunately, this technique cannot be extended
to the case of
p
= 2, because one needs to consider the number of nonzero
ω
which is 2
n−
1
.
Theorem 3.
For any
0
<τ <p
n
, we have the following bound for the autocor-
relation of the sequence
S
=
s
i
}
i
=0
defined by (1) with
A
(
x
)=
Tr
(
x
p
n
−
2
)
{
<
4(2
n−
1
p
n/
2
+1)(1+ln
p
)
n−
1
+2
p
n/
2
+3
.
|
C
S
(
τ
)
|
−
1)(2
·
·
