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where
⎧
⎨
p
−
(
τ
k
+
ω
k
)
,ω
k
+1
=0
,
1
≤
k<n,
a
k
=
τ
k
+
ω
k
,
ω
k
+1
=1
,
1
≤
k<n,
⎩
p,
k
=
n.
Henceforth we denote the additive canonical character
ψ
1
by
ψ
for simplicity.
Lemma 5 ([7]).
With the notation as above, we have
q
(1 + ln
p
)
n−
1
.
ψ
(
ηξ
)
≤
η∈
F
p
n
ξ∈P
ω
∈
F
p
n
,let
g
ξ
(
x
)
be
Theorem 2.
For any polynomial
f
(
x
)
over
F
p
n
,andany
ξ
the nondegenerate polynomial associated with
f
(
x
)
−
f
(
x
+
ξ
)
.If
deg(
g
ξ
)
>
1
∈
F
p
n
, then for any
0
<τ <p
n
, we have the following bound for the
autocorrelation of the sequence
S
=
for any
ξ
{
s
i
}
i
=0
defined by (1) with
A
(
x
)=
Tr
(
f
(
x
))
2(2
n−
1
1)
p
n/
2
(1 + ln
p
)
n−
1
+max
ξ
=0
1)
p
n/
2
,
|
C
S
(
τ
)
|≤
−
1) max
ξ
=0
(deg(
g
ξ
)
−
(deg(
g
ξ
)
−
where
ln(
·
)
is the natural logarithm.
Proof.
For any 0
<τ <p
n
,wehave
p
n
p
n
−
1
−
1
e
2
πi
(
s
i
+
τ
−s
i
)
/p
=
C
S
(
τ
)=
ψ
(
f
(
ξ
i
+
τ
)
−
f
(
ξ
i
))
i
=0
i
=0
=
ξ∈
F
p
n
ψ
(
f
(
ξ
+
ξ
τ
)
−
f
(
ξ
))
+
ω
=0
(
ψ
(
f
(
ξ
+
ξ
τ
+
ω
))
−
ψ
(
f
(
ξ
+
ξ
τ
)))
ψ
(
−
f
(
ξ
))
.
ξ∈P
ω
By Lemma 2,
ξ
=0
(deg(
g
ξ
)
−
1)
p
n/
2
+
ω
=0
|C
S
(
τ
)
|≤
max
ψ
(
f
(
ξ
+
ξ
τ
+
ω
)
− f
(
ξ
))
ξ∈P
ω
+
ω
=0
ψ
(
f
(
ξ
+
ξ
τ
)
−
f
(
ξ
))
.
(2)
ξ∈P
ω
Now we derive an upper bound for
ψ
(
f
(
ξ
+
ξ
τ
+
ω
)
−
f
(
ξ
))
.
ξ
∈
P
ω
