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∈
F
p
n
, suppose that the nondegenerate polynomial
g
(
x
) asso-
Proof.
For any
α
f
(
x
+
α
)isoftheform
c
p
ciated with
f
(
x
)
−
−
c
for some
c
∈
F
p
n
. Then there
f
(
x
+
α
)=
h
p
(
x
)
exists
h
(
x
)
∈
F
p
n
[
x
] such that
f
(
x
)
−
−
h
(
x
). It follows that
α
x
(
x
+
α
)
=
h
p
(
x
)
−
h
(
x
)
.
By Lemma 3, it is a contradiction.
Corollary 1.
Let
f
(
x
)=
x
p
n
−
2
.If
p
n
>
4
, then the least period of the sequence
S
=
s
i
}
i
=0
defined by (1) with
A
(
x
)=
Tr
(
f
(
x
))
is
p
n
.
{
Proof.
By Theorem 1 and Lemma 4, the result follows.
4 Autocorrelation
In this section, we investigate the autocorrelation of the sequence
S
defined in
(1). For any 0
τ<p
n
, the autocorrelation of
S
at shift
τ
is defined by
≤
p
n
p
n
−
1
−
1
e
2
πi
(
s
i
+
τ
−s
i
)
/p
=
e
2
πi
(
A
(
ξ
i
+
τ
)
−A
(
ξ
i
))
/p
.
C
S
(
τ
)=
i
=0
i
=0
In order to study
C
S
(
τ
), we need to know the relationship between
ξ
i
and
ξ
i
+
τ
.
For any 0
i, τ < p
n
,let
≤
i
=
i
1
+
i
2
p
+
...
+
i
n
p
n−
1
,
0
≤
i
k
<p, k
=1
,
2
, ..., n,
and
τ
=
τ
1
+
τ
2
p
+
...
+
τ
n
p
n−
1
,
0
≤
τ
k
<p, k
=1
,
2
, ..., n.
Put
ω
1
=0,andfor1
≤
1 define recursively
ω
k
+1
=
1
,
if
i
k
+
τ
k
+
ω
k
≥
k
≤
n
−
p
;
0
,
otherwise.
Then we have
ξ
i
+
τ
=
ξ
i
+
ξ
τ
+
ω,
where
ω
=
k
=2
ω
k
α
k
. There are at most 2
n−
1
choices for
ω
.Forfixed
τ
,we
define the sets
P
ω
=
{
ξ
i
∈
F
p
n
|
ξ
i
+
τ
=
ξ
i
+
ξ
τ
+
ω
}
F
p
n
.Forfixed
ω
=
k
=2
ω
k
α
k
,
for all possible
ω
. Then the sets
P
ω
is a partition of
the set
P
ω
canbewrittenintheform
n−
1
n
P
ω
=
1
,k
=1
,
2
, ..., n
,
(
p
−
(
τ
k
+
ω
k
))
α
k
+
u
k
α
k
|
0
≤
u
k
≤
a
k
−
k
=1
,ω
k
+1
=1
k
=1
