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Lemma 2 ([8]).
Let
ψ
be a nontrivial additive character of
F
p
n
,andletthe
expression
f
(
x
)
/g
(
x
)
be a rational function over
F
p
n
.Let
v
be the number of
distinct roots of the polynomial
g
(
x
)
in the algebraic closure
F
p
n
of
F
p
n
.Suppose
that
f
(
x
)
/g
(
x
)
is not of the form
h
p
(
x
)
−
h
(
x
)
,where
h
(
x
)
is a rational function
F
p
n
.Then
over
ψ
f
(
ξ
)
g
(
ξ
)
(max(deg(
f
)
,
deg(
g
)) +
v
∗
−
2)
p
n/
2
+
δ,
≤
ξ
∈
F
p
n
,g
(
ξ
)
=0
where
v
∗
=
v
and
δ
=1
if
deg(
f
)
≤
deg(
g
)
,and
v
∗
=
v
+1
and
δ
=0
otherwise.
Niederreiter and Winterhof proved the following result which will be used in the
next section.
Lemma 3 ([9]).
Let
f
(
x
)
/g
(
x
)
be a rational function over
F
p
n
such that
g
(
x
)
is not divisible by the
p
th power of a nonconstant polynomial over
F
p
n
,
f
(
x
)
=0
,
and
deg(
f
)
−
deg(
g
)
≡
0mod
p
or
deg(
f
)
<
deg(
g
)
.Then
f
(
x
)
/g
(
x
)
is not of
the form
h
p
(
x
)
−
h
(
x
)
,where
h
(
x
)
is a rational function over
F
p
n
.
3P od
In this section we will derive some conditions under which the least period of
S
defined by (1) is
p
n
.
For any polynomial
f
(
x
)over
s
i
}
i
=0
given by (1) with
A
(
x
)=
Tr
(
f
(
x
)), we only need to consider the case
f
(
x
)
F
p
n
, in order to investigate the period of
{
−
f
(
x
+
α
n
)where
α
n
is the element in the basis
{
α
1
, ..., α
n
}
.
Theorem 1.
For any polynomial
f
(
x
)
over
F
p
n
,let
g
(
x
)
be the nondegenerate
=
c
p
polynomial associated with
f
(
x
)
−
f
(
x
+
α
n
)
.If
g
(
x
)
−
c
for any
c
∈
F
p
n
,then
s
i
}
i
=0
defined by (1) with
A
(
x
)=
Tr
(
f
(
x
))
the least period of the sequence
S
=
{
is
p
n
.
=
p
n
.Thenwehave
Proof.
Let
T
denote the least period of
S
. Suppose that
T
p
n−
1
because
p
n
is a period of
S
. Hence
s
i
=
s
i
+
p
n
−
1
for any 0
i<p
n
.By
T
|
≤
the definition of
S
,wehave
i<p
n
.
Tr
(
f
(
ξ
i
)) =
Tr
(
f
(
ξ
i
+
p
n
−
1
))
,
for any 0
≤
Hence
i<p
n
.
Tr
(
f
(
ξ
i
)
−
f
(
ξ
i
+
α
n
)) = 0
,
for any 0
≤
∈
F
p
n
such that
g
(
x
)=
c
p
Then there exists
c
−
c
. It is a contradiction. Thus
the least period of
S
is
p
n
.
Lemma 4.
Let
f
(
x
)=
x
p
n
−
2
.If
p
n
>
4
, then for any
α
∈
F
p
n
, the nondegen-
f
(
x
+
α
)
is not of the form
c
p
erate polynomial
g
(
x
)
associated with
f
(
x
)
−
−
c
for any
c
∈
F
p
n
.
