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order. Then a sequence generated by
A
(
x
) via the additive order may not have
good pseudorandom properties.
In this paper, we study the least period and the autocorrelation of the sequence
S
defined by (1). Some conditions are given under which they have the maximum
period. For the autocorrelation, we present some nontrivial upper bounds when
p
is large. When
p
=2and
A
(
x
)=
Tr
(
x
2
n
−
2
)or
A
(
x
)=
Tr
(
x
2
n
−
4
), where
Tr
(
·
)
is the trace function from
F
2
, some numerical results are given. The data
show that
S
has good autocorrelation in these two cases. One conjecture on the
upper bound of the autocorrelation in these cases is also given.
This paper is organized as follows. In Section 2, we present some background
which will be used in this paper. In Section 3, we study the least period of
the sequence
S
defined by (1). In Section 4, we present some results on the
autocorrelation of the sequence
S
defined by (1). Finally, Section 5 concludes
this paper.
F
2
n
to
2 Preliminaries
Definition 1.
For any polynomial
f
(
x
)=
i
=0
f
i
x
i
∈
F
p
n
[
x
]
,if
f
i
=0
for any
p
|
i
,then
f
(
x
)
is called nondegenerate.
For any polynomial
f
(
x
)
∈
F
p
n
[
x
], we can find a nondegenerate polynomial
g
(
x
)
∈
F
p
n
[
x
] such that
Tr
(
f
(
ξ
)) =
Tr
(
g
(
ξ
))
,ξ
∈
F
p
n
.
Such
g
(
x
) is called the nondegenerate polynomial associated with
f
(
x
).
Let
Γ
(
n
) denote the set of cosetleaders of
p
n
−
1 with respect to
p
. For any
k
Γ
(
n
), we denote the coset containing
k
by
C
k
. The the function
A
(
x
)in(1)
can be written as (see [3])
A
(
x
)=
k∈Γ
(
n
)
∈
Tr
n
1
(
A
k
x
k
)+
A
p
n
−
1
x
p
n
−
1
,
where
Tr
n
1
(
·
) is the trace function from
F
p
n
k
to
F
p
,
n
k
is the cardinality of
C
k
,
A
k
∈
F
p
n
k
,and
A
p
n
−
1
∈
F
p
.
For any
a
∈
F
p
n
,themap
ψ
a
(
x
)=
e
2
πiT r
(
ax
)
/p
is an additive character of
F
p
n
,
F
p
n
canbewritteninthisform[4].If
a
and all the additive characters of
=0,
then
ψ
a
is called a nontrivial character.
Lemma 1 ([4]).
Let
ψ
be a nontrivial additive character of
F
p
n
. Then for any
a ∈
F
p
n
, we have
ψ
(
aμ
)=
p
n
,
if
a
=0;
0
,
if
a
=0
.
μ∈
F
p
n
The following bound for exponential sums generalizes that for the well known
Weil exponential sums.
