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few examples that did not fit into the known families. Except for a single case
with
m
=8and
d
= 7, all decimations leading to at most four-valued cross
correlation are such that
m
=2
k
and
d
(2
l
+1)
2
i
(mod 2
k
≡
−
1) for some
integer
l
with 0
≤
l<k
and
i
≥
0. Note that the Kasami case satisfies this
general condition on
d
taking
l
= 0, three-valued cases from [4] are achieved
when gcd(
l, k
) = 1, and the four-valued pairs found in [5] hold with
k
=3
t
and
l
=
t
. The main result of this paper is finding the cross-correlation distribution
for the above decimations. Moreover, for any given
τ
0
,
1
,...,
2
k
we
exactly compute the corresponding cross-correlation value (see Corollary 3). We
also conjecture that we have a characterization of all decimations leading to at
most four-valued cross correlation of
m
-sequences with the described parameters
except for a single case.
In the remaining part of this section, we present preliminaries needed to prove
our main results. In Sect. 2, we give the distribution of the number of zeros of
a particular ane polynomial
A
a
(
x
) and find the zeros. This information is
useful when obtaining the cross-correlation values and their distribution. Sect. 3
determines the cross-correlation distribution of our family.
Let GF(
q
) denote a finite field with
q
elements and let GF(
q
)
∗
=GF(
q
)
∈{
−
2
}
.
The finite field GF(
q
l
) is a subfield of GF(
q
m
) if and only if
l
divides
m
.The
trace and norm mappings from GF(
q
m
)toGF(
q
l
) are defined respectively by
\{
0
}
m/l−
1
m/l−
1
x
q
li
x
q
li
Tr
l
N
l
(
x
)=
and
(
x
)=
.
i
=0
i
=0
In the case when
l
=1,weusethenotationTr
m
(
x
) instead of Tr
1
(
x
).
Let
m
=2
k
and
α
be an element of order
p
=2
m
1inGF(2
m
). Then
−
the
m
-sequence
{
s
t
}
of length
p
=2
m
−
1 can be written in terms of the trace
mapping as
s
t
=Tr
m
(
α
t
)
.
Let
β
=
α
2
k
+1
be an element of order 2
k
−
1. The sequence
{
u
t
}
of length
2
k
1 (which is used in the construction of the well known Kasami family) is
obtained as
−
u
t
=Tr
k
(
β
t
)
.
In this paper, we consider the cross correlation between the
m
-sequences
{
s
t
}
and
{
v
t
}
=
{
u
dt
}
at shift
τ
defined by
p−
1
1)
s
t
+
v
t
+
τ
C
d
(
τ
)=
(
−
,
t
=0
where gcd(
d,
2
k
1) = 1 and
τ
=0
,
1
,...,
2
k
−
−
2.
Take any
a
∈
GF(2
k
)
∗
and let
1)
Tr
m
(
ax
)+Tr
k
(
x
d
(2
k
+1)
)
S
(
a
)=
−
.
(
(1)
x∈
GF(2
m
)
The following moment identities are known to hold for
S
(
a
).
