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If
s
is also odd, then
3
t
+1
2
3
rk
2
+
3
s
+1
2
=
,
and, 2
t
+2
(3
t
+1)
/
2
2
s
+2
(3
s
+1)
/
2
1(mod2
k
−
1
≡
−
−
1).
If
s
is even, then
3
t
+1
2
3
rk
+3
s
+1
2
3
rk
+
k
+
s
2
=
(3
r
+1)
k
+
s
2
=
=
(3
r
+1)
k
2
+
s
=
2
,
and consequently 2
t
+2
(3
t
+1)
/
2
2
s
+2
s/
2
1(mod2
k
−
1
≡
−
−
1)
.
As a corollary we have:
Corollary 1.
Let
F
=
F
2
m
and
d ∈
Gold
∪
Kasami
∪
Welch
∪
Niho
,then
Θ
d
(1)
satisfies (5).
3 Decomposition of Gold Exponent
In this section we let
m
=
kk
,
K
=
F
2
k
,
d
be an AB exponent such that it
x
d
,and
f
:
K
x
d
.
Rewriting Corollary 1, we have
W
f
(1) =
W
f
(1) 2
m
−
2
k
.
Actually if
d
is a
Gold exponent, we can generalize the above equality. Since, as mentioned before,
the Gold spectrum is completely known, the following theorem is not new. But
it emphasizes an interesting property of Gold exponents, which is not shared
by other exponents, even by Kasami exponents; and our proof has a corollary,
which is also a generalization of a result in [1].
Theorem 4.
Let
d
=2
e
+1
is AB on all subfields,
f
:
F
2
m
→
F
2
m
,x
→
→
K, x
→
∈
Gold
and
K
⊥
=
{
x
∈
F
:
Tr
(
xy
)=0
,
∀
y
∈
K
}
.
K
⊥
and
a
If
γ
∈
∈
K
,then:
W
f
(
α
+
γ
)=
W
f
(
α
)2
m
−
2
δ
(
γ
)
,
(6)
and, in particular,
2
k
,
W
f
(
α
)=
W
f
(
α
)2
m
−
k
(7)
2
where
δ
(
γ
):=
sign
W
f
(1+
γ
)
W
f
(1)
.
Proof.
We have
W
f
(1) =
x∈
F
1)
Tr
(
x
2
e
+1
+
x
)
=
1)
Tr
(
(
u
+
v
)
2
e
+1
+
u
+
v
)
(
−
(
−
u∈K,v∈K
⊥
1)
Tr
(
u
2
e
+1
+
u
2
e
v
+
v
2
e
u
+
v
2
e
+1
+
u
+
v
)
=
(
−
u∈K,v∈K
⊥
=
u∈K
1)
Tr
(
u
d
+
u
)
1)
Tr
(
v
d
)
,
(
−
(
−
v∈K
⊥
