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1
(mod 3) can be proved similarly (essentially, the roles of
P
(
z
2
k
)and
Q
(
z
2
k
)are
exchanged). If
k
In what follows, we treat the case
k
≡
0 (mod 3), the case (
m
−
k
)
≡
≡
0(mod3),wehave2
k
−
1
≡
2 (mod 3) and from Lemma 5
P
(
z
2
k
)=2
kn
ω
c
δ
ψ
(
z
2
k
)+
a
(3)
for some
c
∈
Z
8
and
a
∈
V
n
.Now
k
≡
0 (mod 3) implies
m
−
k
≡
m
(mod 3),
so 2(
m
0 (mod 3) or 1 (mod 3). Hence by Lemma 5
Q
(
z
2
k
)=
2
(
m−k
)
n
ω
d
(
−
k
)
≡
1)
b·φ
(
z
2
k
)
−
if
m
≡
0(mod3)
(4)
2
(
m−k
)
n
ω
d
(
1)
b·φ
(
z
2
k
)
i
−
wt(
ψ
(
z
2
k
))
−
if
m
≡
2(mod3)
for some
d ∈
Z
8
and
b ∈ V
n
. Combining (2), (3), and (4), we arrive at
N
(
f
)(
u
1
,...,u
2
m
)
⎧
⎨
ω
c
+
d
1)
h
(
z
2
k
)+
z
2
k
·u
2
k
+
b·φ
(
z
2
k
)
i
wt(
z
2
k
)
δ
ψ
(
z
2
k
)+
a
(
−
if
m
≡
0(mod3)
z
2
k
∈V
n
=
⎩
ω
c
+
d
1)
h
(
z
2
k
)+
z
2
k
·u
2
k
+
b·φ
(
z
2
k
)
δ
ψ
(
z
2
k
)+
a
−
if
m
≡
2(mod3
.
(
z
2
k
∈
V
n
In either case the term inside the sum is zero unless
z
2
k
=
ψ
−
1
(
a
). Therefore,
|N
(
f
)(
u
1
,...,u
2
m
)
|
= 1, as was claimed.
Example 6.
Take
m
=2and
k
=1inTheorem4.Then
f
reads
f
(
x
1
,x
2
,y
1
,y
2
)=
y
1
·
ψ
(
x
1
)+
φ
(
x
1
)
·
y
2
+
y
2
·
x
2
+
h
(
x
1
)
.
In this way we can construct bent-negabent functions in 4
n
variables of degree
ranging from 2 to
n
.
In general, whenever
m
1 (mod3),wecanuseTheorem 4 to construct bent-
negabent functions in 2
mn
variables of degree ranging from 2 to
n
. This yields
bent-negabent functions in 2
t
variables for every
t
≡
≥
2and
t
≡
1(mod6);if
t
≡
1(mod3),wecantake
n
=1and
m
=
t
,andif
t
≡
1(mod3)and
t
≡
1
(mod 6), we can take
n
=2and
m
=
t/
2.
In the remainder of this section we apply Theorem 3 to construct further
bent-negabent functions by taking a partial dual of
f
givenin(1).Wetherefore
have to prove that the partial dual of
f
exists with respect to certain subspaces
of
V
and to find an explicit expression for this function.
Write
V
=
U
⊕
W
, where dim
W
=
k
and
k
≤
mn
. Suppose that we have a
function
τ
:
V
→
V
.Wecanseparate
τ
on
U
and
W
by defining
|
W
|
functions
τ
z
:
U
→
U
and
|
U
|
functions
τ
x
:
W
→
W
such that
τ
(
x, z
)=(
τ
z
(
x
)
,τ
x
(
z
))
,
x
∈
U, z
∈
W.
