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With the above preparations, the rank distribution of
Π
δ,γ
(
x
)canbedetermined
as follows. Since
S
(0
,
0
,
0) =
p
n
, by Proposition 10 and the values of
S
(
γ,δ,
0)
corresponding to the rank of
Π
γ,δ
(
x
), one has
p
2
(
)+
p
2
+
d
(
)+
p
n
=
|
R
0
,
1
|−|
R
0
,−
1
|
|
R
2
d,
1
|−|
R
2
d,−
1
|
S
(
γ,δ,
0)
,
γ∈
F
p
m
δ∈
F
p
n
1)
p
−
2
p
n
+
d
p
n
+
p
n
+2
d
+
p
2
n
=
S
(
γ,δ,
0)
2
,
|
R
0
|
+(
−
|
R
d
|
|
R
2
d
|
γ∈
F
p
m
δ∈
F
p
n
p
3
2
(
)+
p
3
2
+3
d
(
)+
p
3
n
=
S
(
γ,δ,
0)
3
.
|
R
0
,
1
|−|
R
0
,−
1
|
|
R
2
d,
1
|−|
R
2
d,−
1
|
γ∈
F
p
m
δ∈
F
p
n
|
R
0
,
1
|
|
R
0
,−
1
|
|
R
d
|
|
R
2
d,
1
|
|
R
2
d,−
1
|
This together with the equalities
+
+
+
+
=
p
n
+
m
−
|
R
i
|
|
R
i,
1
|
|
R
i,−
1
|
(
i
=0
,
2
d
)aswellasProposition9give
1,
=
+
p
d
(
p
m
+1)(
p
n
=
(
p
n
+
d
−
2
p
n
+
p
d
)(
p
m
−
1)
−
1)
|
R
0
,
1
|
=
,
|
R
0
,−
1
|
,
2(
p
d
+1)
2(
p
d
−
1)
(14)
(
p
m
−
d
−
1)(
p
n
−
1)
|
R
2
d,
1
|
=0
,
|
R
2
d,−
1
|
=
.
p
2
d
−
1
Therefore, we have the following result.
Proposition 11.
When
(
γ,δ
)
runs through
F
p
m
×
F
p
n
\{
(0
,
0)
}
, the rank dis-
tribution of the quadratic form
Π
γ,δ
(
x
)
is given as follows:
⎧
⎨
⎩
(
p
m
+
n
+2
d
+
p
n
+
p
m
+
d
p
m
+
n
p
m
+
n
+
d
p
2
d
)
/
(
p
2
d
n,
−
−
−
−
1) times
,
p
m−d
(
p
n
n
−
d,
−
1) times
,
2
d,
(
p
m−d
1)(
p
n
1)
/
(
p
2
d
n
−
−
−
−
1) times
.
4 Weight Distribution of the Nonbinary Kasami Codes
This section determines the weight distribution of the nonbinary Kasami codes
C
k
. Further, we also give the distribution of
S
(
γ,δ,
), which will be used to derive
the correlation distribution of the sequence families proposed in next section.
Since the weight of the codeword
c
(
γ,δ,
)isequalto
p
n
−
1
−
(
N
γ,δ,
(0)
−
1) =
p
n
N
γ,δ,
(0), it is sucient to find
N
γ,δ,
(0) for any given
γ,δ,
.
Under the basis
−
i
=1
i
α
i
with
i
∈
F
p
.
n
{
α
1
,α
2
,
···
,α
n
}
of
F
p
n
over
F
p
,let
=
Then,
Tr
1
(
x
)=
Λ
T
CX
where
Λ
T
=(
1
,
2
,
p
and the matrix
C
=
···
,
n
)
∈
F
(
Tr
1
(
α
i
α
j
))
1
≤i,j≤n
, which is nonsingular since
{
α
1
,α
2
,
···
,α
n
}
is a basis of
F
p
n
F
p
.Thenlet
X
=
BY
as in Section 2 and
Λ
T
CB
=(
b
1
,b
2
,
over
···
,b
n
), one has
i
=1
a
i
y
i
+
i
=1
b
i
y
i
.
n
n
Π
γ,δ
(
x
)+
Tr
1
(
x
)=
Y
T
B
T
ABY
+
Λ
T
CBY
=
We calculate
N
γ,δ,
(
ρ
)(
ρ
∈
F
p
)and
S
(
γ,δ,
) as follows:
Case 1.
(
γ,δ
)=(0
,
0): the weight of
c
(
γ,δ,
)is(
p
1)
p
n−
1
for
−
=0,and0
=0,and
p
n
for
=0.
for
=0.Then
S
(0
,
0
,
)=0for
