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p
. A function
f
(
x
)on
to the
F
p
n
is a
quadratic form
if it can
be written as a homogeneous polynomial of degree 2 on
F
p
-vector space
F
p
,namelyoftheform
F
f
(
x
1
,
···
,x
n
)=
a
ij
x
i
x
j
,a
ij
∈
F
p
.
The
rank
of
f
(
x
) is defined as the
1
≤
i
≤
j
≤
n
F
p
-vector space
V
f
=
z
codimension of the
∈
F
p
n
|
f
(
x
+
z
)=
f
(
x
) for all
x
∈
F
p
n
,
denoted by rank(
f
). Then
=
p
n−
rank(
f
)
.
For the quadratic form
f
(
x
), there exists a symmetric matrix
A
such that
f
(
x
)=
X
T
AX,
where
X
T
=(
x
1
,x
2
,
|
V
f
|
p
denotes the transpose of
a column vector
X
.The
determinant
det(
f
)of
f
(
x
) is defined to be the de-
terminant of
A
,and
f
(
x
)is
nondegenerate
if det(
f
)
···
,x
n
)
∈
F
= 0. By Theorem 6.21 of
[14], there exists a nonsingular matrix
B
such that
B
T
AB
is a diagonal matrix.
Making a nonsingular substitution
X
=
BY
with
Y
T
=(
y
1
,y
2
, ···,y
n
), one has
n
f
(
x
)=
Y
T
B
T
ABY
=
a
i
y
i
,a
i
∈
F
p
.
(3)
i
=1
The
quadratic character
of
F
q
is defined by
η
(
x
)=1if
x
is a square element in
F
q
,-1if
x
is a non-square element in
F
q
,and0if
x
=0.
Lemma 1 (Theorems 6.26 and 6.27 of [14]).
For odd
q
,let
f
be a nonde-
generate quadratic form over
F
q
in
l
indeterminates, and a function
υ
(
x
)
over
F
q
∈
F
q
. Then for
ρ
is defined by
υ
(0) =
q
−
1
and
υ
(
x
)=
−
1
for
x
∈
F
q
the number
,x
l
)=
ρ
is
q
l−
1
+
q
l
−
2
η
(
det(
f
)
1)
l
−
2
ρ
of solutions to the equation
f
(
x
1
,
···
−
·
for odd
l
,and
q
l−
1
+
υ
(
ρ
)
q
l
−
2
η
(
1)
2
det(
f
)
for even
l
.
−
Lemma 2 (Theorem 5.15 of [14]).
Let
ω
be a complex primitive
p
-th root of
η
(
k
)
ω
k
=
(
p−
1
k
=1
1)
p
−
2
p
.
unity. Then
−
p
with dimension
l
.The
A
p
-ary
[
m, l
]
linear code
C
is a linear subspace of
F
Hamming weight
of a codeword
c
1
c
2
···
c
m
of
C
is the number of nonzero
c
i
for
1
≤
i
≤
m
.
=
1
be a family of
Mp
-ary sequences
p
n
−
2
t
=0
Let
F
{
s
i
(
t
)
}
|
0
≤
i
≤
M
−
of period
p
n
−
1. The
periodic correlation function
of the sequences
{
s
i
(
t
)
}
and
p
n
t
=0
−
2
ω
s
i
(
t
)
−s
j
(
t
+
τ
)
,0
p
n
{
s
j
(
t
)
}
in
F
is
C
i,j
(
τ
)=
≤
τ
≤
−
2, and
{
s
i
(
t
)
}
and
<p
n
{
s
j
(
t
)
}
are
cyclicly inequivalent
if
|
C
i,j
(
τ
)
|
−
1 for any
τ
.The
maximum
magnitude
of the correlation values is
C
max
=max
{|
C
i,j
(
τ
)
|
:
i
=
j
or
τ
=0
}
.
From now on, we always assume that the prime
p
is odd, and
n
=2
m
≥
4.
k
Without loss of generality, the integer
k
in the definition of code
C
can be
assumed to satisfy 1
≤
k<n/
2. For an odd integer
t
relatively prime to
m
,the
integer
k
=
m
t
satisfies Equation (2), and
d
=1.Inparticular,for
t
=1,
gcd(
m, k
)=gcd(
m
−
1 corresponds to the
binary Kasami code [9], and for this reason, we call these
p
-ary [
p
n
−
k,
2
k
) = 1. The parameter
k
=
m
−
−
1
,
5
m
] linear
k
with
k
satisfying Equation (2)
the nonbinary Kasami codes
.
The main result of this paper is stated as the following theorem.
codes
C
