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in Theorem 14 of [9]. By choosing cyclicly inequivalent codewords from the
Kasami code, the large set of binary Kasami sequences was obtained in [19]. The
minimum distance bound of
1
C
was established by evaluating the exponential
1)
Π
γ,δ
(
x
)+
Tr
1
(
x
)
in [12] and [16], and the weight distribution and
then the minimum distance were completely determined in [22]. For even
n/
2,
the binary code
(
−
sums
x
∈
F
2
n
1
has the same weight distribution as the Kasami code [9,22].
Furthermore, for any
k
with gcd(
k, n
)=2if
n/
2isoddor1if
n/
2iseven,the
weight distribution of binary codes
C
k
was also determined, and these codes were
used to construct families of generalized Kasami sequences, which have the same
correlation distribution and family size as the large set of Kasami sequences [26].
This paper discusses the code
C
k
in the nonbinary case, namely we assume
p
is odd, for a wide range of
k
that satisfies
C
gcd(
n/
2
,k
)=gcd(
n/
2
−
k,
2
k
)=
d
being odd
.
(2)
Applying the techniques developed in [1], we describe some properties of the
roots to the equation
δ
p
n
−
k
y
p
n/
2
−
k
+1
+
γy
+
δ
=0with
γδ
= 0. Based on these
properties and the theory of quadratic form over finite fields of odd characteristic,
we completely determine the weight distribution. These codes are also used to
construct a class of nonbinary sequence families
k
. The correlation function
F
k
of sequences in
F
takes 5
p
+ 2 values. Let
k
=
n/
2
−
t
for any odd integer
k
have the maximum
magnitude
p
2
+1
+ 1. Some families of
p
-ary sequences of period
p
n
t
relatively prime to
n/
2, then
d
= 1 and the families
F
1 are listed
in Table 1. The maximum magnitude of the proposed family deviates from the
optimal correlation value [23], however, it has a large family size.
−
Table 1.
Families of
p
-ary sequences for odd prime
p
Family
n
Period Family size Maximum magnitude
2
2
+1
p
n
−
1
Kumar, Scholtz, Welch [11]
even
p
p
2
2
+1
p
n
−
1
Liu, Komo[13]
even
p
p
2
2
+1
p
n
−
1
Moriuchi, Imamura [15]
even
p
p
2
+1
p
n
−
1
p
n
Sidelnikov [18]
even or odd
p
2
+1
p
n
−
1
p
n
+1
Kumar, Moreno [10]
(2
m
+1)
k
p
n
+1
2
p
n
−
1
p
n
+1
p
Trachtenb erg [20]
odd
+1
n
+
k
2
p
n
−
1
p
n
+1
Tang, Udaya, Fan [21]
(2
m
+1)
k
p
+1
3
2
2
+1
+1
p
n
−
1
The proposed family
even
p
p
2 Preliminaries and Main Result
The field
F
q
is an
n
-dimensional vector space over
F
p
. For any given basis
{
α
1
,α
2
,
···
,α
n
}
of
F
q
over
F
p
,eachelement
x
∈
F
q
can be uniquely represented
i
=1
n
as
x
=
x
i
α
i
with
x
i
∈
F
p
. Under this representation, the field
F
q
is identical
