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A Class of Nonbinary Codes and Sequence
Families
Xiangyong Zeng
1
,
,NianLi
1
,andLeiHu
2
1
The Faculty of Mathematics and Computer Science, Hubei University,
Xueyuan Road 11, Wuhan 430062, P.R. China
xzeng@hubu.edu.cn
2
The State Key Laboratory of Information Security,
Graduate School of Chinese Academy of Sciences,
Beijing 10049, P.R. China
hu@is.ac.cn
Abstract.
Inthispaper,foraneveninteger
n ≥
4 and any positive
integer
k
with gcd(
n/
2
,k
)=gcd(
n/
2
− k,
2
k
)=
d
beingodd,aclass
of
p
-ary codes
C
k
is defined and the weight distribution is completely
determined, where
p
is an odd prime. A class of nonbinary sequence
families is constructed from these codes, and the correlation distribution
is also determined.
Keywords:
Linear code, weight distribution, exponential sum, quadratic
form, correlation distribution.
1
Introduction
Nonlinear functions have important applications in coding theory and cryptog-
raphy [7,17]. Linear codes constructed from functions with high nonlinearity
[2,6,11] can be good and have useful applications in communications [9,10,19] or
cryptography [3,4,5,24].
Throughout this paper, let
F
q
be the finite field with
q
=
p
n
elements for
F
q
=
aprime
p
and a positive integer
n
,andlet
F
q
\{
0
}
. For an even integer
k
denote the [
p
n
k
n
≥
4, let
C
−
1
,
5
n/
2] cyclic code given by
C
=
{
c
(
γ,δ,
)=
Π
γ,δ
(
x
)+
Tr
1
(
x
)
x∈
F
p
n
|
γ
∈
F
p
n/
2
,δ,
∈
F
p
n
}
constructed from the function
Π
γ,δ
(
x
)=
Tr
n/
2
(
γx
p
n/
2
+1
)+
Tr
1
(
δx
p
k
+1
)
,
(1)
1
=
n/
2, and
Tr
1
(
where 1
≤
k<n
with
k
·
) is the trace function from
F
p
l
to
F
p
.
k
have been extensively studied for various
values of the parameter
k
. The binary code
Several classes of binary codes
C
n/
2
±
1
C
is exactly the Kasami code
