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5 A Class of Nonbinary Sequence Families
k
, a family of nonbinary
By choosing cyclicly inequivalent codewords from
C
sequences is defined by
k
=
∈
F
p
n
,
s
a,b
(
α
t
)
F
{
}
0
≤t≤p
n
−
2
|
a
∈
F
p
m
,b
(19)
where
s
a,b
(
α
t
)=
Tr
1
(
aα
(
p
m
+1)
t
)+
Tr
1
(
bα
(
p
k
+1)
t
+
α
t
), and
α
is a primitive
element of
m
+1
were discussed in [25],
F
p
n
. The possible correlation values of
F
but the correlation distribution remains unsolved.
It is easy to verify that for two sequences
s
a
1
,b
1
and
s
a
2
,b
2
,
C
a
1
b
1
,a
2
b
2
(
τ
)=
a
2
α
(
p
m
+1)
τ
,
λ
2
=
b
1
−
b
2
α
(
p
k
+1)
τ
,
λ
3
=1
α
τ
.
−
1+
S
(
λ
1
,λ
2
,λ
3
)
,
where
λ
1
=
a
1
−
−
k
With this, the correlation distribution of
F
can be described in terms of the
exponential sum
S
(
λ
1
,λ
2
,λ
3
).
k
Theorem 14.
Let
F
be the family of sequences defined in (19) and
ϕ
=
(
1)
p
−
2
.Then,
k
is a family of
p
3
2
nonbinary sequences with period
p
n
1
,
and its maximum correlation magnitude is equal to
p
2
+
d
+1
.Further,thecor-
relation distribution is given as follows, where
ρ
=1
,
2
,
−
F
−
···
,p
−
1
:
Correlation value
Frequency
3
2
p
n
−
1
p
3
2
(
p
n
−
2)(1 +
p
3
2
−
d
3
2
−
2
d
+
p
3
2
−
3
d
− p
n
−
2
d
)
−
1
p
− p
3
2
+
d
2
n
−
2
2
)+1)
2
2
p
(
p
+1)
((
p
n
−
2)(
p
n
−
1
+
p
p
−
1
− p
2(
p
d
+1)
2
3
2
+
d
(
p
2
+1)(
p
n
−
2)(
p
n
−
1
− p
n
−
2
2
)
/
(2(
p
d
+1))
ω
ρ
−
1
p
p
3
2
2
n
+
d
n
d
2
+
p
n
−
2
2
)+1)
p
(
p
−
2
p
+
p
)
((
p
n
−
2)(
p
n
−
1
− p
−p
−
1
2
2(
p
+1)(
p
d
−
1)
2
ω
ρ
−
1
p
2
n
−
1
(
p
n
+
d
−
2
p
n
+
p
d
)(
p
n
−
2)
/
(2(
p
d
−
1))
−p
n
+
d
2
p
2
n
−
d
((
p
n
−
2)
p
n
−
d
−
1
+1)
/
2
±p
ϕ −
1
n
+
d
2
n
−
d
−
1
ϕω
ρ
−
1
p
2
n
−
d
(
p
n
−
2)(
p
n
−
d
−
1
+
η
(
−ρ
)
p
p
)
/
2
2
n
+
d
2
n
−
d
−
1
ϕω
ρ
−
1
p
2
n
−
d
(
p
n
−
2)(
p
n
−
d
−
1
− η
(
−ρ
)
p
−p
)
/
2
2
3
2
−
d
2
n
−
2
d
n
−
2
2
)+
p
d
)
2
+
d
2
+
p
p
−
p
((
p
n
−
2)(
p
n
−
d
−
1
− p
−p
−
1
p
2
d
−
1
2
+
d
3
2
)(
p
n
−
2)(
p
n
−
2
d
−
1
+
p
2
−
d
−
1
)
/
(
p
2
d
−
1)
ω
ρ
−
1
(
p
2
n
−
d
− p
−p
Proof.
For any fixed (
a
2
,b
2
)
∈
F
p
m
×
F
p
n
,when(
a
1
,b
1
) runs through
F
p
m
×
F
p
n
and
τ
varies from 0 to
p
n
−
2, (
λ
1
,λ
2
,λ
3
) runs through
F
p
m
×
F
p
n
×{
F
p
n
\{
1
}}
one
k
is
p
3
2
times as that of
S
(
γ,δ,
)
time. Thus, the correlation distribution of
F
−
1
when (
γ,δ,
) runs through
. By Proposition 9, Equation
(14) and the possible values of
S
(
γ,δ,
0) corresponding to (
γ,δ
), the distribution
of
S
(
γ,δ,
0)
F
p
m
×
F
p
n
×{
F
p
n
\{
1
}}
−
1 is obtained when (
γ,δ
) runs through
F
p
m
×
F
p
n
. This together
with Proposition 13 give the distribution of
S
(
γ,δ,
)
−
1as(
γ,δ,
) runs through
×
F
p
n
.Noticethat
S
(
γ,δ,
)=
S
(
γ
−
(
p
m
+1)
,δ
−
(
p
k
+1)
,
1) for any fixed
F
p
m
×
F
p
n
∈
F
p
n
,andthenforanygiven
∈
F
p
n
,when(
γ,δ
) runs through
×
F
p
n
,the
distribution of
S
(
γ,δ,
) is the same as that of
S
(
γ,δ,
1). Thus, the distribution of
F
p
m
