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where
(
x
)
denotes the real part of complex-valued variable
x
. Specifically, when
L
is odd,
R
2
L−
1
L−
1
1)
ϕ
(
a
(
t
))+
ϕ
(
b
(
t
))
=2
ω
a
(
t
)
−b
(
t
)
)
.
(
−
·
R
(
t
=0
t
=0
3 Correlation Distribution of Family
A
Throughout this paper let
n
be an odd integer and
{
η
0
,η
1
,
···
,η
2
n
−
1
}
an enu-
i<
2
n−
1
,
meration of the elements in
T
, satisfying
η
i
+2
n
−
1
=1+
η
i
(mod 2),0
≤
and
Tr
1
(
η
i
)=
0(mod2)
,
0
≤ i<
2
n−
1
1(mod2)
,
2
n−
1
i<
2
n
.
≤
The well-known sequence Family
A
is defined in [1].
A
{
a
i
,i
=0
,
1
,
···
,
2
n
}
Definition 1 ([1]).
The sequence Family
=
of length
2
n
−
1
is defined by
a
i
(
t
)=
Tr
1
((1 + 2
η
i
)
β
t
)
,
0
i<
2
n
≤
2
Tr
1
(
β
t
)
,
i
=2
n
.
In this paper, we will focus on a smaller set comprised of the first 2
n
sequences.
For convenience, we still call it Family
,
2
n
A
, i.e.,
A
=
{
a
i
,i
=0
,
1
,
···
−
1
}
.
,
2
n−
1
a
i
,i
=2
n−
1
,
,
2
n
Let
A
1
=
{
a
i
,i
=0
,
···
−
1
}
and
A
2
=
{
···
−
1
}
.
Specifically, we are interested in the correlation distribution between
A
i
and
A
j
,
1
2, which is crucial to derive our main result on the correlation distri-
bution of the family of Kerdock sequences. First of all, consider the correlation
function between
a
i
∈A
≤
i, j
≤
and
a
j
∈A
at the shift
τ
as
2
n
−
2
ω
Tr
1
((1+2
η
i
)
β
t
+
τ
(1+2
η
j
)
β
t
)
.
−
R
i,j
(
τ
)=
t
=0
Define
θ
τ
=(1+2
η
i
)
β
τ
−
(1+2
η
j
)=
b
τ
+2
c
τ
,
b
τ
,c
τ
∈T
.If
b
τ
= 0, which implies
=
j
)and
R
i,j
(
τ
)=2
n
τ
=0,then
R
i,j
(
τ
)=
−
1when
c
τ
= 0 (that is
i
−
1when
c
τ
c
τ
= 0 (that is
i
=
j
). Otherwise let
U
τ
=
b
τ
∈T
, the correlation function then
becomes
2
n
−
2
ω
Tr
1
((1+2
U
τ
)
β
t
)
.
R
i,j
(
τ
)=
t
=0
Next, the following lemma is important to determine the distribution of
U
τ
when
a
i
,a
j
range over
A
2
.
Lemma 2 ([5]).
Let
0
<τ <
2
n
A
1
or
−
1
. Then for any
c
∈T
the number of solutions
η
i
,
η
j
satisfying
U
τ
=
c
is
2
n−
2
if
i
runs from
2
n−
1
δ
1
to
2
n−
1
(1 +
δ
1
)
−
1
,and
j
runs from
2
n−
1
δ
2
to
2
n−
1
(1 +
δ
2
)
−
1
, respectively, where
δ
1
,δ
2
∈{
0
,
1
}
.
