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On the other hand, the correlation sum
Δ
(
c
)=
2
n
−
2
t
=0
ω
Tr
1
((1+2
c
)
β
t
)
has the
following weight distribution.
Lemma 3 ([1], Theorem 5).
As
c
is varying over
T
, the correlation sum
Δ
(
c
)
assumes values as
⎧
⎨
1+2
n
−
1
+2
n
−
2
ω,
2
n−
2
+2
n
−
3
−
times
2
2
1+2
n
−
1
2
n
−
2
ω,
2
n−
2
+2
n
−
3
−
−
times
2
2
Δ
(
c
)=
2
n
−
1
+2
n
−
2
ω,
2
n−
2
2
n
−
3
⎩
−
1
−
−
times
2
2
2
n
−
1
2
n
−
2
ω,
2
n−
2
2
n
−
3
−
1
−
−
−
times.
2
2
Combining Lemma 2 and 3, we therefore have the following correlation distri-
bution between the sequences in
A
2
.
Theorem 4.
The correlation distribution of Family
A
1
and
A
1
and
A
2
are as follows:
i, j <
2
n−
1
or
2
n−
1
i, j <
2
n
,
1. for
0
≤
≤
⎧
⎨
−
1+2
n
,
2
n−
1
times
−
1
,
2
n−
1
(2
n−
1
−
1)
times
1+2
n
−
1
+2
n
−
2
ω,
2
n−
2
(2
n
2)(2
n−
2
+2
n
−
2
)
times
−
−
2
R
i,j
(
τ
)=
1+2
n
−
1
2
n
−
2
ω,
2
n−
2
(2
n
2)(2
n−
2
+2
n
−
2
)
times
⎩
−
−
−
2
2
n
−
1
+2
n
−
2
ω,
2
n−
2
(2
n
2
n
−
2
)
times
2)(2
n−
2
−
1
−
−
−
2
2
n
−
1
2
n
−
2
ω,
2
n−
2
(2
n
2
n
−
2
)
times
;
2)(2
n−
2
−
1
−
−
−
−
2
i<
2
n−
1
j<
2
n
or
0
j<
2
n−
1
i<
2
n
,
2. for
0
≤
≤
≤
≤
⎧
⎨
2
2
n−
2
−
1
,
times
1+2
n
−
1
+2
n
−
2
ω,
2
n−
2
(2
n
2)(2
n−
2
+2
n
−
2
)
times
−
−
2
1+2
n
−
1
2
n
−
2
ω,
2
n−
2
(2
n
2)(2
n−
2
+2
n
−
2
)
times
R
i,j
(
τ
)=
−
−
−
2
⎩
2
n
−
1
+2
n
−
2
ω,
2
n−
2
(2
n
2
n
−
2
)
times
2)(2
n−
2
−
1
−
−
−
2
2
n
−
1
2
n
−
2
ω,
2
n−
2
(2
n
2
n
−
2
)
times.
2)(2
n−
2
−
1
−
−
−
−
2
In particular,
R
i,j
(
τ
)=2
n
−
1
if and only if
τ
=0
and
i
=
j
,and
R
i,j
(
τ
)=
−
1
if and only if
τ
=0
and
i
=
j
.
4 Correlation Distribution of Kerdock Sequences
The following 2-adic expression of the quaternary sequences in trace form is
helpful to obtain the counterparts of the sequences in Family
A
under the Gray
maps.
Lemma 5 ([3]).
Assuming that
η
, the 2-adic representation of
a
(
t
)=
Tr
1
((1 + 2
η
)
β
t
)
, i.e.,
a
(
t
)=2
b
(
t
)+
c
(
t
)
, is given by
b
(
t
)=
tr
1
(
α
t
)
,
∈T
(5)
c
(
t
)=
tr
1
(
ζα
t
)+
p
(
α
t
)
(6)
where
ζ
=
μ
(
η
)
and
p
(
x
)=
n
−
1
2
l
=1
tr
1
(
x
2
l
+1
)
.
