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where
ζ
i−
2
n
−
1
+
ζ
j−
2
n
−
1
=
ζ
i
+
ζ
j
,2
n−
1
i, j <
2
n
, come from the fact that
≤
ζ
i
=1+
ζ
i−
2
n
−
1
and
ζ
j
=1+
ζ
j−
2
n
−
1
.
Case 3
.(0
i<
2
n−
1
j<
2
n
or 0
i<
2
n−
1
j<
2
n
)and
τ
=0.
≤
≤
≤
≤
For this case,
(
x∈
F
2
n
1) +
x∈
F
2
n
1)
tr
1
((
ζ
i
+
ζ
j
)
x
)
1)
tr
1
((
ζ
i
+
ζ
j
)
x
)
R
i,j
(
τ
)=
−
(
−
−
(
−
−
1
=0
,
wherewealsousethefactthat
ζ
i
=1+
ζ
i−
2
n
−
1
for 2
n−
1
i<
2
n
,or
ζ
j
=
≤
1+
ζ
j−
2
n
−
1
for 2
n−
1
j<
2
n
.
≤
i, j <
2
n−
1
and
τ
Case 4
.0
=0.
From Lemma 1 and Lemma 7 it holds that
≤
2
n
+1
−
3
1)
v
i
(
t
+
τ
)+
v
j
(
t
)
R
i,j
(
τ
)=
(
−
t
=0
=
2
(
2
n
−
2
·
R
ω
a
i
(
t
+
τ
1
)
−a
j
(
t
)
)
,
τ
=2
τ
1
t
=0
(
2
n
−
2
t
=0
ω
a
i
+2
n
−
1
(
t
+
τ
1
+2
n
−
1
)
−a
j
(
t
)
)
,τ
=2
τ
1
+1
.
2
·
R
As a direct consequence of Theorem 4, when
τ
=2
τ
1
we have
R
i,j
(
τ
)=
2+2
n
+
2
,
2
n−
1
(2
n
2)(2
n−
2
+2
n
−
2
)
times
−
−
2
n
+
2
,
2
n−
1
(2
n
2
n
−
2
)
times
2)(2
n−
2
−
2
−
−
−
as
τ
1
ranges over 0
<τ
1
<
2
n
1and
i, j
varyfrom0to2
n−
1
−
−
1, respectively.
If
τ
=2
τ
1
+1 = 2
n
2, which occurs 2
2
n−
2
−
1, then
R
i,j
(
τ
)=
−
times.
Otherwise from Theorem 4, the other correlation distribution is
R
i,j
(
τ
)=
2+2
n
+
2
,
2
n−
1
(2
n
2)(2
n−
2
+2
n
−
2
)
times
−
−
2
n
+
2
,
2
n−
1
(2
n
2
n
−
2
)
times
2)(2
n−
2
−
2
−
−
−
=2
n−
1
1variesfrom0to2
n
2, and
i, j
rangesfrom0to2
n−
1
,
as
τ
1
−
−
respectively.
Case 5
.2
n−
1
i, j <
2
n
and
τ
≤
=0.
Then,
R
i,j
(
τ
)=
R
i−
2
n
−
1
,j−
2
n
−
1
(
τ
)
,τ
=2
τ
1
−
R
i−
2
n
−
1
,j−
2
n
−
1
(
τ
)
,τ
=2
τ
1
+1
.
Hence, it is immediate from case 4 that the correlation distribution is
⎧
⎨
2
2
n−
2
times
2
,
2+2
n
+
2
,
2
n−
1
(2
n
2)(2
n−
2
+2
n
−
2
)
times
−
−
2
n
+
2
,
2)(2
n−
2
+2
n
−
2
)
times
2
n−
1
(2
n
R
i,j
(
τ
)=
2
−
−
⎩
2
n
+
2
,
2
n−
1
(2
n
2
n
−
2
)
times
2)(2
n−
2
−
2
−
−
−
2+2
n
+
2
,
2
n
−
2
)
times
2
n−
1
(2
n
2)(2
n−
2
−
−
