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L<
2
n−
1
, from Theorem 1 we know that for any two sequences
S
,
For any 1
≤
S
∈A
(
L
),
d
H
(
S
,
S
)
≥
4. Hence we have
A
(
L
)
∩
(
A
(
L
)+
E
t
)=
∅
,
(30)
A
(
L
)
∩
(
A
(
L
)+
E
i,j
)=
∅
,
and
(31)
(
A
(
L
)+
E
t
)
∩
(
A
(
L
)+
E
i,j
)=
∅
,
(32)
for all
E
t
∈
E
1
and
E
i,j
∈
E
2
.
We enumerate the disjoint sets in equation (29) and obtain the counting func-
tion when 1
L<
2
n−
2
using the fact that
d
H
(
S
,
S
)
≤
≥
8 from Theorem 1 for
any two sequences
S
,
S
∈A
(
L
).
Theorem 5.
If
w
H
(2
n
L<
2
n−
2
, then the sets
−
L
)
≥
≤
A
(
L
)
,
A
(
L
)+
3
and
1
E
i
,
E
i
∈
E
1
,and
A
(
L
)+
E
i,j
,
E
i,j
∈
E
2
, are disjoint and
N
2
(
L
)=
2
n
2
+2
n
+1
2
L−
1
.
We enumerate the disjoint sets in equation (29) and give the counting function
when 2
n−
1
L<
2
n
.
Theorem 6.
Let
w
H
(2
n
≤
−
L
)
≥
3
where
2
n
(2
n−r
1
+2
n−r
2
)
<L<
2
n
(2
n−r
1
+2
n−r
2
−
1
)
,
−
−
(33)
for some
r
1
,r
2
satisfying
1
<r
1
≤
r
2
≤
n
−
1
. Define the sets
2
n−r
1
+1
D
1
(
L
)=
{
E
i
:0
≤
i
≤
−
1
}
and
2
n−r
1
+1
D
2
(
L
)=
{
E
i,j
:0
≤
i<j
≤
−
1
}
.
,
2
n−r
2
For
u
=0
,
···
−
1
define the sets
1
−
1
{
E
i,i
+2
n
−
r
1
:
i
=
u
+
t
2
n−r
2
D
u
(
L
)=
}
(34)
1
≤t≤
2
r
2
−
r
1
and
2
r
2
−
r
1
−
1
2
{
E
i,j
,
E
i,j
+2
n
−
r
1
:
i
=
u
+
t
1
2
n−r
2
,j
=
u
+
t
2
2
n−r
2
D
u
(
L
)=
}
.
(35)
t
2
=1
0
≤t
1
<t
2
Consider the set
D
(
L
)
formed from sets in equations
(6)
,
(34)
,and
(35)
by
2
n
−
r
2
−
1
1
2
D
(
L
)=
D
2
(
L
)
−
(
D
u
(
L
)
∪D
u
(
L
))
.
(36)
u
=0
Then the sets
A
(
L
)
,
A
(
L
)+
E
i
,
E
i
∈
D
1
(
L
)
,and
A
(
L
)+
E
i,j
,
E
i,j
∈D
(
L
)
,are
disjoint. Furthermore,
N
2
(
L
)=
2
n−r
1
+1
2
1) + 2
n−r
1
+1
+1
2
L−
1
.
2
n−r
2
(2
2
r
2
−
2
r
1
−
−
(37)
