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in equation (45) result in 2 n−r 2 (2 r 2 −r 1
1) doubly counted sets in
A
( L )+
D 2 ( L )
1
, 2 n−r 2
, 2 n−r 2
enumerated as
D
u ( L ), u =0 ,
···
1. For each u =0 ,
···
1
we have
=2 2 r 2 −r 1
2
.
1
u ( L )
=2 r 2 −r 1
2
u ( L )
|D
|
1and
|D
|
(46)
Note that any L such that 2 n− 1
L< 2 n and w H (2 n
L )
3, satisfies
equations (30) and (32). From Lemma 5 and equation (38) we have
A
( L )
A
( L )+ E i,j )=
, E i,j D 2 ( L ) .
(
(47)
Thus, from equations (29), (30)-(32), (41)-(45), and (47), 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. The number of
A
( L )+
E 2 is equal to
|D
( L )
|
disjoint sets in
. From equations (36) and (46)
we have
2 n r 2
1
1
2
|D
( L )
|
=
| D 2 ( L )
|−
(
|D
u ( L )
|
+
|D
u ( L )
|
)
(48)
u =0
= 2 n−r 1 +1
2
2 n−r 2 2 r 2 −r 1
1+2 2 r 2 −r 1
2
.
2 n . Hence the counting function
in equation (37) follows from equations (29),(30)-(32), (39),(47), and (48). This
completes the proof of the theorem.
=2 L− 1 ,1
From Lemma 2 we have
|A
( L )
|
L
Using Remark 1 with r 1 =1and r 1 <r 2 in the statement of Theorem 6, we get
the characterization when 2 n− 2
L< 2 n− 1 .
We also characterized 2 n -periodic binary sequences with fixed 3-error linear
complexity L when w H (2 n
= 2. Using the characterization we obtained the
corresponding counting function. Due to space constraints we state our results
here without proofs and also use some of the notation established in the state-
ment of Theorem 6. The approach used is the same as that used for the 2-error
case and the proofs use Lemma 5, Theorem 2, Lemma 7, and some intermediate
findings of Theorem 6. It is straightforward to see that
L )
N 3 (0) = 2 n
3
+ 2 n
2
+2 n +1 ,
A 3 (0) =
E 1 E 2 E 3 ∪{ 0 }
and
A 3 (2 n )=
N 3 (2 n )=0 ,
and
and
N 3 ( L )=0 for L =2 n
2 t ,
A 3 ( L )=
and
0
t<n.
Theorem 7. Let L< 2 n be a positive integer such that w H (2 n
L )
3 .Then
A 2 ( L )
.
A 3 ( L )=
(
A
( L )+ E i,j,k )
E i,j,k E 3
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