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In-Depth Information
2
n
,let
cardinality of
A
k
(
L
). Let
0
denote the zero sequence. For any 1
≤
t
≤
1, denote the 2
n
-periodic binary sequence of
weight
t
with a 1 at positions with subscripts
i
1
,
2
n
E
i
1
,··· ,i
t
,0
≤
i
1
<
···
<i
t
≤
−
···
,i
t
in the first period and 0
A
(
L
)+
E
i
1
,··· ,i
t
{
S
+
E
i
1
,··· ,i
t
:
S
∈A
(
L
)
}
elsewhere. We denote by
the set
.For
R
of 2
n
-periodic binary sequences, by
A
(
L
)+
R
the rest of the paper, for any set
,
2
n
,
{A
(
L
)+
R
:
R
∈R}
E
t
,
t
=1
,
···
denote the set of sets
. We define sets
which are used for the rest of the paper. Let
E
t
=
{
E
i
1
,··· ,i
t
:0
≤
i
1
<i
2
<
···
<
2
n
i
t
≤
−
1
}
. It is straightforward to see that
N
2
(0) =
2
n
2
+2
n
+1
.
A
2
(0) =
E
1
∪
E
2
∪{
0
}
and
(26)
From Lemmas 3 and 4 we have
A
2
(2
n
)=
N
2
(2
n
)=0
.
∅
and
(27)
For any 2
n
-periodic binary sequence
S
, from [6, Proposition 1] we know that for
k
2,
L
k
(
S
) is different from 2
n
2
t
for every integer
t
with 0
≥
−
≤
t<n
. Hence
we get
L
=2
n
2
t
,
A
2
(
L
)=
∅
and
N
2
(
L
)=0
for
−
0
≤
t<n.
(28)
A characterization of 2
n
-periodic binary sequences with fixed 2-error linear com-
plexity
L
such that
w
H
(2
n
L
) = 0 or 1 is given in equations (26)-(28). Next,
we give the characterization when
w
H
(2
n
−
3. We start with a result that
will be used for the rest of the section. The proof is provided in the appendix.
Theorem 3.
Let
−
L
)
≥
{
i
1
,
···
,i
t
1
}
and
{
j
1
,
···
,j
t
2
}
denote two sets of subscripts
2
n
where
0
≤
i
l
,j
m
≤
−
1
,
l
=1
,
···
,t
1
,
m
=1
,
···
,t
2
.Then
A
(
L
)+
E
i
1
,··· ,i
t
1
)
∩
A
(
L
)+
E
j
1
,··· ,j
t
2
)=
∅
(
(
or
(
L
)+
E
j
1
,··· ,j
t
2
.
We need the following generalization of [3, Theorem 4] to obtain the basic char-
acterization of
A
(
L
)+
E
i
1
,··· ,i
t
1
=
A
A
2
(
L
).
Lemma 7.
Let
S
be a
2
n
-periodic binary sequence. Consider any two positive
integers
u
,
v
such that
0
<v
u
and
u
+
v<merr
(
S
)
. Then for any
2
n
-periodic
binary sequence
E
such that
w
H
(
E
)=
v
we have
≤
L
u
(
S
+
E
)=
L
(
S
)
.
The basic characterization can be obtained by using the definition of
k
-error
linear complexity and Lemma 7.
Theorem 4.
If
w
H
(2
n
−
L
)
≥
3
,then
⎛
⎞
(
L
)
⎝
⎠
.
A
2
(
L
)=
A
(
A
(
L
)+
E
i
)
(
A
(
L
)+
E
i,j
)
(29)
E
i
∈
E
1
E
i,j
∈
E
2
