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Furthermore,
(i) If
1
L<
2
n−
2
, then the sets
≤
A
(
L
)
,
A
(
L
)+
E
i
,
E
i
∈
E
1
,
A
(
L
)+
E
i,j
,
E
i,j
∈
E
2
,and
A
(
L
)+
E
i,j,k
,
E
i,j,k
∈
E
3
are disjoint and
N
3
(
L
)=
2
n
3
+
2
n
2
+2
n
+1
2
L−
1
.
(ii) Let
L
be such that
2
n
(2
n−r
1
+2
n−r
2
)
<L<
2
n
(2
n−r
1
+2
n−r
2
−
1
)
,
−
−
for some
r
1
,r
2
satisfying
1
<r
1
≤
r
2
≤
n
−
1
.Let
D
(
L
)
be as in equation
3
,
2
n−r
2
,and
(36)
. Define the sets
D
3
(
L
)
,
D
u
(
L
)
,
u
=0
,
···
E
(
L
)
by
2
n−r
1
+1
D
3
(
L
)=
{
E
i,j,k
:0
≤
i<j<k
≤
−
1
}
,
0
≤g
1
<g
2
<
2
r
2
−
r
1
{
E
i
1
,j
1
,i
2
,
E
i
1
,j
1
,j
2
,
E
i
1
,i
2
,j
2
,
E
i
2
,j
1
,j
2
:
i
t
=
u
+
g
t
2
n−r
2
,j
t
=
i
t
+2
n−r
1
,t
=1
,
2
3
D
u
(
L
)=
}
,
and
2
n
−
r
2
u
=0
D
−
1
3
E
(
L
)=
D
3
(
L
)
−
u
(
L
)
.
Then the sets
A
(
L
)
,
A
(
L
)+
E
i
,
E
i
∈
D
1
(
L
)
,
A
(
L
)+
E
i,j
,
E
i,j
∈D
(
L
)
,and
A
(
L
)+
E
i,j,k
,
E
i,j,k
∈E
(
L
)
are all disjoint and
N
2
(
L
)+
2
n−r
1
+1
3
2
n−r
2
2
r
2
−r
1
2
2
L−
1
.
N
3
(
L
)=
−
4
·
Using Remark 1 with
r
1
=1and
r
1
<r
2
in the statement of Theorem 7, we get
the characterization when 2
n−
2
≤
L<
2
n−
1
.
4Con lu on
In this paper, we characterize 2
n
-periodic binary sequences with fixed 2-error
or 3-error linear complexity
L
when
w
H
(2
n
− L
)
= 2. First, we derive some
properties of 2
n
-periodic binary sequences with fixed linear complexity. We use
the Games-Chan algorithm to find the exact form of four symbol changes that
can be made in a 2
n
-periodic sequence so that the resulting sequence has the
same linear complexity as the original sequence. We use these properties to
obtain the characterizations and the corresponding counting functions.
We believe that our approach in Theorem 2 can be used to generalize the re-
sults to the case when the number of changes made is any power of 2. We empha-
size that we can also obtain the characterization in the case when
w
H
(2
n
L
)=2
by further analysis of the Games-Chan algorithm when the linear complexity is
of the form
L
=2
n
−
(2
i
+2
j
), 0
1. Statistical properties like
expected value and variance can also be considered for
L
k
(
S
)forseveralsmall
k
.Extensionto
p
n
-periodic sequences over
−
≤
i<j
≤
n
−
F
p
can also be considered.
