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Remark 3.
If condition (6) is satisfied then (9) will be zero for all multisequences
S
∈M
(
m
)
q
(
f
). As gcd(deg(
r
i
)
,m
) can at most be
m
the largest possible relative
distance between joint linear complexity and generalized joint linear complexity
of an
m
-fold multisequence is given by (
m
1)
/m
.
We give two examples illustrating our results.
Example 1.
Let
N
,
m
be positive integers and consider the
N
-periodic
m
-fold
multisequences over
−
F
q
. Equivalently, let
f
=
x
N
−
1
∈
F
q
[
x
] and we can consider
(
m
)
q
the multisequences in
M
(
f
). Let
p
be the characteristic of the finite field
F
q
1)
p
v
,andthe
and
N
=
p
v
n
with gcd(
n, p
)=1.Thenwehave
x
N
1=(
x
n
−
−
canonical factorization of
x
n
−
1in
F
q
[
x
]isgivenby
k
r
i
(
x
)w th
r
i
(
x
)=
j∈C
i
x
n
α
j
)
,
−
1=
(
x
−
i
=1
where
C
1
,...,C
k
are the different cyclotomic cosets modulo
n
relative to powers
of
q
and
α
is a primitive
n
th root of unity in some extension field of
F
q
.Let
S
be an
N
-periodic
m
-fold multisequence over
F
q
with minimal polynomial
d
=
r
ρ
1
r
ρ
2
···r
ρ
k
,where0
≤ ρ
i
≤ p
v
. Then using Theorem 2 we have
2
k
l
i
gcd(
l
i
,m
)
,
L
(
S
)
≥
ρ
i
(13)
i
=1
where
l
i
denotes the cardinality of the cyclotomic coset
C
i
. Equation (13) coin-
cides with the corresponding result in [8, Theorem 2].
Example 2.
Let
r
1
,...,r
k
∈
F
q
[
x
] be distinct irreducible polynomials and let
e
1
,...,e
k
be positive integers. For a positive integer
m
,let
f
=
r
e
1
r
e
2
...r
e
k
,
(
m
)
q
and consider the multisequences in
M
(
f
). It is not dicult to observe that
(
m
)
q
there exists a multisequence
S
∈M
(
f
) with joint linear complexity
L
(
m
)
(
S
)=
t
if and only if
t
can be written as
q
t
=
i
1
deg(
r
1
)+
i
2
deg(
r
2
)+
···
+
i
k
deg(
r
k
)
,
(14)
where 0
≤
i
1
≤
e
1
, ..., 0
≤
i
k
≤
e
k
are integers. Let
ξ
=(
ξ
1
,...,ξ
m
)bean
ordered basis of
F
q
m
over
F
q
.Let0
≤
i
1
≤
e
1
, ..., 0
≤
i
k
≤
e
k
be chosen
(
m
)
q
integers. Consider the nonempty subset
T
(
i
1
,...,i
k
)of
M
(
f
) consisting of
such that
L
(
m
)
S
)=
t
,where
t
is as in (14). Using the methods of this
paper we obtain that, among the multisequences in
(
S
q
T
(
i
1
,...,i
k
), there exists a
)=
t
if and
S
with generalized joint linear complexity
L
q
m
,
ξ
(
S
multisequence
only if
t
can be written as
deg(
r
1
)
gcd(deg(
r
1
)
,m
)
+
i
2
j
2
deg(
r
2
)
gcd(deg(
r
2
)
,m
)
+
deg(
r
k
)
gcd(deg(
r
k
)
,m
)
,
t
=
i
1
j
1
···
+
i
k
j
k
where 1
≤
j
1
≤
gcd(deg(
r
1
)
,m
), ..., 1
≤
j
k
≤
gcd(deg(
r
k
)
,m
) are integers.