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Let
a
1
,a
2
,...,a
k
be integers with
0
≤
a
i
≤
e
i
for
1
≤
i
≤
k
.Let
m
≥
2
be an integer and
ξ
=(
ξ
1
,...,ξ
m
)
be an ordered basis of
F
q
m
over
F
q
.There
(
m
)
q
exists an
m
-fold multisequence
S
∈M
(
f
)
over
F
q
such that its joint minimal
polynomial
d
is
d
=
r
a
1
r
a
2
...r
a
k
,
k
and its generalized joint linear complexity
L
q
m
,
ξ
(
S
)
is
k
deg(
r
i
)
gcd(deg(
r
i
)
,m
)
.
L
q
m
,
ξ
(
S
)=
a
i
i
=1
Proof.
By reordering
r
1
,...,r
k
suitably, we can assume without loss of gener-
ality that there exists an integer
l
,1
≤
l
≤
k
,withgcd(
m,
deg(
r
i
)) =
u
i
≥
2
for 1
≤
i
≤
l
and gcd(
m,
deg(
r
i
)) = 1 for
l
+1
≤
i
≤
k
. Indeed otherwise
gcd(
m,
deg(
r
i
)) = 1 for 1
k
and hence the result is trivial by Theorem 1.
Using Proposition 1 we obtain that the canonical factorizations of
r
i
,1
≤
i
≤
≤
i
≤
l
,
into irreducibles over
F
q
m
are of the form
r
i
=
t
i,
1
t
i,
2
...t
i,u
i
.
(1)
q
m
(
f
) corresponding to the polynomial
Let
S
be the sequence in
M
l
f
d
(
t
i,
2
...,t
i,u
i
)
a
i
G
=
∈
F
q
m
[
x
]
i
=1
and let
h
1
,h
2
,...,h
m
∈
F
q
[
x
] be the uniquely determined polynomials in
F
q
[
x
]
such that
l
(
t
i,
2
...,t
i,u
i
)
a
i
=
ξ
1
h
1
+
ξ
2
h
2
+
···
+
ξ
m
h
m
.
(8)
i
=1
(
m
)
q
(
f
)bethe
m
-fold multisequence such that the
sequence
σ
i
corresponds to
g
i
=
h
i
f/d
Let
S
=(
σ
1
,...,σ
m
)
∈M
∈
F
q
[
x
]for1
≤
i
≤
m
.Weobservethat
we have
S
=
S
(
S
,
ξ
)and
l
k
L
q
m
,
ξ
(
S
)=
a
i
deg(
t
i,
1
)+
a
i
deg(
r
i
)
.
i
=1
i
=
l
+1
Moreover
d
is the joint minimal polynomial of
S
. Indeed, otherwise using (8) we
obtain that there exists 1
≤
i
≤
k
with
l
(
t
i,
2
...,t
i,u
i
)
a
i
in
r
i
|
F
q
m
[
x
]
.
i
=1
This is a contradiction, which completes the proof.
