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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.
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