Information Technology Reference
In-Depth Information
=deg( r u )= deg( r )
u
deg( r 1 )=
···
.
Proof. This is just a restatement of [6, Theorem 3.46]. We refer to [6] for a
proof.
Proposition 2. Let m be a positive integer, let
ξ
=( ξ 1 ,...,ξ m ) be an ordered
basis of
F q m over
F q ,andlet h 1 ,...,h m F q [ x ] be arbitrary polynomials. For
h
F q [ x ] ,thereexists s
F q m [ x ] such that
sh = ξ 1 h 1 +
···
+ ξ m h m
if and only if there exist s 1 ,...,s m F q [ x ] such that
s i h = h i for 1
i
m.
Proof. For a polynomial s
F q m [ x ]let s 1 ,...,s m F q [ x ] be the uniquely deter-
mined polynomials in
F q [ x ] such that
s = ξ 1 s 1 +
···
+ x m s m .
Then
sh = ξ 1 s 1 h +
···
+ x m s m h
is the unique representation in the basis
ξ
of the polynomial sh and the claim
immediately follows.
Finally we recall an important definition from [4]. For a monic polynomial f
F q [ x ] and a positive integer m we let Φ ( m )
( f ) denote the number of m -fold
q
F q with minimal joint polynomial f .Notethat Φ ( m )
multisequences over
( f )can
q
be considered as a function on the set of monic polynomials in
F q [ x ]. In [4,
Section 2] several important properties of Φ ( m q ( f ) have been derived, which we
will use in this paper. We refer to [4] for further details.
3 Generalized Joint Linear Complexity
In this section we obtain our main results and we give illustrative examples. The
following three lemmas will be used in the proof of the next theorem.
Lemma 1. For an integer n
2 ,let H n ( x ) be the real valued function on
R
defined by
H n ( x )= x n
1) n .
1
( x
For a real number x> 1 , we have H n ( x ) > 0 .
Search WWH ::




Custom Search