Information Technology Reference
In-Depth Information
(2
l
4 Polynomials over the Galois Ring
GR
,m
)
Recall that
R
=
GR
(2
l
,m
). A polynomial
d
c
j
x
j
f
(
X
)=
∈
R
[
X
]
j
=0
is called
canonical
if
c
j
= 0 for all even
j
. Given an integer
D
≥
4, define
S
D
=
{
f
(
X
)
∈
R
[
X
]
|
D
f
≤
D, f
is canonical
}
,
where
D
f
is the weighted degree of
f
.Observethat
S
D
is an
GR
(2
l
,m
)
module.
Recall [7, Lemma 4.1]. For a weaker condition on
D
see [6, Theorem 6.13].
−
Lemma 4.1.
For any integer
D
≥
4
, we have:
=2
(
D−D/
2
l
)
m
,
|
S
D
|
where
x
is the largest integer
≤
x
.
Recall the following property of the weighted degree [8, Lemma 3.1].
R
∗
=
R
Lemma 4.2.
Let
f
(
X
)
∈
R
[
X
]
and
α
∈
\
2
R
is a unit of
R
and let
g
(
X
)=
f
(
αX
)
∈
R
[
X
]
.Then
D
g
=
D
f
,
where
D
f
,
D
g
are respectively the weighted degrees of the polynomials
f
(
X
)
and
g
(
X
)
.
We will need the following technical result.
Lemma 4.3.
Let
f
(
X
)
∈
R
[
X
]
and assume that
f
∈
S
D
withanon-zerolin-
ear term. If we fix any
r
integers
0
≤
s
1
<s
2
< ... < s
r
=
n
−
1
then
f
(
ξ
s
1
X
)
,f
(
ξ
s
2
X
)
,...,f
(
ξ
s
r
X
)
, are linearly independent over
Z
2
l
, i.e. for any
integers
j
1
,j
2
,...,j
r
the equality
j
1
f
(
ξ
s
1
X
)+
j
2
f
(
ξ
s
2
X
)+
...
+
j
r
f
(
ξ
s
r
X
)=0
implies
j
1
=
j
2
=
...
=
j
r
=0
.
Proof.
Suppose
f
(
X
)
∈
R
[
X
]isofdegree
d
≤
D
and
f
(
X
)=
α
1
X
+
...
+
α
d−
1
X
d
,
where
α
1
=0and
α
k
=0foreven
k
. Fix the integers
s
1
,s
2
,...,s
r
, then for any
r
integers
j
1
,j
2
,...,j
r
let
r
j
i
f
(
ξ
s
i
X
)
.
g
(
X
)=
g
j
1
...j
r
(
X
)=
i
=0