Information Technology Reference
In-Depth Information
α
2
i
C
α
:=
{
:
i
∈{
0
,...,m
−
1
}}
.Wecanview
F
=
F
2
m
as a vector space of
dimension
m
over
1)-dimensional linear subspace
of this vector space, and can be defined as one of the subsets
H
α
:=
F
2
.A
hyperplane
is an (
m
−
{
x
∈
F
:
Tr
(
αx
)=0
}
F
, for all nonzero
α
∈
F
. The Trace-0 hyperplane
H
1
, is denoted
of
simply by
H
. Obviously for all
α
∈
F
∗
,
H
α
=
α
−
1
H
.Notethat
F
is the union of
F
2
m
=
k|m
C
k
,where
cyclotomic cosets of length
k
whenever
k
|
m
.Thatis:
C
k
is
C
k
:=
α∈
F
,|C
α
|
=
k
C
α
.
the set of field elements in a cyclotomic coset of length
k
, i.e.
The cardinalities
c
k
:=
|C
k
|
can be calculated as follows:
c
1
=2
,
since
C
1
=
{
0
,
1
}
,
(1)
c
k
=2
k
−
c
l
.
(2)
l|k,l
=
k
When
m
is odd, the intersections of the Trace-0-hyperplane with the subfields
of
F
2
m
have the following cardinalities:
Lemma 1.
Let
m
be odd,
k
|
m
,and
H
∩
F
2
k
denote the intersection of the
hyperplane
H
with the subfield
F
2
k
of
F
2
m
.Then
=2
k−
1
.
|
H
∩
F
2
k
|
Moreover, the intersection with the cyclotomic cosets of length
k
is
c
k
2
.
|H ∩C
k
|
=
(3)
Proof.
Let
α
∈
F
2
k
F
2
m
.Then
Tr
1
(
α
)=
Tr
1
(
Tr
k
(
α
)) =
Tr
1
m
k
α
=
m
k
Tr
1
(
α
)=
Tr
1
(
α
)
.
=2
k−
1
. We prove the second claim by an easy induction on the
Hence
|
H
∩
F
2
k
|
+
e
r
of the prime decomposition
m
=
p
e
1
p
e
r
r
of
m
.
length
l
=
e
1
+
···
···
If
l
=0,then
H
∩
F
2
=
{
0
}
,and
|
H
∩
F
2
|
=
c
1
/
2. Also for
l
=1,wehave
c
m
=2
m
=2
m−
1
−
2and
|
H
∩C
k
|
−
1. Now if all
n
|
m
,
n
=
m
satisfy (3), then
we have (using (2))
c
k
2
c
m
2
=2
m−
1
|
H
∩C
m
|
−
=
,
k
|
m,k
=
m
which was to be shown.
It is well known that the crosscorelation spectrum of
Θ
d
is related to the cardi-
nality of intersections
H
d
αH
if
gcd
(
d, q
∩
−
1) = 1,
Θ
d
(
α
)=
x∈
F
1)
Tr
(
x
d
+
αx
)
=
1)
Tr
(
x
d
)
1)
Tr
(
x
d
)
(
−
(
−
−
(
−
x∈α
−
1
H
x∈α
−
1
H
α
−
1
H
1)
Tr
(
x
d
)
=2
H
d
=2
x∈α
−
1
H
α
−
1
H
−
H
d
(
−
∩
∩
2
m
+4
H
d
α
−
1
H
,
=
−
∩
