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In-Depth Information
G
A
.Then
N
(
a,
3
a
;
b
)=
0
if
b
=
a,
3
a
8. Let
a, b
∈
1
otherwise.
=
b
.Then
N
(
a, a
;
b
)=
2
if
tr
(
b
)=
tr
(
a
)
0
otherwise.
9. Let
a, b
∈
G
A
,
a
10. Let
a, b, c
∈
G
A
,
a
=
b,
3
b
.Then
⎧
⎨
1
if
c
=
a, b
2
if
c
=
a, b, tr
(
ab
+
ac
+
bc
)=
tr
(
c
)
0
otherwise.
N
(
a, b
;
c
)=
⎩
We need the following definition in the next section.
1+2
β
k
,k
=
,
2
n
Definition 1.
Let
γ
=(1+2
a
)
∈{
∞
,
0
,
···
−
2
}
. Then the
trace number
of
γ
is defined as the value of
tr
(
μ
(
a
))
. The trace number is
always 1 or 0.
2.2 Sequence Families A, B and C
A
q
ary sequence family made up of
M
cyclically distinct sequences of length
N
is defined to be the collection of vectors
−
Z
q
,
1
{
s
1
,...,
s
M
}
,
s
i
∈
≤
i
≤
M
with
s
1
=(
s
1
(0)
,...,s
1
(
N
1)), where
Z
q
is the ring of integers modulo
q.
Here we restrict ourselves to quaternary sequences, i.e.,
q
=4
.
The (periodic)
correlation function between sequences
i
and
j
at relative shift
τ
is defined as
−
N−
1
ω
s
i
(
t⊕τ
)
−s
j
(
t
)
C
i,j
(
τ
)=
t
=0
where
ω
=
exp
(2
πj/
4) =
√
−
1 is a primitive fourth root of unity and where
⊕
denotes addition modulo
N.
Given a sequence family such as above, the Welch [12] and Sidelnikov [7] lower
bounds determine how good such a family can be. For example, if
M
N
then
the maximum nontrivial correlation magnitude (sometimes called the maximum
sidelobe)
≈
C
max
=max
{|C
i,j
(
τ
)
|
:
i
=
j
or
τ
=0
}
√
N
for nonbinary se-
quences.
Family A
[1,6] is a large sequence family which delivers the promised
improvement for
C
max
for the practically significant (due to the widespread use
of quaternary PSK modulation)
q
=4value.
Family A
comprises a set of
M
=2
n
+ 1 cyclically distinct sequences over
Z
4
with length
N
=2
n
√
2
N
for binary sequences and
C
max
≥
obeys
C
max
≥
−
1
,
which obey a common linear recurrence whose
characteristic polynomial is a primitive basic irreducible polynomial of degree
n
over the ring
Z
4
[
x
]
.
Each element
s
i
of
Family A
can be expressed as
s
i
(
t
)=
T
(
γβ
t
)where
β
is a generator of the Teichmuller set, and
γ
=0
.
In fact the
enumeration of representatives
Γ
ν
=
{
2
}∪
G
A
,
