Information Technology Reference
In-Depth Information
We refer to the integer
l
as the
subspace-size exponent
(sse) associated with the
constellation parameter
(c-p)
m
.Thus
l
lies in the range 0
≤
l
≤
(
r
−
1). Let
μ
2
r−
1
+1,maps
m
denote the function that, given c-p
m
in the range 1
≤
m
≤
to the corresponding sse
l
given above, i.e.,
μ
(
m
)=
l
.
Treating
F
q
as a vector space over
F
2
of dimension
r
,let
W
r−
1
denote the
F
q
of dimension (
r
−
subspace of
1) corresponding to the elements of trace = 0.
Let
W
l
denote a subspace of
W
r−
1
having dimension
l
.Let
ζ
be an element in
F
q
having trace 1 and let
V
l
denote the subspace
V
l
=
W
l
∪{
of size 2
l
+1
.
Noting that every element in the coset
W
l
+
ζ
of
W
l
has trace 1, we select
as the elements
W
l
+
ζ
}
m−
1
k
=1
2
{
δ
k
}
to be used in the construction of family
I
SQ−B
,an
2
l
elements selected from the set
W
l
+
ζ
.
Next, we partition
W
r−
1
into the 2
r−l−
1
cosets
W
l
+
g
of
W
l
.Witheach
coset, we associate a distinct user. To this user, we assign the coecient set
{
arbitrary collection of (
m
−
1)
≤
m−
1
k
=1
belong to the coset
W
l
+(
g
+
ζ
)of
W
l
. Thus, in general, each user is assigned
m
coecients, with one
coecient
g
belonging to the coset
W
l
+
g
of
W
l
lying in
W
r−
1
and the remaining
drawn from the coset
W
l
+
g
+
ζ
of
W
l
.Since
V
l
=
W
l
∪
g, g
+
δ
1
,g
+
δ
2
,...,g
+
δ
l
}
. The coecients
{
g
+
δ
k
}
(
W
l
+
ζ
), all
m
coecients
taken together belong to the coset
V
l
+
g
of
V
l
.Notethat
V
l
+
g
=
V
l
+
g
implies
W
l
+
g
W
l
+
ζ
+
g
.
Butthisisimpossiblesince
g, g
belong to different cosets of
W
l
and
g, g
have
trace zero, whereas,
tr
(
ζ
) = 1. It follows that the coecient sets of distinct users
belong to different cosets of
V
l
and are hence distinct.
Let
G
be the set of all such coset representatives of
W
l
in
W
r−
1
. Since each
user is associated to a unique coset representative, the number of users is given
by
G
=2
r−l−
1
. When combined with (2), we obtain
2
r
4(
m
{
W
l
+
g
}∪{
W
l
+
ζ
+
g
}
=
{
}∪{
}
2
r
2(
m
<
|
G
|≤
1)
.
(3)
−
1)
−
Thus, the size of
G
is at most a factor of 4 smaller than the best possible
suggested by the Hamming bound (1). The reader is referred to [1] for more
details on the construction of
G
and
H
.
2.2 Sequence Definition
We describe our interleaved construction for sequences in the family
2
I
SQ
M
2
−B
.
Let
{
τ
1
,τ
2
,...,τ
m−
1
}
be a set of non-zero, distinct time-shifts with
1
,α
τ
1
,α
τ
2
,...,α
τ
m−
1
where the set
{
}
being a linearly independent set. Let
κ
=
m−
1
2
(
κ
0
,κ
1
,...,κ
m−
1
)
∈
Z
4
×
F
.
2
Family
I
SQ
M
2
−B
is then defined as
SQ−B
=
s
(
g, κ, t
)
κ
g ∈ G
2
m−
1
2
I
∈
Z
4
×
F
so that each user is identified by an element of
G
.
Each user is assigned the collection
s
(
g, κ, t
)
m
−
1
|
κ
∈
Z
4
×
F
2
