Information Technology Reference
In-Depth Information
1.2 Preliminaries
The
M
2
-QAM constellation is the set
{
a
+
ib
|−
M
+1
≤
a, b
≤
M
−
1
,a,b
odd
}
.
When
M
=2
m
, this constellation can alternately be described as
√
2
ı
m−
1
2
k
ı
a
k
,
a
i
∈
Z
4
k
=0
where
√
2
ı
denotes the element (1+
ı
)=
√
2exp(
ı
2
8
). The 16-QAM constellation
is shown in Fig. 1. This representation suggests that quaternary sequences can
be used in the construction of low correlation sequences over these constellations.
Fig. 1.
16-QAM Constellation
Galois rings are Galois extensions of the prime ring
Z
p
.Let
R
=
GR
(4
,r
)
denote a Galois extension of
Z
4
of degree
r
.
R
is a commutative ring with
identity and contains a unique maximal ideal
M
=2
R
generated by the element
2. The quotient
R/M
is isomorphic to
F
q
, the finite field with
q
=2
r
elements.
As a multiplicative group, the set
R
∗
of units of
R
has the following structure:
R
∗
=
Z
2
r
−
1
×
F
2
×
F
2
...
×
F
2
.
r
times
Let
ξ
be a generator for the multiplicative cyclic subgroup isomorphic to
Z
2
r
−
1
0
,
1
,ξ,...,ξ
2
r
contained within
R
∗
.Let
−
2
T
T
{
}
T
is called
the set of Teichmueller representatives (of
F
q
in
R
). More details on Galois rings
can be found in [6,7,8].
denote the set
=
.
