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Here, we restrict our attention to DS-CDMA. It is algebraically convenient to
design sequence families with good periodic correlation properties and there are
benchmarks to measure how good such a design is, namely the Welch [12] and
Sidelnikov bounds [7]. The aperiodic correlation properties also play a significant
part in system performance for the case of Galois ring sequences, the aperiodic
correlation was investigated in [11]. Another significant contributor to the per-
formance, especially in the current wireless environment where longer and longer
sequence periods are necessary to support an increasing number of users (the
family size is typically an increasing function of the period), is the partial period
correlation, which is the main focus of this paper.
In this paper we have obtained new results on the partial period correlations
of families A, B and C. This substantially extends the results we have obtained
in [2].
The paper is organised as follows. In Section 2, we provide a brief overview
of the structure of Galois rings and properties of the Galois ring trace function,
after introducing some definitions and notation for general sequence designs.
This is followed by the definition of
Families A, B and C
. This section con-
cludes with a discussion of a related Cayley table and its properties. In Section
3 we obtain the first moment of the partial correlation function of the Galois
ring
m
sequences in families A, B, and C. In Section 4 we obtain so-called lo-
cal and global second moments of the partial correlation function of
Family A
.
Section 5 concludes the paper.
−
2 Rings, Trace Functions and Sequences
2.1 Galois Ring Preliminaries
We will be quite informal, highlighting the details we need, and ask the reader
unfamiliar with the topic to consult [1].
We denote the Galois ring as
R
=
GR
(4
,n
) and note that it is a Galois
extension of
Z
4
, defined by
R
=
Z
4
[
β
]where
β
has multiplicative order 2
n
1
and is a root of a primitive basic irreducible polynomial (i.e., a basic irreducible
polynomial whose modulo 2 reduction is a primitive polynomial over
Z
2
). It is
always possible to construct such a polynomial. Note that the ring
R
contains
4
n
elements, and
R
=
−
1
,β,...,β
n−
1
as a
Z
4
−
module.
R
has a unique 2-adic representation
c
=
a
+2
b,
where
a, b
belong to the Teichmuller set
Every element
c
∈
0
,
1
,β,...,β
2
n
−
2
T
=
{
}
a
is given by
α
(
c
)=
c
2
n
.
Given
c
, after determining
a,
as above,
b
can then be solved for. If we denote the modulo 2 reduction
function by
μ
and extend it to polynomials in the obvious way, then
μ
(
and the map
α
:
c
→
T
)=
0
,
1
,θ,...,θ
2
n
−
2
=
GF
(2
n
). The set of invertible elements of
R
is denoted
{
}
R
∗
=
R
2
R
where 2
R
is the set of zero divisors and is the unique maximal ideal
in
R.
Every element in
R
∗
has a unique representation of the form
β
r
(1+2
z
)
,
0
\
≤
