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and near-optimality [8]. However, no design method for M -ary LCZ sequence
sets for general M has been reported.
In this paper we present a new method to construct an M -ary LCZ sequence
set from an M -ary sequence with good autocorrelation by using the interleaved
technique, where M is an even integer. Our construction method is flexible in the
sense of period, LCZ size, and alphabet size. We also show that the constructed
LCZ sequence sets are optimal or nearly optimal with respect to the Tang-Fan-
Matsufuji bound by giving some construction examples.
The outline of the paper is as follows. Section II gives some preliminaries
for our presentation. In Section III, we present a new design of an M -ary LCZ
sequence set from an M -ary sequence with good autocorrelation. In Section IV,
we discuss the optimality of the parameters obtained from our construction.
Finally, some concluding remarks are given in Section V.
2 Preliminaries
Let
. It is called an
N -periodic M -ary sequence or an M -ary sequence of period N if it satisfies
{
s ( t )
}
be an M -ary sequence over
Z M =
{
0 , 1 , ..., M
1
}
s ( t + N )= s ( t )
for all t and some positive integer N .Let
be two M -ary
sequences of period N . If there is no integer τ such that s 2 ( t )= s 1 ( t + τ ) for all
t ,theyaresaidtobe cyclically distinct . Otherwise, they are said to be cyclically
equivalent .The crosscorrelation C s 1 ,s 2 ( τ )of
{
s 1 ( t )
}
and
{
s 2 ( t )
}
{
s 1 ( t )
}
and
{
s 2 ( t )
}
is defined as
N− 1
s 1 ( t ) s 2 ( t + τ ) ,
C s 1 ,s 2 ( τ )=
t =0
and ω M =exp 2 π 1
M
.If s 1 ( t )= s 2 ( t ) for all t ,then
where s ( t )= ω s ( t )
M
C s 1 ,s 1 ( τ ) is called the autocorrelation of
{
s 1 ( t )
}
, simply denoted by C s 1 ( τ ).
If an N -periodic binary sequence
{
b ( t )
}
has
C b ( τ )= N, τ
0mod N
δ, τ
=0 mod N,
{
is called a binary sequence with two-level autocorrelation . In the particular
case that N
b ( t )
}
≡−
1mod4, f C b ( τ )=
1 for any τ
=0 mod N ,
{
b ( t )
}
is called
a binary sequence with ideal autocorrelation .
Let
S
=
{{
s i ( t )
}|
0
i
L
1
}
be a set of L sequences with period N .Fora
positive integer Z
2, let the crosscorrelation C s i ,s j ( τ ) between two sequences
{
s i ( t )
}
and
{
s j ( t )
}
in
S
satisfy
|
C s i ,s j ( τ )
|≤
η
for 0 <
|
τ
|
<Z when i = j ,and
|
τ
|
<Z when i
= j .Thenwecall
S
an
( N, L, Z, η ) LCZ sequence set, and Z the LCZ size of
S
.
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