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1=
2
T−
1
4
d
−
respectively. If 0
≤
i
=
j
≤
f
−
1, we have
2
d
≤
2
ad,
2
T
−
2
ad
≤
2
T
−
2
d,
2
d
+1
≤
2
bd
+1
,
2
T
−
2
bd
−
1
≤
2
T
−
2
d
−
1
.
Thus
|
C
(
i,m
)
,
(
j,n
)
(
τ
)
|≤
2
≤|
τ
|
<
2
d
and
i
=
j
.If
i
=
j
,wehave
for 0
2
ad
≡
2
T
−
2
ad
≡
0mod2
T,
2
d
+1
≤
2
bd
+1
,
2
T
−
2
bd
−
1
≤
2
T
−
2
d
−
1
.
Therefore, it is easily checked that
|
C
(
i,m
)
,
(
i,n
)
(
τ
)
|≤
2
for 0
<
|
τ
|
<
2
d
and
m
=
n
,andfor0
≤|
τ
|
<
2
d
and
m
=
n
.
T−
1
The case that
d
|
may be proved in a similar way.
2
For any even integer
M
, we showed that an
M
-ary LCZ sequence set of period 2
T
can be constructed from a
T
-periodic
M
-ary sequence with good autocorrelation.
In addition, it is possible to select any even integer 2
≤
Z
≤
T
−
1astheLCZ
size.
Remark
: It is easily checked that the construction in [8] is equivalent to replac-
ing the component sequences in (2) by
s
(
u,
0)
(2
t
)=
s
(2
t
−
u
)
,s
(
u,
0)
(2
t
+1)=
s
(2
t
+
u
)
,
s
(
u,
1)
(2
t
)=
s
(2
t
−
u
)+1
,s
(
u,
1)
(2
t
+1)=
s
(2
t
+
u
)
}
with ideal autocorrelation. Thus, our construction is a modified generalization
of the construction in [8] in the sense that the alphabet size and period are more
flexible.
≤
u<
2
n−
1
−
2anda(2
n
−
{
s
(
t
)
for some integers 1
1)-periodic binary sequence
4 Optimality of Constructed LCZ Sequence Sets
Tang, Fan, and Matsufuji derived a bound on LCZ sequence sets as in the fol-
lowing theorem [12].
Theorem 3 (Tang, Fan, and Matsufuji, [12]).
Let
S
be an
(
N, L, Z, η
)
LCZ
sequence set. Then we have
N
−
1
LZ
−
1
≤
η
2
/N
.
1
−
