04675749

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 12, DECEMBER 2008 5729
A New Systematic Construction of Zero Correlation
Zone Sequences Based on Interleaved Perfect
Sequences
Xiaohu Tang, Member, IEEE, and Wai Ho Mow, Senior Member, IEEE
Abstract—In the literature, many constructions of zero correla-
tion zone (ZCZ) sequences have been reported. While most of them
are suboptimal with respect to the known upper bound, some con-
structed by Matsufuji et al. and Torii et al. respectively, are almost
optimal (or even optimal). In this paper, we propose a systematic
construction of almost optimal ZCZ sequence sets which general-
izes the aforementioned constructions so that more flexible rela-
tionships between the set size and the sequence length are allowed.
In particular, the obtained almost optimal ZCZ sequence sets of
size mand length mnare new for 1 < gcd(m;n) < min(m;n). In
addition, their alphabets can be binary or nonbinary since our con-
struction is only based on interleaving perfect sequences according
to a certain orthogonal matrix.
Index Terms—Orthogonal matrices, perfect autocorrelation,
quasi-synchronous code-division multiple access (QS-CDMA),
shift sequence, zero correlation zone (ZCZ) sequence.
I. INTRODUCTION
TO implement an interference-free asynchronous code-di-
vision multiple-access (A-CDMA) communication appli-
cations, the ideal set of spreading sequences should have perfect
impulse-like periodic autocorrelation functions as well as all
zero pairwise cross-correlation functions. Unfortunately, such
an ideal sequence set does not exist as implied by the famous
Welch bound. One possible way to realize such an interfer-
ence-free system is to use the so-called orthogonal set of com-
plementary sequences [1]. However, this approach may be ex-
pensive to implement as it requires each user to be assigned mul-
tiple transmission channels.
Recently, a new class of sequences with the so-called zero
correlation zone (ZCZ) property (i.e., the sequence set has an
ideal periodic auto- and cross-correlation properties only within
Manuscript received May 11, 2006; revised April 12, 2008. Current version
published November 21, 2008. The work of X. Tang was supported by the Pro-
gram for New Century Excellent Talents in University (NCET) under Grants
04-0888, and the Specialized Research Fund for the Doctoral Program of Higher
Education (SRFDP) under Grants 20040613023. The work of W. H. Mow was
supported by the Hong Kong Research Grants Councils under project numbers
HKUST6113/04E and 617706. The material in this paper was presented in part
at the Second International Workshop on Sequence Design and Its Applications,
Shimonoseki, Japan, October 2005.
X. Tang is with the Provincial Key Lab of Information Coding and Transmis-
sion, the Institute of Mobile Communications, Southwest Jiaotong University,
Chengdu, China (e-mail: xhutang@ieee.org).
W. H. Mow is with the Department of Electrical and Electronic Engineering,
The University of Science and Technology, Clear Water Bay, Kowloon, Hong
Kong S.A.R., China (e-mail: w.mow@ieee.org).
Communicated by G. Gong, Associate Editor for Sequences.
Digital Object Identifier 10.1109/TIT.2008.2006574
a finite zone of time shifts) were introduced to implement an
interference-free quasi-synchronous code-division multiple-ac-
cess (QS-CDMA) communication system [2], [3], in which the
relative time delays between the signals of different users are re-
stricted to within a few chips. Therefore, by adding appropriate
cyclic prefix or postfix, such ZCZ sequences can be employed
to eliminate both multiple access interference and multipath in-
terference in a QS-CDMA system, provided that the ZCZ width
is longer than the multipath spread plus the multiple access rela-
tive delay [3]. To accommodate many multiple access users and
to ease the synchronization requirement, it is very desirable to
construct a ZCZ sequence set of a given length having both the
set size and the ZCZ width as large as possible. However,
the theoretical bound in [4] implies that . It
is therefore desirable to maximize the ratio .
In the literature, there are two main types of constructions
of ZCZ sequence sets. The first type is based on complemen-
tary sequence sets [5]–[9]. The second type is based on perfect
sequences [10], [11].1 The constructions presented in [10] and
[11] achieve , but must satisfy the con-
straints that or with the sequence
length , where the set size is and the length
of the perfect sequence involved is . Besides, these parameters
are limited by the existence of perfect sequences for a given al-
phabet. For example, 4-phase perfect sequences are unlikely to
exist for any lengths other than 2, 4, 8, and 16, according to a
well-supported conjecture [14] (cf. [15]).
In this paper, we interpret a ZCZ sequence of the second type
as an interleaved sequence whose component sequences are all
cyclically shifted versions of a perfect sequence. This allows us
to reduce the construction of a ZCZ sequence of length to
the construction of a length- -ary shift sequence that speci-
fies how a given perfect sequence of length is shifted to pro-
duce all of the required component sequences. A ZCZ sequence
set of size can then be obtained by scaling the component se-
quences differently according to an orthogonal matrix.
Our key contribution here is to propose a general construction
of shift sequences which can result in almost optimal ZCZ se-
quence sets, for all such integers and . In particular, our
construction includes as special cases the two known construc-
tions in [10], [11] with and , respectively,
1As pointed out by an anonymous reviewer, some new constructions of ZCZ
sequence sets based on perfect sequences were published in [12], [13] during the
review of this paper. We note that their main contributions are to produce novel
good ZCZ sequence sets with a wider range of parameters, and do not overlap
with our new constructions of ZCZ sequence sets which are almost optimal as
defined in Section II.
0018-9448/$25.00 © 2008 IEEE

5730 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 12, DECEMBER 2008
and produce new almost optimal ZCZ sequence sets for all other
choices of and .
In what follows, the essential notation and background will
be introduced in Section II. Section III proposes our system-
atic construction of ZCZ sequence sets from interleaved perfect
sequences and deduces three classes of shift sequences from
known constructions of almost optimal ZCZ sequence sets in
[10], [11]. Our new general construction of shift sequences are
presented in Section IV and are classified in Section V. Finally,
Section VI contains the conclusion.
II. NOTATION AND BACKGROUND
Given two length- complex sequences
and , the pe-
riodic cross-correlation between and is defined by
where the subscript addition is performed modulo and
* denotes the complex conjugation.
Definition 1: A sequence is said to
be a perfect sequence if for .
Definition 2: Let be a set of complex sequences of
length . Its zero correlation zone width is defined as
Moreover, is said to be an -ZCZ sequence set.
Lemma 1: (Tang–Fan-Matsufuji bound [4]) For any
-ZCZ sequence set, we have
(1)
Definition 3: An -ZCZ sequence set is said to be
optimal if , and almost optimal if .
Note that this definition of optimality is in a much stronger
sense than that of requiring the ratio to approach
one asymptotically, and any almost optimal ZCZ sequence set
can achieve for large enough
. Note also that, by definition, any optimal ZCZ sequence set
is almost optimal.
Following closely the notation in [16], we now introduce the
representation of an interleaved sequence and its associated shift
sequence. Let be a complex sequence
of length and let be an -ary sequence
of length defined over . Let be the
left cyclic shift operator such that denotes the -element
left cyclically shifted version of . An matrix can be
obtained by defining its th column as for
, i.e.
By concatenating the rows of matrix , one generates an inter-
leaved sequence of length . Mathematically,
(2)
where denotes the interleaving operator, and and are called
the component and shift sequences of , respectively.
Let us consider a left cyclically shifted version of defined as
. It was
shown in [16] that is just another interleaved sequence with
the same component sequence but a different shift sequence
. Namely, we have
with
if
if
(3)
Since two distinct shift sequences and may result in two
equivalent interleaved sequences and which are cyclically
shifted versions of each other, it is necessary to introduce the
corresponding notion of equivalence for two shift sequences.
Definition 4: Two shift sequences and
with are equivalent if there exist inte-
gers and with and such that
and satisfy (3).
Let denote another interleaved sequence with the compo-
nent sequence and the shift sequence
. Then the cross-correlation function at
shift between the
interleaved sequences and is given by
(4)
Finally, let us define a new operation. Let
be an matrix. Also let be an
matrix and be the th row of . Then is defined
as the set of matrices , where
and is the th column of for . Further,
if is orthogonal, is referred to as the orthogonality-
preserving transformation of by .
III. SYSTEMATIC CONSTRUCTION OF ZCZ SEQUENCE SETS
AND ALMOST OPTIMAL SHIFT SEQUENCES
Given a perfect sequence of length and an or-
thogonal matrix , our systematic procedure for constructing
ZCZ sequence sets consists of four steps, as follows.
Step 1: Generate an appropriate shift sequence of length .

TANG AND MOW: NEW SYSTEMATIC CONSTRUCTION OF ZERO CORRELATION ZONE SEQUENCES 5731
Step 2: From the perfect sequence and the shift sequence
, generate
(5)
Step 3: Form the set of matrices
by performing orthogonality-preserving transfor-
mation on .
Step 4: Concatenate the successive rows of ,
to produce the th interleaved se-
quence , resulting in the ZCZ sequence set
.
In the following, we state two theorems concerning the auto-
and cross-correlation properties of the sequences generated
by the above procedure. Their proofs are not difficult and are
omitted due to limitation of space.
Theorem 1: Suppose that is a perfect sequence of length
and is the interleaved sequence given by (2). Let be
the maximal nonnegative integer such that the shift sequence
satisfies the condition:
if
if
(6)
for any positive integer with
. Then is an -ZCZ sequence.
Theorem 2: Let be a perfect sequence of length and
be an orthogonal matrix. The sequence set produced
by the above procedure is an -ZCZ sequence set if
the shift sequence generated in Step 1 satisfies the condition
(6).
It is clear from Theorem 2 that the key to construct good ZCZ
sequence sets (with ) is to find appropriate shift
sequence that satisfies the condition (6) with as close to the
maximum value as possible. Parallel to the notions of op-
timal and almost optimal ZCZ sequence sets stated in Definition
3, we define the counterparts for shift sequences as follows.
Definition 5: The shift sequence is said to be almost optimal
(optimal) if it satisfies (6) with .
Next, we deduce three classes of almost optimal shift se-
quences from the almost optimal ZCZ sequence sets introduced
in [10] and [11], respectively. Note that the shift sequences in
case 1 of Theorem 3 is actually optimal.
Theorem 3: Let be the ZCZ sequence set generated by the
procedure in this section.
1) For , by setting is
an -ZCZ sequence set, where denotes
the multiplicative inverse of modulo .
2) For , by setting is an
-ZCZ sequence set.
3) For , by setting is an
-ZCZ sequence set.
The Proof of Theorem 3 follows from a detailed verification
of the condition (6) for each of the three cases.
Finally, we end this section with some remarks on the corre-
spondences between Theorem 3 and the constructions in [10],
[11], which are not at all obvious. The case 1) of Theorem 3 are
equivalent to Matsufuji et al.’s construction in [10], and the con-
structed ZCZ sequence set is actually optimal. The cases 2) and
3) of Theorem 3 are equivalent to Theorems 1 and 2 of Torii et
al. in [11], respectively, and both classes of ZCZ sequence sets
are almost optimal.
IV. NEW SYSTEMATIC CONSTRUCTION OF ALMOST OPTIMAL
SHIFT SEQUENCES
Let and be two nonnegative integers such that
with . Clearly, we have , where
denotes the largest integer no larger than , and for
and , otherwise. For given and , define
otherwise.
(7)
It is easy to see that as follows. Obviously,
for . When , we have =1 and
hence ; When , we have which leads to
.
Further, Let and . Define a function
with for
and . Then forms
a monotonic increasing sequence, which is shown in (8) at the
bottom of the page. Let . For ,
the subsequence
in the th row of (8) after computed modulo
enumerates all the elements of the set . Consequently,
the sequence (8) computed modulo enumerates the elements
of .
Base on , we introduce
. Then, is a permutation of the elements of ,
and its inverse is denoted by , i.e.,
for and . Now it is ready to define our
new almost optimal shift sequence as
(9)
Example 1: Six examples of almost optimal shift sequences
generated by (9) for various combinations of and are given
in Table I. Note that sequences 1 and 4 are optimal.
...
(8)

TABLE I
EXAMPLES OF ALMOST OPTIMAL SHIFT SEQUENCES
The next theorem presents a closed-form expression of the
shift sequences generated from (9).
Theorem 4: Given two integers and , let
and , where is defined in (7). Further let
denote the shift sequence produced
by (9). For and
, we have
or and
otherwise
(10)
where as defined in the beginning of this
section.
Proof: Applied to (9). Clearly if
or ( and . Otherwise
(11)
otherwise
(12)
since .
Let . Then the theorem follows from the
fact that , where the
second equality holds because .
To prove that the constructed shift sequences are indeed al-
most optimal, we need the following lemma which characterizes
the difference between any two distinct elements of the shift se-
quence.
Lemma 2: Suppose and are two distinct elements of the
sequence obtained from (9), for . Then
1) either or if .
2) . Furthermore,
(2.1) or only if , and
(2.2) or only if
, or ,
where is an integer satisfying .
Proof: Let and
, and or . Without
loss of generality, assume that which implies
or ( and ).
Notice that unless ,
which leads to item (1). It can then be deduced from (11) and
the definition of that
if and
otherwise
(13)
and
(14)
where
By (10), because . On the
other hand, if and , then
by (12). Otherwise,
from (13).
Based on the above, it is proved that .
In what follows, we derive the necessary conditions for four
specific cases, i.e., or .
Our discussion will be based on (8), (13), (14) and their analyses
presented.
1. .
It is impossible that and otherwise
. Then from
, we have and . Then
for .
2. .
I. When is the minimal value and
.
i) If , then and
.
ii) If , then from ,
and from , where
, and .
II. When or follows from
.
i) If , it is manifest that
and . That is .
ii) If , then or
, where .
3. .
Obviously and , otherwise
. Hence, or from
. Same as the Case 2-II,
we have
I. If , then and ,
i.e., and .
II. If , then or
where .

TANG AND MOW: NEW SYSTEMATIC CONSTRUCTION OF ZERO CORRELATION ZONE SEQUENCES 5733
4. .
It follows that and , which occurs
only for .
Summarizing the results in all cases, we complete the proof
of item (2).
Theorem 5: The shift sequence obtained from (9) is almost
optimal. In particular, it is optimal for .
Proof: It suffices to consider and in condition (6)
in the slightly larger range (i.e.,
but not ) for two cases.
Case 1: .
If , then or
because . Note that ; otherwise,
it implies that and contradicts the assumption
. By Lemma 2 item (2), we have
which together with indicates that and
or . It then follows from Lemma 2 item
(2.1) that .
Case 2: .
If , then
or . Since and
, we have or . Then equals
or for , or equals
or for . By Lemma 2 items (2.1) & (2.2), we
obtain , or , for . If
, then or from Lemma
2 item (1), which results in one of the two cases below:
1) or , and ;
2) or , and .
However, they contradicts all the corresponding Cases 1, 2-II-i,
3-I, and 4 in the proof of Lemma 2. To summarize, we have
and which follows from the fact that
or with .
Concluding from the above, the shift sequence obtained
from (9) satisfies the condition (6) for all
with and . In other words, for
the constructed shift sequences,
otherwise
where the last equality comes from the definition of in (7)
and the fact that . This completes the Proof of
Theorem 5.
Note that almost optimal ZCZ sequence sets can be obtained
by applying Theorem 2 to almost optimal shift sequences.
Note also that (9) can generate almost optimal shift sequences
for any two positive integers and . In particular, for
, the generated almost optimal
shift sequences and -ZCZ sequence sets are
new (cf. Theorems 3).
V. CLASSIFICATION OF ALMOST OPTIMAL SHIFT SEQUENCES
In the previous section, a general class of almost optimal shift
sequences have been introduced. Our computer search results
suggest that many more almost optimal shift sequences satis-
fying condition (6) can be found. In this section, we catego-
rize these shift sequences into four classes and discuss their
uniqueness.
Class 1: For , there are many almost optimal
shift sequences but there seems to be only one optimal shift
sequence , which
is given in Case 1 of Theorem 3 and corresponds to the optimal
ZCZ sequences of Matsufuji et al. in [10].
Since and , we have
and . In this case,
and . By substituting
and ,
we obtain for
which coincides with (10). Therefore, the optimal shift sequence
obtained from (9) is the same as
.
Class 2: For , up to the cyclically shift equivalence (cf.
Definition 4), there appears to be one almost optimal shift se-
quence , which is given in Case 2 of Theorem
3 and corresponds to one construction of almost optimal ZCZ
sequence sets of Torii et al. in [11].
Let be the almost optimal shift
sequence obtained from (9). From (10), the expression of is
and . It can
be seen that the almost optimal shift sequence obtained from
(9) is equivalent to
with respect to Definition 4 for and .
Class 3: For , there are many almost optimal shift se-
quences. One class of almost optimal shift sequence sets
is given in Case 3 of The-
orem 3, which corresponds to another construction of almost
optimal ZCZ sequence sets of Torii et al. in [11]. In the fol-
lowing, we prove that the almost optimal shift sequence
obtained from (9) is new and the two classes
of shift sequences are inequivalent for .
Suppose that there are two integers and
satisfying (3). Immediately, we have , and
if or
if . Because
, . Thus
if , or if . Then
has to be since or from (12), which implies
. The condition (3) results in the following one-to-one
correspondence between the elements of the two sequences:
However, as or . Thus
there does not exist satisfying (3) and
it can be concluded that the two shift sequences are inequivalent
with respect to Definition 4.

Example 2: Let and . We obtain two in-
equivalent shift sequences and
.
Class 4: For , there are many
almost optimal shift sequences and (9) only produces one of
them.
Example 3: For and
and are
two inequivalent almost optimal shift sequences, where is
generated from (9).
VI. CONCLUSION
We introduced a systematic construction of almost optimal
ZCZ sequence sets of size and length based on inter-
leaving properly shifted versions of a length- perfect se-
quence according to a shift sequence as well as an or-
thogonal matrix. This reduces the problem of constructing al-
most optimal ZCZ sequence sets to that of constructing almost
optimal shift sequences. A general construction of almost op-
timal shift sequences for all positive integers and have
been derived. It is also proved that the constructed almost op-
timal -ary shift sequences of length are new for and
for . In particular, our results lead
to almost optimal ZCZ sequence sets with
which have not been reported before, to the best of
our knowledge. Finally, it is noteworthy that the alphabet size
of the new ZCZ sequences depends on those of the perfect se-
quence and the orthogonal matrix used.
ACKNOWLEDGMENT
The authors wish to thank the referees for their helpful com-
ments.
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