Binary Almost-Perfect Sequence Sets - Semantic Scholar
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Binary Almost-Perfect Sequence Sets Kai Cai, Guobiao Weng, and Xueqi Cheng, Member, IEEE
Abstract—Sequence set with lower correlation values is highly desired for engineering applications. However, p theoretical results (e.g., Welch bound) show that max N in general, that is, the maximum out-of-phase autocorrelation and cross-correlation magnitudes of a sequence set is not less than the square root of the sequence period. In this paper, we propose a new concept, namely almost perfect sequence set (APSS), which has the property max c except for at most m shifts, where c and m are predefined small integers. A uniform method is presented to construct APSS and then the properties of such APSS are discussed. Moreover, a distance inequality on the APSS with m = 1 is obtained and several APSS families such as (2p; 8p +2; 6; 4)-APSS and 3p; 64p3 +8 ; 9; 9 -APSS for any prime p 5 are constructed based on Paley and Paley partial sequences. Finally, it shows that the APSS can be used to construct LCZ sequences and the properties of such LCZ sequences are presented. Index Terms—Almost perfect sequence set (APSS), correlation function, LCZ sequences, Paley and Paley partial sequences.
I. INTRODUCTION
L
ET and be two binary sequences of period , where , for . The (periodic) cross-correlation function between and is defined as1
where . When , it is being known as the (periodic) . autocorrelation function of and denoted by Sequences with good autocorrelation property are widely used in the engineering applications. However, some applications, e.g., direct-sequence code-division multiple access Manuscript received November 16, 2007; revised February 06, 2010. Current version published June 16, 2010. The material in this paper was partly based on the Ph.D. dissertation by K. Cai, “Difference sets and sequences with good correlation properties,” School of Mathematical Sciences, Peking University, Beijing, China, 2004. The work of K. Cai was supported by the NSF of China (No. 60872036 and No. 60873243) and by the 863 Project of China (No. 2008AA01Z140). The work of G. Weng was supported by the NSF of China (No. 10826072). The work of X. Cheng was supported by the NSF of China (No. 60933005). K. Cai is with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]; caikai@software. ict.ac.cn). G. Weng was with the School of Mathematical Sciences, Peking University, Beijing 100871, China. He is now with the Department of Applied Mathematics, Dalian University of Technology, Liaoning, China (e-mail: [email protected]. cn). X. Cheng is with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China. Communicated by G. Gong, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2010.2048490 1It
is also defined as C
w
( )=
0
( 1)
in some literatures.
(DS-CDMA) system require sequences with both good autocorrelation and cross-correlation properties. Many sequences have been constructed to meet such demands, including which, Gold sequences, Gold-like sequences [1], [2], and Kasami sequences [3] are the most well-known families. For the latest advances on this topic, the reader can refer to [4] and [5] and their references. The well-known Sarwate bound [6] (I.1) uncovers the relation between the peak cross-correlation magniand the peak out-of-phase autocorrelation magnitude tude for sequences of period . Let . By (I.1), one can easily deduced that for . The result indicates that it is impossible to construct sequence set with both very low autocorrelation magnitudes and very low cross-correlation magnitudes. As an example, we give a classical result comprising the works of Gold and Kasami [7]. Theorem 1.1: Let be m-sequences with period , be the -decimation sequence (see Section II-B and let , or , for the definition) of , where is odd, then takes three values such that as follows:
Note that in Theorem 1.1, when , the maximal magnitude of takes on its smallest value . One method to get “lower” correlation sequences is to construct low correlation zone (LCZ) sequence set [8], [9]. -LCZ sequence set is defined as a sequence A sequences with period such that set which contains for and for for some small value . The LCZ sequences is useful in approximately synchronized CDMA (AS-CDMA) systems. Another probable way to get “lower” correlation sequences is to construct sequences with both low autocorrelation magnitudes and low cross-correlation magnitudes except for a few (as small as possible) exceptional shifts. In this paper, we propose this concept as almost perfect sequence set (APSS), which is formally defined as follows. is called a Definition I.2 (APSS): Sequence set -APSS, if contains shift-distinct sequences of period and satisfies the following two conditions: and each , 1. For each except for at most . 2. For each and each , except for at most .
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Moreover, when , is called a -perfect seand , is called a -pair. quence set (PSS); when In [10], Wolfmann proposed the notion of the almost perfect autocorrelation sequence, which has optimal autocorrelation magnitudes except for one possible shift. By Definition I.2, -APSS. In what follows, his construction is initial a we shall only concern on the binary APSS. As to the general complex-valued sequences, one can give discussions similarly. The main contributions of this paper can be illustrated as follows. We proposed a uniform method to construct APSS. The main idea of this method is to find PSS (or specifically, -pair) first, and then construct APSS based on the technique of sequence interleaving. It shows that this easy-to-implement method can effectively produce a lot of APSS families. Since our construction is PSS-based, a necessary condition on the existence of the PSS is also presented. We obtain several APSS families such as -APSS, -APSS, and -APSS based the perfect
-pairs and
-APSS, -APSS based on Paley and Paley partial -pairs. Note that the parameters and of these sequences do not increase even the period and the size of the sequence set approach infinity. Beyond these constructions, there are many potential ways to construct APSS. For example, the interleaving method proposed in [11] and the subfield decomposition method proposed in [12]. We also give discussions on these existed APSS families. We use the APSS to construct LCZ sequence sets. It shows that the APSS can provide a large number of LCZ sequence sets with a variety of lower correlation zones. Although these LCZ sequences are not optimal, we believe that these families may be useful in some particular applications since they provide more choices of LCZ sequences. The rest of the paper is organized as follows. In Section II, we introduce some basic notations and necessary results for the APSS construction. In Section III, we propose the PSS-interleaving based method and also a necessary condition on the existence of the PSS. In Section IV, we construct several families of APSS and also give discussions on the existed families. In Section V, we construct LCZ sequences by using the APSS, and finally we conclude this paper in Section VI. II. PRELIMINARIES
. represents the inverse of , , where ,
• Let i.e., for
.
B. Multiplier and Decimation Let
be a subset of is defined by
. The characteristic sequence if otherwise,
where is the -th component of . Meanwhile, is called the support of . A (Hall) multiplier of is a automorphism of such that , for some , where and . The multiplier group of is composed of all the multipliers of , a subgroup of and denoted . , be a binary sequence of Let , and be an integer relatively prime to . period is called the -decimation sequence of . By abusing the denotation to represent either an integer relatively prime to or an automorphism of , the following result holds. Theorem 2.1 ([13]): Let group of . Suppose that
•
denotes the left shift operation on sequences. Let . Then , and for are called the shifts of . and . • Let denotes that , for . and are called shift-equivalent if there exists an integer such that . Otherwise, and are called shift-distinct. Note that the sequences in the APSS need to be shift-distinct from each other.
and
be the multiplier
be the coset decomposition of with respect to , and . Here is the Euler -funcwhere tion. Let be the characteristic sequence of . Then, compose all the decimation se1) quences of ; is , for . 2) The support of C. Difference Function of (where The difference function ) is defined as , where and . The following theorem shows that the cross-correlation function of sequences and can be determined by the difference function of their supports. , and Theorem 2.2 ([13]): Let be two binary sequences of period with , Then supports and , respectively. If
In this section, we briefly introduce some notations and results, which will be useful in the construction of the APSS. A. Notations
of
(II.1) When and
as
and, hence, and
, by denoting , respectively, (II.1) becomes , which is a well-known result
deduced by Ding in [14]. Two sequences are called to have the same autocorrelation spectrum if their autocorrelation functions have same images and each value in the images occurs the same time. Theorem 2.3: Let be a binary sequence with period , and be an integer relatively prime to , then and have the same autocorrelation spectrum.
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D. Cyclotomic Classes be the finite field with elements, and , i.e., for some inbe a primitive element of , deand , for . These ’s are called cyclotomic classes of order . The cyclotomic with respect to are defined as numbers of order . The following two families of cyclotomic numbers (refer to [15]) can be used to construct Paley and Paley partial sequences. (1) If for even , then and Let divide let teger . Let fine
(2) If
for odd , then
Definition 2.4: Let prime. The characteristic sequence of Paley (Paley partial) sequence with period .
different interleaving order yields It should be noted that: may be difdifferent results, e.g., generally, ; and need not be ferent with different for , e.g., it is permitted to interleave a sequence itself for several times. The correlation functions of the interleaved sequence can be represented by the correlation functions of their base sequences. , Lemma 3.1: Let sequences with period , and let . Then, for , we have
be , and
and be a is called the
III. PSS-INTERLEAVING-BASED CONSTRUCTION In this section, we propose a uniform method to construct APSS based on the technique of sequence interleaving and the assumption of the existence of the PSS. A. Sequences Interleaving The idea of sequence interleaving exists widely in the area of sequences design [16]. For example, Gong proposed the notion of -ary interleaved sequences and shown that a large number of popular sequences, such as multiplexed sequences, clock-controlled sequences, Kasami sequences, GMW sequences, geometric sequences, mapping sequence, and No sequences have particular interleaving structures [17], [18]. More literatures using interleaving techniques can be found in almost difference sets design [19] and LCZ sequences design [11]. In fact, most known LCZ sequences have interleaving structures [12]. In the mentioned literatures, the base sequences used for interleaving usually involve an ideal autocorrelation sequence and its shifts [17], [18] and/or its inverse [11], [19]. Unlike these sequences, the APSS constructed in this paper is based on the PSS, for which one needs to concern the autocorrelations and the cross-correlations between each sequence pair. Therefore, we first give the correlation properties for the interleaved sequences of a more general form. , for be Let sequences with period . The interleaved sequence of is a sequences defined as with period
Generally, sequences) of .
is
denoted by and are called as the base sequences (or column
Proof: We just prove item (4). For the other items, one can prove them following a similar way.
B. PSS-Interleaving Method -PSS , we will construct an Given an arbitrary for each . Note that the construcAPSS, namely, tion is based on the assumption of the existence of PSS. Here, we set a large to ensure this assumption and leave the detailed discussions later. be a -perTheorem 3.2: Let fect sequence set. Then
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is a
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-APSS with
.
and ; , for ; are shift-distinct sequences if and are shift-distinct; is a shift of if and only if for 4) .
Proof: Let
, and . Let , be the exceptional value of the cross-correlation function between and , and be the excepand tional value of the cross-correlation function between , for . Let
.. .
.. .
..
.
.. .
.. .
.. .
..
.
.. .
Proof: 1) is obviously. 2) can be checked straightforward from
For item 3), we prove that for
is shift-distinct with . Let One can see that any shift of has the form of either or for some and . If , then , has the least period . If which is contradict to that , then , which is contradict to that is shift-distinct from . The contradictions yield that are shift-distinct sequences, which proves item 3). be a shift of , then has For item 4), let the form of either or . If , then we have and . If , then we have and . On , then , the other hand, if which is obviously a shift of . Hence, 4) is proved. By all the discussions above, the lemma follows.
Note that the elements of and represent the positions of , the exceptional shifts, and for , . only conBy Lemma 3.1, it is not difficult to get that and only contribute to tributes to , for . Thus, except possible shifts, which are listed as the values of the entries of , and possible shifts, which are listed as the values of the entries of (see the equation shown at the bottom of the and take their absolute values less than . page), both By the definition, the sequences in an APSS need to be shiftgendistinct. Hence, it is difficult to give the exact size of . erally, but we can see that it is upper bounded by By above discussions, the theorem is proved.
With the help of this lemma, we now can get the exact size of . Theorem 3.4: Let quence set with the least period
By Theorem 3.2, given a -PSS, we can always ob-APSS for each . Obviously, a small tain a leads to a small and also a small number of the exceptional shifts as we expected. However, one can see that may be a quite loose bound. In the following, we deduce the for and . exact size of 1) The Case of : In this case, by Theorem 3.2, one can construct APSS with 4 exceptional shifts. To decide the exact size of the constructed APSS, we first need a lemma. Lemma 3.3: Let , , least period . Then
and
if and only if
1) 2) 3)
is a
. Let
-APSS for odd -APSS for even Proof: Let
be binary sequences with the
be a perfect se. Then
.. .
.. .
..
.
.. .
.. .
.. .
..
.
.. .
.
and/or a
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By the definition (III.1) By Lemma 3.3, it is obvious that for and for , where denotes the integral part of a real number. equals the sum of enThus, the number of sequences in tries of the upper triangular matrix
.. .
.. .
..
.
That is, for odd , which proves the theorem.
and
.. . . In other words, for even
2) The Case of : In this case, one can construct APSS with 9 possible exceptional shifts. To decide the exact size of , we first need a lemma. Lemma 3.5: Let least period . Then, 1) ; 2) for
and if and only if
be sequences with the ,
, and
and (III.2) It is not difficult to check that (III.1) and (III.2) are equivalent , they have no solution. Thus, equations. When each equivalence class of contains 3 sequences. When , they have a solution , , and when , they have a solution , . , each equivalence class of contains 3 Thus, when sequences except one class which contains one sequence. That is , contains shift-distinct sequences and to say, when when , contains shift-distinct sequences. The lemma is proved. By the above discussions, it can be seen that the number of shift-distinct sequences in equals the number of elements in , where the equivalent relation is decided by , for . As for . an example, we give Example 3.7: For , consists elements, which are illustrated as , in the fol-th position is means that lowing table. Note that the belongs to the equivalent class . For example, , , and etc.
Proof: It can be proved through straightforward checking. Lemma 3.6: Let be shift-distinct sequences with the lest period . Then, have 1) shift-distinct sequences; 2) have shift-distinct sequences; 3) have shiftdistinct sequences for , and shift-distinct . sequences for Proof: By Lemma 3.5, in cases 1) and 2), different choices of and shall result in shift-distinct sequences. Thus, all and have shift-distinct sequences. Now we prove 3). First, note that there is an equivalent resuch that for , if and only if lation on and are shift-equivalent. Thus, the number of shift-distinct sequences equals the number of equivalence classes in . By Lemma 3.5, , , are equivalent seand quences in . Consider the following equations:
One can check that except , each , for contains three elements. All the elements in one correspond to one class of shift-equivalent sequences in . For example, corresponds to the shift-equivalent sequences , and . , consists elements, For which are illustrated in the following table. Note that except , contains 3 elements. each , for
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For , consists elements, which are illustrated in the following table. Note that each , for contains three elements.
C. Necessary Condition on the Existence of PSS In Section III-B, a large number of APSS can be obtained by using the technique of sequence interleaving. However, according to Theorem 3.2, the constructed APSS is based on some -PSS. Hence, the existence of -PSS is an important issue for APSS constructions. In the following, we give -PSS. a necessary condition on the existence of the Lemma 3.10: Let be a -pair with period . Then, such that there exists some or , where denotes the Hamming distance2 between and . Proof: Let
, and
, for
. Using
Theorem 3.8: Let quence set with the least period
be a perfect se. Then
. Let
we have
The inequality can be rewritten as is a
-APSS when
and/or a
-APSS when
.
be shift-distinct sequences. ObProof: Let , and each of them contains viously, shift-distinct sequences. Similarly, let be mutually shift-distinct. Then and each of sequences. Note that them contains and for . Thus, by Lemma 3.6, when
Thus, we have
It means that That is to say,
(Distance Inequality): Let be a -PSS. Then, for , there exists some such all or . that A binary sequence is called perfect if it has a 2-level autocorrelation function with the off-peck autocorrelation coefficients (Note that sequences with rarely exist, see [20] for reference). By using this definition, we have the following result, which is a direct corollary of Lemma 3.10.
,
The theorem is proved. Using the same notations as above, the following theorem can be obtained directly by Lemma 3.6. Theorem 3.9: Let period . Then
is a
.
By this lemma, the following theorem is obvious. Theorem
when
or or
3.11
. If Theorem 3.12: Let be a PSS such that is perfect, then . , The theorem shows that except the trivial -pair exists. For the sake of conveno other type of PSS with nience, we call it as the perfect -pair.
be a -pair with the least IV. APSS FAMILIES
-APSS when -APSS when
and/or a .
In this section, we discuss the APSS families constructed by the perfect -pairs, by the Paley and Paley partial -pairs and the other families already in the existed literatures. 2the Hamming distance between two binary sequences of equal length is the number of positions for which the corresponding symbols are different.
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A. APSS Based on Perfect -pairs
where
Let be a perfect sequence with period . According to last froms a perfect -pair. By Theorems 3.4 and 3.9 section, and using the same notations, the following APSS families can be obtained directly.
. By Theorem Proof: The multiplier group of is decimation sequences, i.e., the 2.1, it has
be a perfect -pair with the
Theorem 4.1: Let least period . Then
is a
and characteristic sequences of . The values of and as and let likely way. Denote
Theorem 4.2: Let least period . Then
. We just calculate can be decided by a . For ,
if if
-APSS.
Proof: Note that the off-peck autocorrelation coefficients for perfect sequences and . By (II.1), , thus and the result follows.
is a
be a prime.
, .
Thus, by (II.1),
be a perfect -pair with the if if if -APSS when
-APSS when
and/or a .
B. APSS Based on Paley and Paley Partial -pairs In this subsection, we first give a family of -pair with for any prime . The APSS families with parameters , , and are then obtained. 1) Paley and Paley Partial -pairs: For any prime , we shall construct a -pair with period and . The construction is based on Paley and Paley partial sequences. Lemma 4.3([13]): Paley sequence with period has two decimation sequences, say and , and their correlation functions are listed as follows:
, ,
The result follows. By Lemmas 4.3 and 4.4, for any prime , there ex. More preciously, when ists a -pair with period and , the two decimation sequences of Paley sequences form such a -pair, namely the Paley -pair; when , the two decimation sequences of Paley partial sequences form such a -pair, namely the Paley partial -pair. Formally, we rewrite this as a theorem. Theorem 4.5: For any prime . period and
, there exists a -pair with
2) Paley and Paley Partial -pair Based Construction: Based on Paley and Paley Partial -pairs, we give the families of APSS using the PSS-Interleaving method of Section III. Theorem 4.6: For any prime -APSS. Proof: Let sequence when
and
, there exists a
be the decimation sequences of Paley and Paley partial sequence when
. Let
where be a prime. Following a similar way in [13], we get the correlation functions of Paley partial sequences. Lemma 4.4: Paley partial sequence with period has two decimation sequences, say and , and their correlation functions are listed as follows.
By Theorem 3.4, for any odd prime -APSS, which proves the theorem. Theorem 4.7: For any prime -APSS.
,
is a , there exists a
Proof: By Theorem 3.9 and using Paley and Paley partial -pairs, such APSS can be directly constructed. The constructions illustrated by Theorems 4.6 and 4.7 can be, respectively, extended when Paley (and Paley partial) -pair is extended to perfect sequence set , as shown follows.
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Theorem 4.8: For any prime -APSS.
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, there exists a
Proof: Let , be the Paley and/or Paley partial -pair is as aforementioned. Then we declare that a perfect sequence set. First, for arbitrary binary sequences and with period , it is obvious that , and for . Secondly, one can easily see that are shift-distinct. Thus, the four sequences in form a PSS. By Theorem 3.4, one can construct a -APSS for any prime . Theorem 4.9: For any prime -APSS.
, there exists a
Proof: Let be the perfect sequence set used in the proof of Theorem 4.8. By Theorem 3.8, the result is directly obtained. A (38,39,6,4)-APSS constructed by Paley -pair is given as follows. Example 4.10: The decimation sequences of Paley sequence with period 19 are C. Other Existed Families and
The corresponding (38,39,6,4)-APSS constructed by and is:
Here, we shall discuss the sequences in the existed literatures that can be considered as APSS, i.e., the construction based on interleaving of perfect sequences [11] and the construction based on subfield decomposition [12], [16]. Both of them are intermediate sequences for constructing LCZ sequences. 1) Perfect Sequence Based Construction: In [11], the authors constructed several families of LCZ sequences by using the interleaving technique. Among which, one of their construction was choosing sequences from a larger sequence set which is in fact an APSS. Now we restate their result in terms of almost perfect sequence sets. Theorem 4.11 ([11], Theorem 3): Let be a perfect sequence . Let with period
Then is a -APSS. sequence when interleaving By Lemma 3.3, it yields and sequences when interleaving for . Thus the number of shift-dis. It can be seen that this tinct sequences in is APSS is in fact a subset of the APSS constructed in Theorem 4.1. 2) Subfield Decomposition Based Constructions: There are a large number of -ary LCZ families constructed by the method of subfield decomposition [12], [21]–[23]. In [12], the authors connected these families by using a unified subfield decomposition method, and they also define the almost low correlation zone (ALCZ) signal set, which permits a high in-phase cross-correlation value. In fact, these constructions can also be regarded as particular almost perfect sequence sets, as stated below.
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for contributes to
. By the proof of Theorem 3.2, only , only contributes to , and only contributes to . Thus, , for and for , which is a LCZ sequence set with low correlation indicates . Obviously, the set size of is upper zone bounded by , and the theorem is proved.
TABLE I APSS CONSTRUCTED BY THEOREM 4.12 WITH SMALL q AND m
Theorem 4.12 ([12], Theorem 2): 3Let be the set of all subfield reducible sequences with the trace representation , where is a fixed function from to with the twotuple balance property and the evaluation of ’s runs through the with the balanced set of all the shift-distinct sequences over . Then is a property of period -APSS with . Table I lists the parameters of the structed by Theorem 4.12 with small and
-APSS con.
Remark 4.13: From Table I, One can see that the APSS constructed by the subfield decomposition method always have a smaller size and more number of exceptional shifts. This is because these ALCZ sequences prefer a larger LCZ rather than a less number of exceptional high correlation shifts, and more importantly, all the exceptional shifts of these ALCZ sequences are for , at same positions, i.e., which is a stronger property than what a general APSS needs. Remark 4.14: All the LCZ sequences constructed in [12], [21]–[23] are the subsets of the corresponding ALCZ sequences of Theorem 4.12. Thus, they are almost perfect sequence sets with the same number of exceptional shifts but a smaller set size. V. LCZ SEQUENCES FROM APSS In this section, we construct LCZ sequence sets based on APSS. A new scene of the construction is that it can provide a large number of LCZ sequence sets with a variety of lower correlation zones. be a -PSS and be the Let be a subset sequence set defined in Theorem 3.2. Let such that for all and of , the following items hold: 1) , for , ; , for , 2) and are the exwhere ceptional values of and , respectively. Using this denotation, we have the following. Theorem 5.1: quence set with
is a
-LCZ se.
Proof: Let , where
, ,
, and
, and ,
3For the definitions of the subfield reducible sequence, balance property, twotuple balance property, trace representation, and evaluation, the reader can refer to [12] for the details.
From Theorem 5.1, one can see that the LCZ of depends on parameters and . Thus one can get particular LCZ sequences by choosing suitable and . Especially, we have the following. Corollary 5.2: is a -LCZ sequence . set when This result indicates that when the base sequences satisfy (it seems easy to be satisfied), one can construct LCZ sequence set with low correlation zone by interleaving the base sequences times. However, the correlation values within the LCZ will increase accordingly. , it is obviously To construct LCZ sequence sets from that one can use exhaustive computer searches beginning with a particular sequence (that is, the exceptional correlation shifts of ) in based on Theorem 5.1. On the its base sequences other hand, we can also have the following constructive results. -perfect sequence set Definition 5.3: A is called normalized if for , i.e., all the exceptional cross-correlation shifts are at position 0. Note that a -pair can be normalized by proper shifting one of the two sequences. Hence, the normalized perfect sequence -normalized perfect seset always exists. Let be a quence set. Take , and for each , let be a set of integers such that and , for and . Denote the collection of all such -integer sets as and let . We have the following result. Lemma 5.4: . be the subset of such that for Proof: Let . Without loss of generality, we calculate . each Obviously, it equals the number of ways for choosing elements from such . It also equals the number that elements from of ways for choosing such that . Let , then it equals the number different elements of ways for choosing from since a set elements in such that corresponds to a of set of elements in such that , and vice versa. The latter is known as . Thus , including which each is multi-calculated times. Hence, there are different ways for choosing from , and we are done.
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Theorem 5.5: Let be a -normalized perfect sequence set. Then, for all and such that , there exist different -LCZ . sequence sets within Proof: Since is normalized, then and , the exceptional shift of for any is . Now we choose integers such that and , for and , where . There are different such -integer -integer set, it can yield difsets. For each -LCZ sequence sets, because there ferent are ways to arrange the sequences, of which , for each one contains base sequences with the form and . Since there are ways to choose , the total number of the resulted LCZ sequence sets is , which proves the theorem. , the exceptional shift for each pair of sequences When and in is completely decided by . By similar arguments as Theorem 5.5, we have be a Corollary 5.6: Let malized perfect sequence set. Then, for all and , there exist -LCZ sequence sets within .
-nordifferent
Remark 5.7: By the definition, unlike the APSS, the sequences within a LCZ sequence set need not be shift-distinct. when So we also consider the shifts of the sequences of using it to construct LCZ sequence sets. Example 5.8: Reconsider the Paly -pair in Example 4.10 with period , where One can see that it is a normalized PSS. Let . Then
,
. and
One can check that there are 19 ways to choose 6 integers with each pair having the distance . Take . There are
ways to arrange into 3 disjoint strings with each string contains 2 numbers. For example, is such an arrangement and is another one. Each arrangement corresponds to different (38,3,4,6)-LCZ sequence , sets. For example, are 2 of 64 such sequence . In all, there sets corresponds to different (38,3,4,6)-LCZ exist , sequence sets based on this Paly -pair. When , and , by similar discussions, one can obtain
different (19,6,2,3)-LCZ sequence sets. There are two issues should be mentioned. The first one is: the LCZ sequence sets constructed in Theorem 5.5 is in general not optimal with respect to Tang–Fan–Matsufuji bound4 [24]; The second one is: a large number of LCZ sequence sets with a variety of different parameters can be constructed from a per, a large fect sequence set, and even for a particular are correnumber of different LCZ sequence sets within sponded. For the first issue, it is an interesting problem to construct optimal LCZ sequence sets based on the perfect sequence set. For the second issue, it may be useful in some particular applications since it provides more choices of LCZ sequences. VI. CONCLUDING REMARKS In this paper, we proposed the concept of almost per, fect sequence sets and constructed , , , and -APSS based on perfect sequences, Paley sequences and Paley partial sequences, respectively. It is important to note that the cross-correlation between the decimation of Paley partial sequences is a 2-level function. It has been an open problem whether this is possible except Kasami cases [5]. The construction of APSS proposed in this paper was based on the existence of PSS. We gave a necessary condition on the existence of such sequence sets. If we define -clique code as a binary sequence set, of which each pair has Hamming distance less than , and define maximal -clique code as a -clique code with maximal number of sequences, i.e., it will not be a -clique code by adding any other sequence, Theorem 3.11 indicates that a PSS is always related to some -clique codes. Unlike the conventional error correction code, which needs a large Hamming distance between the codewords to guarantee the performance, a perfect sequence set corresponds to some -clique codes. On the other hand, any -clique code which contains a perfect sequence will result some almost perfect sequence set. This “dual” relation of the perfect sequence sets with the error correction codes is an interesting issue. In Section V, the APSS was used to construct LCZ sequences. Meanwhile, some known LCZ families can be regarded as APSS as stated in Remark 4.14. Thus, the relation between LCZ sequences and APSS is a valuable research topic. There are a number of problems need further investigations. One of them is to derive a bound on the proposed APSS. Another one is to decide the structure of the APSS. Beyond these issues, to find new families of APSS and PSS is also an valuable topic. Moreover, the performance of APSS, for example in real CDMA systems, is an important issue needs further investigations. ACKNOWLEDGMENT The authors would like to thank the associate editor and the anonymous reviewers for their constructive comments. REFERENCES [1] R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” IEEE Trans. Inf. Theory, vol. 13, no. 4, pp. 619–621, Oct. 1967. 4For
N; M; L; )-LCZ sequence set, it holds that ML 0 1
a(
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[2] R. Gold, “Maximal recursive sequences with 3-valued recursive cross-correlation function,” IEEE Trans. Inf. Theory, vol. 14, no. 1, pp. 154–156, Jan. 1968. [3] T. Kasami, Weight distribution formula for some classes of cyclic codes Coord. Sci. Lab., Univ. Illinois Urbana-Champaign, Urbana, 1966, Tech. Rep. R-285 (AD 637524). [4] H. Dobbertin, P. Felke, T. Helleseth, and P. Rosendahl, “Niho type cross-correlation functions via Dickson polynomials and Kkloosterman sums,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 613–627, Feb. 2006. [5] T. Helleseth, A. Kholosha, and G. J. Ness, “Characterization of m-se1 and 2 1 with three-valued cross-corquences of lengths 2 relation,” IEEE Trans. Inf. Theory, vol. 53, no. 6, pp. 2236–2245, Jun. 2007. [6] D. V. Sarwate, “Bounds on crosscorrelation and autocorrelation of sequences,” IEEE Trans. Inf. Theory, vol. 25, no. 6, pp. 720–724, Nov. 1979. [7] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [8] P. Z. Fan, N. Suehiro, N. Kuroyanagi, and X. M. Deng, “A class of binary sequences with zero correlation zone,” Electron. Lett., vol. 35, pp. 777–779, 1999. [9] P. Z. Fan, N. Suehiro, and N. Kuroyanagi, “A novel interference-free CDMA system,” in PIMRC’99, Osaka, Japan, 1999. [10] J. Wolfmann, “Almost perfect autocorrelation sequences,” IEEE Trans. Inf. Theory, vol. 38, no. 4, pp. 1412–1418, Jul. 1992. [11] Y.-S. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New design of low correlation zone sequence sets,” IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4607–4616, Oct. 2006. [12] G. Gong, S. W. Golomb, and H.-Y. Song, “A note on low correlation zone signal sets,” IEEE Trans. Inf. Theory, vol. 53, pp. 2575–2581, Jul. 2007. [13] K. Cai, R. Feng, and Z. Zheng, “Cross-correlation properties of cyclotomic sequences,” IEICE Trans. Fundamentals, vol. E90-A, no. 1, pp. 281–286, Jan. 2007. [14] C. Ding, T. Helleseth, and H. M. Martinsen, “New families of binary sequences with optimal three-level autocorrelation,” IEEE Trans. Inf. Theory, vol. 47, pp. 428–433, Jan. 2001. [15] T. Storer, Cyclotomy and Difference Sets. Chicago, IL: Markham, 1967. [16] S. W. Golomb and G. Gong, Signal Designs with Good Correlation: for Wireless Communications, Cryptography and Radar Applications. Cambridge, U.K: Cambridge Univ. Press, 2005. [17] G. Gong, “Theory and application of q -ary interleaved sequences,” IEEE Trans. Inf. Theory, vol. 41, no. 2, pp. 400–411, Mar. 1995. [18] G. Gong, “New design for signal sets with low cross-correlation, balance property and large linear span GF (p) case,” IEEE Trans. Inf. Theory, vol. 48, pp. 2847–2867, 2002.
0
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[19] K. T. Arasu, C. Ding, T. Helleseth, P. V. Kumar, and H. M. Martinsen, “Almost difference sets and their sequences with optimal autocorrelation,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2934–2943, Nov. 2001. [20] D. Jungnickel and A. Pott, “Perfect and almost perfect sequences,” Discr. Appl. Math., vol. 95, pp. 331–359, 1999. [21] X. Tang and P. Fan, “A class of pseudonoise sequences over GF (p) with low correlation zone,” IEEE Trans. Inf. Theory, vol. 47, pp. 1644–1649, May 2001. [22] J.-W. Jang, J.-S. No, H. Chung, and X. H. Tang, “New sets of optimal p-ary low-correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 53, pp. 815–821, Feb. 2007. [23] S.-H. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New constructions of quaternary low correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 51, pp. 1469–1477, Apr. 2005. [24] X. H. Tang, P. Z. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, no. 6, pp. 551–552, 2000. Kai Cai received the M.S. and Ph.D. degrees in mathematics from Peking University, Beijing, China, in 2001 and 2004, respectively. From July 2004 to June 2006, he was a Research Fellow with the Department of Electronic Engineering, Tsinghua University, Beijing. From June 2006 to July 2007, he was a Research Associate with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China. He is now with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing. His research interests are in the areas of sequences design, coding theory, and network information theory.
Guobiao Weng received the M.S. and Ph.D. degrees in mathematics from Peking University, Beijing, China, in 2003 and 2006, respectively. He is now with the Department of Applied Mathematics, Dalian University of Technology, Liaoning, China.
Xueqi Cheng (M’99) received the M.S. and Ph.D. degrees in computer science from Northeastern University and Chinese Academy of Science, respectively. He is currently a Professor and Director of the Key Lab of Network Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, Beijing. His research interests are in the areas of network science and social computing, information security, web search, and mining. He has published approximately 100 referred journal and conference proceedings papers in his research areas, including papers published in ACM SIGIR, ACM CIKM, WWW, New Journal of Physics and the Journal of Statistical Mechanics. Dr. Cheng is a member of ACM.
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 7, JULY 2010
Binary Almost-Perfect Sequence Sets Kai Cai, Guobiao Weng, and Xueqi Cheng, Member, IEEE
Abstract—Sequence set with lower correlation values is highly desired for engineering applications. However, p theoretical results (e.g., Welch bound) show that max N in general, that is, the maximum out-of-phase autocorrelation and cross-correlation magnitudes of a sequence set is not less than the square root of the sequence period. In this paper, we propose a new concept, namely almost perfect sequence set (APSS), which has the property max c except for at most m shifts, where c and m are predefined small integers. A uniform method is presented to construct APSS and then the properties of such APSS are discussed. Moreover, a distance inequality on the APSS with m = 1 is obtained and several APSS families such as (2p; 8p +2; 6; 4)-APSS and 3p; 64p3 +8 ; 9; 9 -APSS for any prime p 5 are constructed based on Paley and Paley partial sequences. Finally, it shows that the APSS can be used to construct LCZ sequences and the properties of such LCZ sequences are presented. Index Terms—Almost perfect sequence set (APSS), correlation function, LCZ sequences, Paley and Paley partial sequences.
I. INTRODUCTION
L
ET and be two binary sequences of period , where , for . The (periodic) cross-correlation function between and is defined as1
where . When , it is being known as the (periodic) . autocorrelation function of and denoted by Sequences with good autocorrelation property are widely used in the engineering applications. However, some applications, e.g., direct-sequence code-division multiple access Manuscript received November 16, 2007; revised February 06, 2010. Current version published June 16, 2010. The material in this paper was partly based on the Ph.D. dissertation by K. Cai, “Difference sets and sequences with good correlation properties,” School of Mathematical Sciences, Peking University, Beijing, China, 2004. The work of K. Cai was supported by the NSF of China (No. 60872036 and No. 60873243) and by the 863 Project of China (No. 2008AA01Z140). The work of G. Weng was supported by the NSF of China (No. 10826072). The work of X. Cheng was supported by the NSF of China (No. 60933005). K. Cai is with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China (e-mail: [email protected]; caikai@software. ict.ac.cn). G. Weng was with the School of Mathematical Sciences, Peking University, Beijing 100871, China. He is now with the Department of Applied Mathematics, Dalian University of Technology, Liaoning, China (e-mail: [email protected]. cn). X. Cheng is with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing, China. Communicated by G. Gong, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2010.2048490 1It
is also defined as C
w
( )=
0
( 1)
in some literatures.
(DS-CDMA) system require sequences with both good autocorrelation and cross-correlation properties. Many sequences have been constructed to meet such demands, including which, Gold sequences, Gold-like sequences [1], [2], and Kasami sequences [3] are the most well-known families. For the latest advances on this topic, the reader can refer to [4] and [5] and their references. The well-known Sarwate bound [6] (I.1) uncovers the relation between the peak cross-correlation magniand the peak out-of-phase autocorrelation magnitude tude for sequences of period . Let . By (I.1), one can easily deduced that for . The result indicates that it is impossible to construct sequence set with both very low autocorrelation magnitudes and very low cross-correlation magnitudes. As an example, we give a classical result comprising the works of Gold and Kasami [7]. Theorem 1.1: Let be m-sequences with period , be the -decimation sequence (see Section II-B and let , or , for the definition) of , where is odd, then takes three values such that as follows:
Note that in Theorem 1.1, when , the maximal magnitude of takes on its smallest value . One method to get “lower” correlation sequences is to construct low correlation zone (LCZ) sequence set [8], [9]. -LCZ sequence set is defined as a sequence A sequences with period such that set which contains for and for for some small value . The LCZ sequences is useful in approximately synchronized CDMA (AS-CDMA) systems. Another probable way to get “lower” correlation sequences is to construct sequences with both low autocorrelation magnitudes and low cross-correlation magnitudes except for a few (as small as possible) exceptional shifts. In this paper, we propose this concept as almost perfect sequence set (APSS), which is formally defined as follows. is called a Definition I.2 (APSS): Sequence set -APSS, if contains shift-distinct sequences of period and satisfies the following two conditions: and each , 1. For each except for at most . 2. For each and each , except for at most .
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Moreover, when , is called a -perfect seand , is called a -pair. quence set (PSS); when In [10], Wolfmann proposed the notion of the almost perfect autocorrelation sequence, which has optimal autocorrelation magnitudes except for one possible shift. By Definition I.2, -APSS. In what follows, his construction is initial a we shall only concern on the binary APSS. As to the general complex-valued sequences, one can give discussions similarly. The main contributions of this paper can be illustrated as follows. We proposed a uniform method to construct APSS. The main idea of this method is to find PSS (or specifically, -pair) first, and then construct APSS based on the technique of sequence interleaving. It shows that this easy-to-implement method can effectively produce a lot of APSS families. Since our construction is PSS-based, a necessary condition on the existence of the PSS is also presented. We obtain several APSS families such as -APSS, -APSS, and -APSS based the perfect
-pairs and
-APSS, -APSS based on Paley and Paley partial -pairs. Note that the parameters and of these sequences do not increase even the period and the size of the sequence set approach infinity. Beyond these constructions, there are many potential ways to construct APSS. For example, the interleaving method proposed in [11] and the subfield decomposition method proposed in [12]. We also give discussions on these existed APSS families. We use the APSS to construct LCZ sequence sets. It shows that the APSS can provide a large number of LCZ sequence sets with a variety of lower correlation zones. Although these LCZ sequences are not optimal, we believe that these families may be useful in some particular applications since they provide more choices of LCZ sequences. The rest of the paper is organized as follows. In Section II, we introduce some basic notations and necessary results for the APSS construction. In Section III, we propose the PSS-interleaving based method and also a necessary condition on the existence of the PSS. In Section IV, we construct several families of APSS and also give discussions on the existed families. In Section V, we construct LCZ sequences by using the APSS, and finally we conclude this paper in Section VI. II. PRELIMINARIES
. represents the inverse of , , where ,
• Let i.e., for
.
B. Multiplier and Decimation Let
be a subset of is defined by
. The characteristic sequence if otherwise,
where is the -th component of . Meanwhile, is called the support of . A (Hall) multiplier of is a automorphism of such that , for some , where and . The multiplier group of is composed of all the multipliers of , a subgroup of and denoted . , be a binary sequence of Let , and be an integer relatively prime to . period is called the -decimation sequence of . By abusing the denotation to represent either an integer relatively prime to or an automorphism of , the following result holds. Theorem 2.1 ([13]): Let group of . Suppose that
•
denotes the left shift operation on sequences. Let . Then , and for are called the shifts of . and . • Let denotes that , for . and are called shift-equivalent if there exists an integer such that . Otherwise, and are called shift-distinct. Note that the sequences in the APSS need to be shift-distinct from each other.
and
be the multiplier
be the coset decomposition of with respect to , and . Here is the Euler -funcwhere tion. Let be the characteristic sequence of . Then, compose all the decimation se1) quences of ; is , for . 2) The support of C. Difference Function of (where The difference function ) is defined as , where and . The following theorem shows that the cross-correlation function of sequences and can be determined by the difference function of their supports. , and Theorem 2.2 ([13]): Let be two binary sequences of period with , Then supports and , respectively. If
In this section, we briefly introduce some notations and results, which will be useful in the construction of the APSS. A. Notations
of
(II.1) When and
as
and, hence, and
, by denoting , respectively, (II.1) becomes , which is a well-known result
deduced by Ding in [14]. Two sequences are called to have the same autocorrelation spectrum if their autocorrelation functions have same images and each value in the images occurs the same time. Theorem 2.3: Let be a binary sequence with period , and be an integer relatively prime to , then and have the same autocorrelation spectrum.
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D. Cyclotomic Classes be the finite field with elements, and , i.e., for some inbe a primitive element of , deand , for . These ’s are called cyclotomic classes of order . The cyclotomic with respect to are defined as numbers of order . The following two families of cyclotomic numbers (refer to [15]) can be used to construct Paley and Paley partial sequences. (1) If for even , then and Let divide let teger . Let fine
(2) If
for odd , then
Definition 2.4: Let prime. The characteristic sequence of Paley (Paley partial) sequence with period .
different interleaving order yields It should be noted that: may be difdifferent results, e.g., generally, ; and need not be ferent with different for , e.g., it is permitted to interleave a sequence itself for several times. The correlation functions of the interleaved sequence can be represented by the correlation functions of their base sequences. , Lemma 3.1: Let sequences with period , and let . Then, for , we have
be , and
and be a is called the
III. PSS-INTERLEAVING-BASED CONSTRUCTION In this section, we propose a uniform method to construct APSS based on the technique of sequence interleaving and the assumption of the existence of the PSS. A. Sequences Interleaving The idea of sequence interleaving exists widely in the area of sequences design [16]. For example, Gong proposed the notion of -ary interleaved sequences and shown that a large number of popular sequences, such as multiplexed sequences, clock-controlled sequences, Kasami sequences, GMW sequences, geometric sequences, mapping sequence, and No sequences have particular interleaving structures [17], [18]. More literatures using interleaving techniques can be found in almost difference sets design [19] and LCZ sequences design [11]. In fact, most known LCZ sequences have interleaving structures [12]. In the mentioned literatures, the base sequences used for interleaving usually involve an ideal autocorrelation sequence and its shifts [17], [18] and/or its inverse [11], [19]. Unlike these sequences, the APSS constructed in this paper is based on the PSS, for which one needs to concern the autocorrelations and the cross-correlations between each sequence pair. Therefore, we first give the correlation properties for the interleaved sequences of a more general form. , for be Let sequences with period . The interleaved sequence of is a sequences defined as with period
Generally, sequences) of .
is
denoted by and are called as the base sequences (or column
Proof: We just prove item (4). For the other items, one can prove them following a similar way.
B. PSS-Interleaving Method -PSS , we will construct an Given an arbitrary for each . Note that the construcAPSS, namely, tion is based on the assumption of the existence of PSS. Here, we set a large to ensure this assumption and leave the detailed discussions later. be a -perTheorem 3.2: Let fect sequence set. Then
CAI et al.: BINARY APSS
is a
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-APSS with
.
and ; , for ; are shift-distinct sequences if and are shift-distinct; is a shift of if and only if for 4) .
Proof: Let
, and . Let , be the exceptional value of the cross-correlation function between and , and be the excepand tional value of the cross-correlation function between , for . Let
.. .
.. .
..
.
.. .
.. .
.. .
..
.
.. .
Proof: 1) is obviously. 2) can be checked straightforward from
For item 3), we prove that for
is shift-distinct with . Let One can see that any shift of has the form of either or for some and . If , then , has the least period . If which is contradict to that , then , which is contradict to that is shift-distinct from . The contradictions yield that are shift-distinct sequences, which proves item 3). be a shift of , then has For item 4), let the form of either or . If , then we have and . If , then we have and . On , then , the other hand, if which is obviously a shift of . Hence, 4) is proved. By all the discussions above, the lemma follows.
Note that the elements of and represent the positions of , the exceptional shifts, and for , . only conBy Lemma 3.1, it is not difficult to get that and only contribute to tributes to , for . Thus, except possible shifts, which are listed as the values of the entries of , and possible shifts, which are listed as the values of the entries of (see the equation shown at the bottom of the and take their absolute values less than . page), both By the definition, the sequences in an APSS need to be shiftgendistinct. Hence, it is difficult to give the exact size of . erally, but we can see that it is upper bounded by By above discussions, the theorem is proved.
With the help of this lemma, we now can get the exact size of . Theorem 3.4: Let quence set with the least period
By Theorem 3.2, given a -PSS, we can always ob-APSS for each . Obviously, a small tain a leads to a small and also a small number of the exceptional shifts as we expected. However, one can see that may be a quite loose bound. In the following, we deduce the for and . exact size of 1) The Case of : In this case, by Theorem 3.2, one can construct APSS with 4 exceptional shifts. To decide the exact size of the constructed APSS, we first need a lemma. Lemma 3.3: Let , , least period . Then
and
if and only if
1) 2) 3)
is a
. Let
-APSS for odd -APSS for even Proof: Let
be binary sequences with the
be a perfect se. Then
.. .
.. .
..
.
.. .
.. .
.. .
..
.
.. .
.
and/or a
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By the definition (III.1) By Lemma 3.3, it is obvious that for and for , where denotes the integral part of a real number. equals the sum of enThus, the number of sequences in tries of the upper triangular matrix
.. .
.. .
..
.
That is, for odd , which proves the theorem.
and
.. . . In other words, for even
2) The Case of : In this case, one can construct APSS with 9 possible exceptional shifts. To decide the exact size of , we first need a lemma. Lemma 3.5: Let least period . Then, 1) ; 2) for
and if and only if
be sequences with the ,
, and
and (III.2) It is not difficult to check that (III.1) and (III.2) are equivalent , they have no solution. Thus, equations. When each equivalence class of contains 3 sequences. When , they have a solution , , and when , they have a solution , . , each equivalence class of contains 3 Thus, when sequences except one class which contains one sequence. That is , contains shift-distinct sequences and to say, when when , contains shift-distinct sequences. The lemma is proved. By the above discussions, it can be seen that the number of shift-distinct sequences in equals the number of elements in , where the equivalent relation is decided by , for . As for . an example, we give Example 3.7: For , consists elements, which are illustrated as , in the fol-th position is means that lowing table. Note that the belongs to the equivalent class . For example, , , and etc.
Proof: It can be proved through straightforward checking. Lemma 3.6: Let be shift-distinct sequences with the lest period . Then, have 1) shift-distinct sequences; 2) have shift-distinct sequences; 3) have shiftdistinct sequences for , and shift-distinct . sequences for Proof: By Lemma 3.5, in cases 1) and 2), different choices of and shall result in shift-distinct sequences. Thus, all and have shift-distinct sequences. Now we prove 3). First, note that there is an equivalent resuch that for , if and only if lation on and are shift-equivalent. Thus, the number of shift-distinct sequences equals the number of equivalence classes in . By Lemma 3.5, , , are equivalent seand quences in . Consider the following equations:
One can check that except , each , for contains three elements. All the elements in one correspond to one class of shift-equivalent sequences in . For example, corresponds to the shift-equivalent sequences , and . , consists elements, For which are illustrated in the following table. Note that except , contains 3 elements. each , for
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For , consists elements, which are illustrated in the following table. Note that each , for contains three elements.
C. Necessary Condition on the Existence of PSS In Section III-B, a large number of APSS can be obtained by using the technique of sequence interleaving. However, according to Theorem 3.2, the constructed APSS is based on some -PSS. Hence, the existence of -PSS is an important issue for APSS constructions. In the following, we give -PSS. a necessary condition on the existence of the Lemma 3.10: Let be a -pair with period . Then, such that there exists some or , where denotes the Hamming distance2 between and . Proof: Let
, and
, for
. Using
Theorem 3.8: Let quence set with the least period
be a perfect se. Then
. Let
we have
The inequality can be rewritten as is a
-APSS when
and/or a
-APSS when
.
be shift-distinct sequences. ObProof: Let , and each of them contains viously, shift-distinct sequences. Similarly, let be mutually shift-distinct. Then and each of sequences. Note that them contains and for . Thus, by Lemma 3.6, when
Thus, we have
It means that That is to say,
(Distance Inequality): Let be a -PSS. Then, for , there exists some such all or . that A binary sequence is called perfect if it has a 2-level autocorrelation function with the off-peck autocorrelation coefficients (Note that sequences with rarely exist, see [20] for reference). By using this definition, we have the following result, which is a direct corollary of Lemma 3.10.
,
The theorem is proved. Using the same notations as above, the following theorem can be obtained directly by Lemma 3.6. Theorem 3.9: Let period . Then
is a
.
By this lemma, the following theorem is obvious. Theorem
when
or or
3.11
. If Theorem 3.12: Let be a PSS such that is perfect, then . , The theorem shows that except the trivial -pair exists. For the sake of conveno other type of PSS with nience, we call it as the perfect -pair.
be a -pair with the least IV. APSS FAMILIES
-APSS when -APSS when
and/or a .
In this section, we discuss the APSS families constructed by the perfect -pairs, by the Paley and Paley partial -pairs and the other families already in the existed literatures. 2the Hamming distance between two binary sequences of equal length is the number of positions for which the corresponding symbols are different.
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A. APSS Based on Perfect -pairs
where
Let be a perfect sequence with period . According to last froms a perfect -pair. By Theorems 3.4 and 3.9 section, and using the same notations, the following APSS families can be obtained directly.
. By Theorem Proof: The multiplier group of is decimation sequences, i.e., the 2.1, it has
be a perfect -pair with the
Theorem 4.1: Let least period . Then
is a
and characteristic sequences of . The values of and as and let likely way. Denote
Theorem 4.2: Let least period . Then
. We just calculate can be decided by a . For ,
if if
-APSS.
Proof: Note that the off-peck autocorrelation coefficients for perfect sequences and . By (II.1), , thus and the result follows.
is a
be a prime.
, .
Thus, by (II.1),
be a perfect -pair with the if if if -APSS when
-APSS when
and/or a .
B. APSS Based on Paley and Paley Partial -pairs In this subsection, we first give a family of -pair with for any prime . The APSS families with parameters , , and are then obtained. 1) Paley and Paley Partial -pairs: For any prime , we shall construct a -pair with period and . The construction is based on Paley and Paley partial sequences. Lemma 4.3([13]): Paley sequence with period has two decimation sequences, say and , and their correlation functions are listed as follows:
, ,
The result follows. By Lemmas 4.3 and 4.4, for any prime , there ex. More preciously, when ists a -pair with period and , the two decimation sequences of Paley sequences form such a -pair, namely the Paley -pair; when , the two decimation sequences of Paley partial sequences form such a -pair, namely the Paley partial -pair. Formally, we rewrite this as a theorem. Theorem 4.5: For any prime . period and
, there exists a -pair with
2) Paley and Paley Partial -pair Based Construction: Based on Paley and Paley Partial -pairs, we give the families of APSS using the PSS-Interleaving method of Section III. Theorem 4.6: For any prime -APSS. Proof: Let sequence when
and
, there exists a
be the decimation sequences of Paley and Paley partial sequence when
. Let
where be a prime. Following a similar way in [13], we get the correlation functions of Paley partial sequences. Lemma 4.4: Paley partial sequence with period has two decimation sequences, say and , and their correlation functions are listed as follows.
By Theorem 3.4, for any odd prime -APSS, which proves the theorem. Theorem 4.7: For any prime -APSS.
,
is a , there exists a
Proof: By Theorem 3.9 and using Paley and Paley partial -pairs, such APSS can be directly constructed. The constructions illustrated by Theorems 4.6 and 4.7 can be, respectively, extended when Paley (and Paley partial) -pair is extended to perfect sequence set , as shown follows.
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Theorem 4.8: For any prime -APSS.
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Proof: Let , be the Paley and/or Paley partial -pair is as aforementioned. Then we declare that a perfect sequence set. First, for arbitrary binary sequences and with period , it is obvious that , and for . Secondly, one can easily see that are shift-distinct. Thus, the four sequences in form a PSS. By Theorem 3.4, one can construct a -APSS for any prime . Theorem 4.9: For any prime -APSS.
, there exists a
Proof: Let be the perfect sequence set used in the proof of Theorem 4.8. By Theorem 3.8, the result is directly obtained. A (38,39,6,4)-APSS constructed by Paley -pair is given as follows. Example 4.10: The decimation sequences of Paley sequence with period 19 are C. Other Existed Families and
The corresponding (38,39,6,4)-APSS constructed by and is:
Here, we shall discuss the sequences in the existed literatures that can be considered as APSS, i.e., the construction based on interleaving of perfect sequences [11] and the construction based on subfield decomposition [12], [16]. Both of them are intermediate sequences for constructing LCZ sequences. 1) Perfect Sequence Based Construction: In [11], the authors constructed several families of LCZ sequences by using the interleaving technique. Among which, one of their construction was choosing sequences from a larger sequence set which is in fact an APSS. Now we restate their result in terms of almost perfect sequence sets. Theorem 4.11 ([11], Theorem 3): Let be a perfect sequence . Let with period
Then is a -APSS. sequence when interleaving By Lemma 3.3, it yields and sequences when interleaving for . Thus the number of shift-dis. It can be seen that this tinct sequences in is APSS is in fact a subset of the APSS constructed in Theorem 4.1. 2) Subfield Decomposition Based Constructions: There are a large number of -ary LCZ families constructed by the method of subfield decomposition [12], [21]–[23]. In [12], the authors connected these families by using a unified subfield decomposition method, and they also define the almost low correlation zone (ALCZ) signal set, which permits a high in-phase cross-correlation value. In fact, these constructions can also be regarded as particular almost perfect sequence sets, as stated below.
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for contributes to
. By the proof of Theorem 3.2, only , only contributes to , and only contributes to . Thus, , for and for , which is a LCZ sequence set with low correlation indicates . Obviously, the set size of is upper zone bounded by , and the theorem is proved.
TABLE I APSS CONSTRUCTED BY THEOREM 4.12 WITH SMALL q AND m
Theorem 4.12 ([12], Theorem 2): 3Let be the set of all subfield reducible sequences with the trace representation , where is a fixed function from to with the twotuple balance property and the evaluation of ’s runs through the with the balanced set of all the shift-distinct sequences over . Then is a property of period -APSS with . Table I lists the parameters of the structed by Theorem 4.12 with small and
-APSS con.
Remark 4.13: From Table I, One can see that the APSS constructed by the subfield decomposition method always have a smaller size and more number of exceptional shifts. This is because these ALCZ sequences prefer a larger LCZ rather than a less number of exceptional high correlation shifts, and more importantly, all the exceptional shifts of these ALCZ sequences are for , at same positions, i.e., which is a stronger property than what a general APSS needs. Remark 4.14: All the LCZ sequences constructed in [12], [21]–[23] are the subsets of the corresponding ALCZ sequences of Theorem 4.12. Thus, they are almost perfect sequence sets with the same number of exceptional shifts but a smaller set size. V. LCZ SEQUENCES FROM APSS In this section, we construct LCZ sequence sets based on APSS. A new scene of the construction is that it can provide a large number of LCZ sequence sets with a variety of lower correlation zones. be a -PSS and be the Let be a subset sequence set defined in Theorem 3.2. Let such that for all and of , the following items hold: 1) , for , ; , for , 2) and are the exwhere ceptional values of and , respectively. Using this denotation, we have the following. Theorem 5.1: quence set with
is a
-LCZ se.
Proof: Let , where
, ,
, and
, and ,
3For the definitions of the subfield reducible sequence, balance property, twotuple balance property, trace representation, and evaluation, the reader can refer to [12] for the details.
From Theorem 5.1, one can see that the LCZ of depends on parameters and . Thus one can get particular LCZ sequences by choosing suitable and . Especially, we have the following. Corollary 5.2: is a -LCZ sequence . set when This result indicates that when the base sequences satisfy (it seems easy to be satisfied), one can construct LCZ sequence set with low correlation zone by interleaving the base sequences times. However, the correlation values within the LCZ will increase accordingly. , it is obviously To construct LCZ sequence sets from that one can use exhaustive computer searches beginning with a particular sequence (that is, the exceptional correlation shifts of ) in based on Theorem 5.1. On the its base sequences other hand, we can also have the following constructive results. -perfect sequence set Definition 5.3: A is called normalized if for , i.e., all the exceptional cross-correlation shifts are at position 0. Note that a -pair can be normalized by proper shifting one of the two sequences. Hence, the normalized perfect sequence -normalized perfect seset always exists. Let be a quence set. Take , and for each , let be a set of integers such that and , for and . Denote the collection of all such -integer sets as and let . We have the following result. Lemma 5.4: . be the subset of such that for Proof: Let . Without loss of generality, we calculate . each Obviously, it equals the number of ways for choosing elements from such . It also equals the number that elements from of ways for choosing such that . Let , then it equals the number different elements of ways for choosing from since a set elements in such that corresponds to a of set of elements in such that , and vice versa. The latter is known as . Thus , including which each is multi-calculated times. Hence, there are different ways for choosing from , and we are done.
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Theorem 5.5: Let be a -normalized perfect sequence set. Then, for all and such that , there exist different -LCZ . sequence sets within Proof: Since is normalized, then and , the exceptional shift of for any is . Now we choose integers such that and , for and , where . There are different such -integer -integer set, it can yield difsets. For each -LCZ sequence sets, because there ferent are ways to arrange the sequences, of which , for each one contains base sequences with the form and . Since there are ways to choose , the total number of the resulted LCZ sequence sets is , which proves the theorem. , the exceptional shift for each pair of sequences When and in is completely decided by . By similar arguments as Theorem 5.5, we have be a Corollary 5.6: Let malized perfect sequence set. Then, for all and , there exist -LCZ sequence sets within .
-nordifferent
Remark 5.7: By the definition, unlike the APSS, the sequences within a LCZ sequence set need not be shift-distinct. when So we also consider the shifts of the sequences of using it to construct LCZ sequence sets. Example 5.8: Reconsider the Paly -pair in Example 4.10 with period , where One can see that it is a normalized PSS. Let . Then
,
. and
One can check that there are 19 ways to choose 6 integers with each pair having the distance . Take . There are
ways to arrange into 3 disjoint strings with each string contains 2 numbers. For example, is such an arrangement and is another one. Each arrangement corresponds to different (38,3,4,6)-LCZ sequence , sets. For example, are 2 of 64 such sequence . In all, there sets corresponds to different (38,3,4,6)-LCZ exist , sequence sets based on this Paly -pair. When , and , by similar discussions, one can obtain
different (19,6,2,3)-LCZ sequence sets. There are two issues should be mentioned. The first one is: the LCZ sequence sets constructed in Theorem 5.5 is in general not optimal with respect to Tang–Fan–Matsufuji bound4 [24]; The second one is: a large number of LCZ sequence sets with a variety of different parameters can be constructed from a per, a large fect sequence set, and even for a particular are correnumber of different LCZ sequence sets within sponded. For the first issue, it is an interesting problem to construct optimal LCZ sequence sets based on the perfect sequence set. For the second issue, it may be useful in some particular applications since it provides more choices of LCZ sequences. VI. CONCLUDING REMARKS In this paper, we proposed the concept of almost per, fect sequence sets and constructed , , , and -APSS based on perfect sequences, Paley sequences and Paley partial sequences, respectively. It is important to note that the cross-correlation between the decimation of Paley partial sequences is a 2-level function. It has been an open problem whether this is possible except Kasami cases [5]. The construction of APSS proposed in this paper was based on the existence of PSS. We gave a necessary condition on the existence of such sequence sets. If we define -clique code as a binary sequence set, of which each pair has Hamming distance less than , and define maximal -clique code as a -clique code with maximal number of sequences, i.e., it will not be a -clique code by adding any other sequence, Theorem 3.11 indicates that a PSS is always related to some -clique codes. Unlike the conventional error correction code, which needs a large Hamming distance between the codewords to guarantee the performance, a perfect sequence set corresponds to some -clique codes. On the other hand, any -clique code which contains a perfect sequence will result some almost perfect sequence set. This “dual” relation of the perfect sequence sets with the error correction codes is an interesting issue. In Section V, the APSS was used to construct LCZ sequences. Meanwhile, some known LCZ families can be regarded as APSS as stated in Remark 4.14. Thus, the relation between LCZ sequences and APSS is a valuable research topic. There are a number of problems need further investigations. One of them is to derive a bound on the proposed APSS. Another one is to decide the structure of the APSS. Beyond these issues, to find new families of APSS and PSS is also an valuable topic. Moreover, the performance of APSS, for example in real CDMA systems, is an important issue needs further investigations. ACKNOWLEDGMENT The authors would like to thank the associate editor and the anonymous reviewers for their constructive comments. REFERENCES [1] R. Gold, “Optimal binary sequences for spread spectrum multiplexing,” IEEE Trans. Inf. Theory, vol. 13, no. 4, pp. 619–621, Oct. 1967. 4For
N; M; L; )-LCZ sequence set, it holds that ML 0 1
a(
.
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[2] R. Gold, “Maximal recursive sequences with 3-valued recursive cross-correlation function,” IEEE Trans. Inf. Theory, vol. 14, no. 1, pp. 154–156, Jan. 1968. [3] T. Kasami, Weight distribution formula for some classes of cyclic codes Coord. Sci. Lab., Univ. Illinois Urbana-Champaign, Urbana, 1966, Tech. Rep. R-285 (AD 637524). [4] H. Dobbertin, P. Felke, T. Helleseth, and P. Rosendahl, “Niho type cross-correlation functions via Dickson polynomials and Kkloosterman sums,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 613–627, Feb. 2006. [5] T. Helleseth, A. Kholosha, and G. J. Ness, “Characterization of m-se1 and 2 1 with three-valued cross-corquences of lengths 2 relation,” IEEE Trans. Inf. Theory, vol. 53, no. 6, pp. 2236–2245, Jun. 2007. [6] D. V. Sarwate, “Bounds on crosscorrelation and autocorrelation of sequences,” IEEE Trans. Inf. Theory, vol. 25, no. 6, pp. 720–724, Nov. 1979. [7] D. V. Sarwate and M. B. Pursley, “Crosscorrelation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, pp. 593–619, May 1980. [8] P. Z. Fan, N. Suehiro, N. Kuroyanagi, and X. M. Deng, “A class of binary sequences with zero correlation zone,” Electron. Lett., vol. 35, pp. 777–779, 1999. [9] P. Z. Fan, N. Suehiro, and N. Kuroyanagi, “A novel interference-free CDMA system,” in PIMRC’99, Osaka, Japan, 1999. [10] J. Wolfmann, “Almost perfect autocorrelation sequences,” IEEE Trans. Inf. Theory, vol. 38, no. 4, pp. 1412–1418, Jul. 1992. [11] Y.-S. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New design of low correlation zone sequence sets,” IEEE Trans. Inf. Theory, vol. 52, no. 10, pp. 4607–4616, Oct. 2006. [12] G. Gong, S. W. Golomb, and H.-Y. Song, “A note on low correlation zone signal sets,” IEEE Trans. Inf. Theory, vol. 53, pp. 2575–2581, Jul. 2007. [13] K. Cai, R. Feng, and Z. Zheng, “Cross-correlation properties of cyclotomic sequences,” IEICE Trans. Fundamentals, vol. E90-A, no. 1, pp. 281–286, Jan. 2007. [14] C. Ding, T. Helleseth, and H. M. Martinsen, “New families of binary sequences with optimal three-level autocorrelation,” IEEE Trans. Inf. Theory, vol. 47, pp. 428–433, Jan. 2001. [15] T. Storer, Cyclotomy and Difference Sets. Chicago, IL: Markham, 1967. [16] S. W. Golomb and G. Gong, Signal Designs with Good Correlation: for Wireless Communications, Cryptography and Radar Applications. Cambridge, U.K: Cambridge Univ. Press, 2005. [17] G. Gong, “Theory and application of q -ary interleaved sequences,” IEEE Trans. Inf. Theory, vol. 41, no. 2, pp. 400–411, Mar. 1995. [18] G. Gong, “New design for signal sets with low cross-correlation, balance property and large linear span GF (p) case,” IEEE Trans. Inf. Theory, vol. 48, pp. 2847–2867, 2002.
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[19] K. T. Arasu, C. Ding, T. Helleseth, P. V. Kumar, and H. M. Martinsen, “Almost difference sets and their sequences with optimal autocorrelation,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 2934–2943, Nov. 2001. [20] D. Jungnickel and A. Pott, “Perfect and almost perfect sequences,” Discr. Appl. Math., vol. 95, pp. 331–359, 1999. [21] X. Tang and P. Fan, “A class of pseudonoise sequences over GF (p) with low correlation zone,” IEEE Trans. Inf. Theory, vol. 47, pp. 1644–1649, May 2001. [22] J.-W. Jang, J.-S. No, H. Chung, and X. H. Tang, “New sets of optimal p-ary low-correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 53, pp. 815–821, Feb. 2007. [23] S.-H. Kim, J.-W. Jang, J.-S. No, and H. Chung, “New constructions of quaternary low correlation zone sequences,” IEEE Trans. Inf. Theory, vol. 51, pp. 1469–1477, Apr. 2005. [24] X. H. Tang, P. Z. Fan, and S. Matsufuji, “Lower bounds on correlation of spreading sequence set with low or zero correlation zone,” Electron. Lett., vol. 36, no. 6, pp. 551–552, 2000. Kai Cai received the M.S. and Ph.D. degrees in mathematics from Peking University, Beijing, China, in 2001 and 2004, respectively. From July 2004 to June 2006, he was a Research Fellow with the Department of Electronic Engineering, Tsinghua University, Beijing. From June 2006 to July 2007, he was a Research Associate with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong, China. He is now with the Institute of Computing Technology, Chinese Academy of Sciences, Beijing. His research interests are in the areas of sequences design, coding theory, and network information theory.
Guobiao Weng received the M.S. and Ph.D. degrees in mathematics from Peking University, Beijing, China, in 2003 and 2006, respectively. He is now with the Department of Applied Mathematics, Dalian University of Technology, Liaoning, China.
Xueqi Cheng (M’99) received the M.S. and Ph.D. degrees in computer science from Northeastern University and Chinese Academy of Science, respectively. He is currently a Professor and Director of the Key Lab of Network Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, Beijing. His research interests are in the areas of network science and social computing, information security, web search, and mining. He has published approximately 100 referred journal and conference proceedings papers in his research areas, including papers published in ACM SIGIR, ACM CIKM, WWW, New Journal of Physics and the Journal of Statistical Mechanics. Dr. Cheng is a member of ACM.
