Optimal Variable-Weight Optical Orthogonal ... - Semantic Scholar
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010
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Optimal Variable-Weight Optical Orthogonal Codes via Difference Packings Dianhua Wu, Hengming Zhao, Pingzhi Fan, Senior Member, IEEE, and Satoshi Shinohara
Abstract—Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, the upper bound on the size of variable-weight OOCs is improved, a cyclic t-(v; W; ; Q) packing is introduced to construct a variable-weight OOC, an upper bound for the number of blocks of t-(v; W; ; Q) packings is obtained, and an equivalence between optimal cyclic packing and optimal variable-weight optical orthogonal code is established. Recursive constructions for optimal 2-CP(W; 1; Q; v )s are also presented. By using skew starters and these constructions, infinite classes of optimal (v; W; 1; f1=2; 1=2g)-OOCs are obtained for W = f3; 4g, and f4; 5g. Index Terms—Cyclic packing, optical orthogonal code, packing design, skew starter, variable-weight optical orthogonal code.
I. INTRODUCTION
O
PTICAL orthogonal codes (OOCs) were introduced by Salehi, as signature sequences to facilitate multiple access in optical fibre networks [13], [14]. OOCs are codes over with constant weight property and can be viewed symbols as constant weight error correcting codes [2] or balanced incomplete designs. The OOCs have found wide range of applications such as mobile radio, frequency-hopping spread-spectrum communications, radar, sonar, collision channel without feedback and neuromorphic networks, the interested reader is referred to [2], [9], and [12]–[16]. Most existing works on OOC’s have assumed that all codewords have the same weight. In general, the code size of OOCs depends upon the weights of codewords, the variable-weight OOCs can generate larger code size than that of constant-weight OOCs [10]. Yang introduced multimedia optical CDMA communication system employing variable-weight OOCs [19] in Manuscript received April 13, 2009; revised November 10, 2009. Date of current version July 14, 2010. The work of D. Wu was supported in part by the NSFC (No. 10961006) and by Guangxi Science Foundation (No. 0991089). The work of P. Fan was supported in part by the NSFC (No. 60772087), the 863 High-Tech Program (No. 2009AA01Z238), the 111 project (No.111-2-14), and the Sino-Swedish Int. Cooperation Program (No. 2008DFA12160). D. Wu is with the Keylab of Information Coding and Transmission, Southwest Jiaotong University, 610031, Chengdu, China, and also with the Department of Mathematics, Guangxi Normal University, 541004, Guilin, China (e-mail: [email protected]). H. Zhao is with the Department of Mathematics, Guangxi Normal University, 541004, Guilin, China (e-mail: [email protected]). P. Fan is with the Keylab of Information Coding and Transmission, Southwest Jiaotong University, 610031, Chengdu, China (e-mail: [email protected]). S. Shinohara is with the Department of Information Science, Meisei University, Hino, Tokyo, 191-8506, Japan (e-mail: [email protected]). Communicated by N. Yu, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2010.2050927
1996. In this CDMA system, the codewords of low code weight can be assigned to the low-Quality of Services (QoS) applications and high code weight codewords to high-QoS applications [10], the subscribers with different code weights will have different bit error rate (BER) performance. Hence, the variable-weight property of the OOCs enables the system to meet multiple QoS requirements. Subsequent to these developments, variable-weight OOCs have attracted much attention [5], [10], [17]–[19]. Based on the notation of [19], throughout this paper, let , , and denote the sets , and , respectively as defined below. Without loss of . generality, we assume that Definition 1: An variable-weight optical or-OOC, is a collection of thogonal code , or binary -tuples such that the following three properties hold: • Weight Distribution: Every -tuple in has a Hamming ; furthermore, there are weight contained in the set codewords of weight , i.e., indicates exactly the fraction of codewords of weight . It is clear that . • Periodic Auto-correlation: For any with Hamming weight , and any integer , ,
where the summation is carried out by treating binary symbols as reals. , • Periodic Cross-correlation: Similarly, for any , , and any integer ,
The notation
-OOC is used to denote an -OOC with the property that . The term variable-weight optical orthogonal code, or variable-weight OOC, is also used if there is no need to list the parameters. , without loss of generality, write , For each , . Let where , are integers and be the least common multiple of , and , then ,
,
. For example, suppose , and
, we have
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,
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IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 8, AUGUST 2010
,
, and
. In the sequel, the will be used.
notation
In [19], Yang gave upper and lower bounds on the size of variable-weight OOCs, and presented several constructions for vari, able-weight OOCs with parameters , , , some of them are optimal. In [10], Gu and Wu had constructed variable-weight OOCs by using pairwise balanced designs (PBDs) and packing designs with a partition. Some combinatorial constructions such as balanced incomplete block designs (BIBDs), difference families for variable-weight OOCs were presented in [5]. It should be noted that not all variable-weight OOCs in [19] are optimal. Further, some of the OOCs have . To the authors knowledge, no infinite classes of optimal variable-weight OOCs had been obtained except for the one in [17]. In this paper, by using cyclic packing designs, new infinite classes of optimal variable-weight OOCs are constructed. The rest of the paper is arranged as follows. In Section II, the upper bound on the size of variable-weight OOCs is packings are introduced improved, cyclic to construct variable-weight OOCs, an upper bound for packings and the number of blocks of both s is obtained, then an equivalence between ops and optimal -OOCs timal is established. In Section III, by using skew starters, two infinite are obtained for classes of optimal and . In Section IV, recursive construcs are presented, the two tions for optimal s infinite classes of optimal cyclic in Section III are then extended, and infinite classes of optimal -OOCs are obtained. Finally, a brief summary is presented in Section V. II. PRELIMINARIES In this section, the upper bound on the size of variable-weight OOCs is improved, brief mathematical background needed for the paper is given, then the equivalence between optimal cyclic packings and optimal variable-weight OOCs is established.
In [19], a -OOC is said to be optimal if the above equality holds. However, the bound is not tight , in some cases. The following is an example. Let , , , , then
. It is clear that there does not exist
a -OOC with 7 codewords since the number of codewords of weight 3 and 4 is equal. In the following, Theorem 1 will be improved. Let
Then the following result is obtained. Theorem 2: Let
Proof: Let
From Theorem 1, . So, for each , . is an integer since it is the number of codewords of weight , and hence , . Since is an integer, then . Thus . This completes the proof. For
A. Bounds
. Then
, ,
The definition of a variable-weight OOC is a generalization of the definition of OOC given in [13] and [14]. The work of Yang [19] contains lower and upper bounds on the size of variableweight OOCs to judge the goodness of the constructions, where the upper bound is given by. Theorem1 ([19]): Let
is a lowing code with 6 codewords
, , , it is easy to see that . The fol-OOC
. Then
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packing , let For a cyclic be a block in . The block orbit containing is defined to be the set of the following distinct blocks: Based on the above fact, from now on, a is said to be optimal if
-OOC
Let
Then, the following result is clear from Theorem 2. Theorem 3:
.
for . If a block orbit has distinct blocks, i.e., its set, then this block orbit wise stabilizer is equal to the identity is said to be full, otherwise short. Choose an arbitrarily fixed block from each block orbit and then call it a base block. Let denote the base blocks of a cyclic packing containing only full block orbits, that is, for any base , its setwise stabilizer . Note that in [20], block is used to denote a . a Since it is difficult to decide the upper bound for the number , the following definition is needed. of blocks in a packing is defined to be a A packing with the property that the fraction of blocks of size is , . Similarly, a is a with the property that the fraction of blocks of is , . size is the maximum number The packing number of blocks in any packings. Let
To construct optimal variable-weight OOCs, cyclic packings will be used. B. Cyclic -Packings and Variable-Weight OOCs Optimal optical orthogonal codes are closely related to some combinatorial configurations. For example, Yin [20] showed that an optimal -OOC is equivalent to an . In [6], it is proved that optimal cyclic packing -OOC is equivalent to an optimal cyclic an optimal -packing. Further information can be found in [6]. In this section, the equivalence between optimal cyclic -packings and optimal variable-weight OOCs will be established. a set of positive integers, Let be a positive integer, and each of which is greater than . If , we will omit the packing design (or packing) with order , braces. A , where is a block sizes from and index , is a pair -set and is a collection of subsets of (called blocks), each cardinality from , such that every -subset of occurs in at , then is allowed to contain most blocks of . If is the maxrepeated blocks. The packing number packing. imum number of blocks in any Consider a packing . Let be a permuta, define tion on . For any block . If , then is called an automorphism of the packing. The set of all such permutations forms a group under composition called the full automorphism group of the packing. Any of its subgroups is called an automorphism group of the packing. packing admitting a cyclic and point-regular A automorphism group is called cyclic. For a cyclic packing, the point set can be identified with , the residue ring of integers modulo . In this case, the packing has an automorphism .
The following result is obtained. Theorem 4:
.
Proof: From the definition of a -
packing, we
have
Then
Similar to the proof of Theorem 2, one can obtain the conclusion. We use base blocks in any -
to denote the maximum number of . Then it is clear that
which implies the following simple upper bound on .
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Let
, where
Applying Theorem 5 with is obtained.
,
, the following result
Lemma 1:
Then, the following result is obtained. Theorem 5: . . It is clear that is called optimal if A . , one can conGiven an optimal cyclic -sequences of length , and weight from a struct a whose nonzero bit positions are exactly base block of size indexed by the base block. It is not difficult to see that the de-OOC. Since rived (0, 1)-sequences form a cyclic , then is optimal. -OOC. For Conversely, suppose is an optimal by taking the each codeword, we construct a -subset of . This creindex set of its nonzero bit positions, where of subsets of . From the correlation propates a family forms a . Since erties of the OOC, , then is optimal. Therefore, we have the the following result. Theorem 6: An optimal is equivalent -OOC. to an optimal According to Theorem 6, in order to construct optimal -OOCs, one needs only to construct its corres. It is not easy to sponding optimal s for . When , construct optimal however, there is a feasible way. is a , for , the list of Suppose differences from is defined to be . Define . The difference leave of , denoted by , is defined to be the set of all nonzero integers in which are not covered by .A is -regular if the difference leave along with zero forms having order , which must be genan additive subgroup of . erated by integer , , suppose that , then For a . Since , the number of blocks of size is , , and hence . So, the number of then is a multiple of . The folblocks in a lowing result is obtained. Theorem 7: A necessary condition for the existence of a -regular is . Proof: Suppose blocks, then
Since
is a
-
, then the result is obtained.
with
Since the number of blocks in a tiple of
is a mul-
. Then the following result is ob-
tained. Theorem 8: If packing is optimal.
, then a -regular -
The following result is clear from Theorem 6. Corollary 1: An optimal an optimal -OOC.
is equivalent to
Example 1: Let
Then forms a . Since , then the packing -OOC in is optimal. The Section II is constructed from this packing. By virtue of Corollary 1, in order to construct optimal -OOCs, one needs only to construct its corres. sponding optimal III. CONSTRUCTIONS OF OPTIMAL CYCLIC PACKINGS In this section, two infinite classes of optimal s for and are constructed. The following concept of a skew starter is needed. Definition 2: Let be an abelian group of order . A skew starter in is a set of unordered pairs which satisfies the following three properties: • ; • ; ; • According to the definition, a skew starter in can exist only if is odd. Skew starters were used to construct optimal constant-weight optical orthogonal codes. See [1], [8], and [11] for the examples. For more details on skew starters, the interested reader may be referred to [3], [4]. The following result was stated in [1].
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Lemma 2 ([1]): There exists a skew starter in for each , is not divisible by positive integer such that 5 or is divisible by 25. There does not exist any skew starter in if . From Lemma 2, it is clear that there exists a skew starter in if . are two positive integers. If , Suppose , is isomorphic to from Chinese Reminder then Theorem in number theory. The one-to-one and onto mapping to is as follows:. from
By using the Chinese Reminder Theorem, one can get the of an optimal base blocks as follows:
One where
,
,
,
can
construct -OOC
the corresponding as follows:
,
. A. The Existence of Optimal -
s
Theorem 9: If there exists a skew starter in , then there , which is also exists a 9-regular optimal. Proof: Since a skew starter exists in , then , and, hence, is isomorphic to . is a skew starter in Suppose that , where . Define the base blocks as follows:
We compute the differences from them. For fixed , is a difference from and , denote . , . So, we need It is easy to see that only to consider for .
B. The Existence of Optimal Theorem 10: If there exists a skew starter in exists a 16-regular also optimal.
s , then there , which is
Proof: Since a skew starter exists in , then , is isomorphic to . hence, Suppose that is a skew starter in , where . Take the base blocks as follows:
Then
Now set , then does cover each element of exactly once, while any element of the additive subgroup is not covered at all. So, forms a 9-regular . and , Since then the packing is optimal from Theorem 8. This completes the proof. Example 2: Let , then skew starter in . One can get
is a
where
is the same as defined in Theorem 9.
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Now set , then does cover each element of exactly once, while any element of the additive subgroup is not covered at all. So, forms a 16-reg. ular and , Since then the packing is optimal. This completes the proof. Example 3: Let , then a skew starter in . One can get
be
cyclic difference matrix ( -CDM matrix whose each entry is an integer of such that for any two distinct rows and , the list of contains each integer of exactly once. It is easy to see that if an -CDM exists, -CDM for any positive integer . then so does an The following result was stated in [20]. For convenience, we is an odd prime, an present the construction also. When -CDM can be constructed by simply taking for and . So, we have the following. An in short) is a
an
By using the Chinese can get the following -
One
can
get
Reminder Theorem, one 16-regular and optimal packing , where
the
corresponding variable-weight -OOC as follows (see the equation at the bottom of the page). IV. RECURSIVE CONSTRUCTIONS FOR OPTIMAL S In this section, recursive constructions for optimal s are presented. More optimal s and optimal variable-weight OOCs are then obtained. In [20], a cyclic difference matrix (DM) was used to the repackings. We use the cursive constructions for cyclic definition in [7].
Lemma 3 ([20]): If is an odd prime, then there exists -CDM. The following result was stated in [7].
is odd, and , then Lemma 4 ([7]): If -CDM. there exists an Similar to the constructions in [20], the following two results are obtained. Theorem 11: Suppose that both a -regular and an optimal exist, then an opexists. Moreover, if the given timal is -regular, then so is the derived . Proof: Let given -regular pose that For each
be the family of base blocks of the , then the difference leave , where . Supis the base blocks of an optimal . , let
Since the existence of a -regular implies by Theorem 7, the difference leave of an opand an optimal have timal forms the same cardinality. So, the family . the desired optimal -
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The second part is clear according to the above construction. The proof is then complete. Theorem 12: Suppose that there exist: ; (1) a -regular -CDM; (2) an . (3) an optimal . Moreover, if Then there exists an optimal is -regular, then so is the derived the given . Proof: From Theorem 11 and condition (3), it suffices to -regular which proceeds as construct a follows. is the base blocks of the given -regular Suppose whose difference leave plus zero consists of the additive subgroup of , where . Let be an -CDM, for and . where -regular will be based Now the desired on , whose difference leave plus zero forms the subgroup of . The required base blocks can be constructed in the following way. , we have . Since there exists an For each -CDM, then the -CDM exists. For each base , where , we take block blocks
a 16-regular is clear. This completes the proof. Lemma 6: -
The following result can be obtained from Lemma 3 and Thein [20]. orem 12. Similar result was obtained for Theorem 13: Suppose that and are two odd primes, and . If both a -regular and an optimal exist, then so does . Moreover, if the given an optimal is -regular, then so is the derived . In the following, we will extend the existence results for ops in Section III by using the above contimal structions. Lemma
5:
are also optimal.
There
exist
both a 9-regular and a 16-regular . Moreover, the two packings
There exist a 9-regular and an optimal , is an integer.
Proof: If , the result comes from Lemma 5. If , the result comes from Theorem 13. This completes the proof. , is an inTheorem 14: If teger, then there exist both a 9-regular and an optimal . Proof: Write , where , are nonnegative . Then is odd and . integers, and , then , the conclusion comes from Lemma 6. If , then a skew starter in exists, and hence a 9-regIf ular and an optimal exist from , then the conclusion is true since a 9-regTheorem 9. If exist. If ular and an optimal , then the conclusion comes from Theorem 13 since a 9-regexist from ular and an optimal Lemma 6. This completes the proof. A -CDM exists from Lemma 3, a 16-regand an optimal ular exist from Lemma 5. Similar to the proof of Lemma 6, the following result is obtained. -
where the additive operation is performed in . We need only to verify that the differences arising from these blocks cover exactly once. It is a routine matter of each integer in checking and is omitted here for simplicity.
. The second part
Lemma 7: There exist a 16-regular and an optimal , is an integer.
Theorem 15: If , is an integer, then there exist both a 16-regular and an optimal . , where is an integer, and Proof: Write . Then is odd and . , then , the conclusion comes from Lemma 7. If , then a skew starter in exists, and hence a 16-regIf exist from ular and an optimal Theorem 10. If , then the conclusion is true since a 16-regexist. If ular and an optimal , the conclusion comes from Theorem 13 since a 16-regexist ular and an optimal from Lemma 7. This completes the proof. From Theorem 6, Theorems 14–15, the following results are obtained. Theorem 16: If , there exists an optimal
is an integer, then -OOC.
Theorem 17: If , then there exists an optimal
is an integer, -OOC.
Proof: Let V. CONCLUSION
Then -
forms ,
a
9-regular forms
In this paper, the upper bound on the size of variable-weight OOCs is improved. An equivalence bes and optimal tween optimal -OOCs is established. Recursive constructions for optimal s are also presented, and new infis and optimal nite classes of optimal -
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-OOCs are obtained for
and
.
ACKNOWLEDGMENT The authors would like to thank Associate Editor N. Y. Yu and the anonymous referees for their comments and suggestions that much improved the quality of this paper.
[16] M. P. Vecchi and J. A. Salehi, “Neuromorphic networks based on sparse optical orthogonal codes,” in Neural Information Processing Systems-Natural and Synthetic. New York: Amer. Inst. Phys, 1988, pp. 814–823. [17] D. Wu, P. Fan, H. Li, and U. Parampalli, “Optimal variable-weight optical orthogonal codes via cyclic difference families,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT’09), Jun. 3, 2009, pp. 448–452. [18] G. C. Yang, “Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements,” IEEE GLOBECOM ’93, vol. 1, pp. 488–492, Dec. 1993. [19] G. C. Yang, “Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements,” IEEE Trans. Commun., vol. 44, pp. 47–55, Jan. 1996. [20] J. Yin, “Some combinatorial constructions for optical orthogonal codes,” Discr. Math., vol. 185, pp. 201–219, 1998.
REFERENCES [1] K. Chen, G. Ge, and L. Zhu, “Starters and related codes,” J. Statist. Plann. Inference, vol. 86, pp. 379–395, 2000. [2] F. R. K. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: Design, analysis, and applications,” IEEE Trans. Inf. Theory, vol. 35, pp. 595–604, May 1989. [3] J. H. Dinitz, “Starters,” in The CRC Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds., Second ed. Boca Raton, FL: Chapman and Hall/CRC, 2006, pp. 622–628. [4] J. H. Dinitz and D. R. Stinson, “Room squares and related designs,” in Contemporary Design Theory, J. H. Dinitz and D. R. Stinson, Eds. New York: Wiley, 1992, pp. 137–204. [5] I. B. Djordjevic, B. Vasic, and J. Rorison, “Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs,” J. Lightw. Technol., vol. 21, pp. 1850–1856, Sep. 2003. [6] R. Fuji-Hara and Y. Miao, “Optical orthogonal codes: Their bounds and new optimal constructions,” IEEE Trans. Inf. Theory, vol. 46, pp. 2396–2406, Nov. 2000. [7] G. Ge, “On (g; 4; 1)-diffference matrices,” Discr. Math., vol. 301, pp. 164–174, 2005. [8] G. Ge and J. Yin, “Constructions for optimal (v; 4; 1) optical orthogonal codes,” IEEE Trans. Inf. Theory, vol. 47, pp. 2998–3004, Nov. 2001. [9] S. W. Golomb, Digital Communication With Space Application. Los Altos, CA: Penisula, 1982. [10] F. R. Gu and J. Wu, “Construction and performance analysis of variable-weight optical orthogonal codes for asynchronous optical CDMA systems,” J. Lightw. Technol., vol. 23, pp. 740–748, Feb. 2005. [11] S. Ma and Y. Chang, “Constructions of optimal optical orthogonal codes with weight five,” J Combin. Des., vol. 13, pp. 54–69, 2005. [12] J. L. Massey and P. Mathys, “The collision channel without feedback,” IEEE Trans. Inf. Theory, vol. 31, pp. 192–204, Mar. 1985. [13] J. A. Salehi, “Code division multiple access techniques in optical fiber networks-Part I fundamental principles,” IEEE Trans. Commun., vol. 37, pp. 824–833, Aug. 1989. [14] J. A. Salehi and C. A. Brackett, “Code division multiple access techniques in optical fiber networks-Part II systems performance analysis,” IEEE Trans. Commun., vol. 37, pp. 834–842, Aug. 1989. [15] J. A. Salehi, “Emerging optical code-division multiple-access communications systems,” IEEE Netw., vol. 3, pp. 31–39, Mar. 1989.
Dianhua Wu was born in Shandong, China, in 1966. He received the Ph.D. degree in mathematics from Suzhou University, China, in 2000. He was with Guangxi Normal University, China, as an Assistant Teacher during 1988–1991, a Lecturer during 1992–1996, and an Associate Professor from 1997 to 2000. Since 2001, he has been a full professor. His research interests include combinatorial design theory, coding theory, communication theory, and their interactions.
Hengming Zhao was born in Guangxi, China, in 1984. He received the B.S. degree in mathematics from Guangxi Normal University, China, in 2007. He is working toward the M.S. degree in mathematics with the Department of Mathematics, Guangxi Normal University.
Pingzhi Fan (SM’99) received the M.S. degree in computer science from the Southwest Jiaotong University, PRC, in 1987, and the Ph.D. degree in electronic engineering from the Hull University, U.K., in 1994. He is currently a professor and director of the Institute of Mobile Communications, Southwest Jiaotong University, PRC, and a Guest Professor of Leeds University, U.K. (1997– present), a guest professor with Shanghai Jiaotong University from (1999–present). He is the inventor of 22 patents, and the author of more than 200 research journal papers and eight books, including six books published by John Wiley and Sons, Ltd., RSP (1996), IEEE Press (2003, 2006), Springer (2004), and Nova Science (2007), respectively. His research interests include spread-spectrum and CDMA technology, information theory and coding, sequence design and applications, radio resource management, etc. Dr. Fan was a recipient of the U.K. ORS Award (1992), and the NSFC Outstanding Young Scientist Award (1998).
Satoshi Shinohara was born in Kanagawa, Japan, in 1971. He received the Ph.D. degree in Management Science and Engineering from University of Tsukuba, Japan, in 2000. He was with Meisei University as a Research Assistant (2000–2003), a Lecturer (2004–2005). From 2006, he has been an Associate Professor. His research interests include combinatorial design theory, finite geometry, coding theory, communication theory, and their interactions.
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Optimal Variable-Weight Optical Orthogonal Codes via Difference Packings Dianhua Wu, Hengming Zhao, Pingzhi Fan, Senior Member, IEEE, and Satoshi Shinohara
Abstract—Variable-weight optical orthogonal code (OOC) was introduced by Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirements. In this paper, the upper bound on the size of variable-weight OOCs is improved, a cyclic t-(v; W; ; Q) packing is introduced to construct a variable-weight OOC, an upper bound for the number of blocks of t-(v; W; ; Q) packings is obtained, and an equivalence between optimal cyclic packing and optimal variable-weight optical orthogonal code is established. Recursive constructions for optimal 2-CP(W; 1; Q; v )s are also presented. By using skew starters and these constructions, infinite classes of optimal (v; W; 1; f1=2; 1=2g)-OOCs are obtained for W = f3; 4g, and f4; 5g. Index Terms—Cyclic packing, optical orthogonal code, packing design, skew starter, variable-weight optical orthogonal code.
I. INTRODUCTION
O
PTICAL orthogonal codes (OOCs) were introduced by Salehi, as signature sequences to facilitate multiple access in optical fibre networks [13], [14]. OOCs are codes over with constant weight property and can be viewed symbols as constant weight error correcting codes [2] or balanced incomplete designs. The OOCs have found wide range of applications such as mobile radio, frequency-hopping spread-spectrum communications, radar, sonar, collision channel without feedback and neuromorphic networks, the interested reader is referred to [2], [9], and [12]–[16]. Most existing works on OOC’s have assumed that all codewords have the same weight. In general, the code size of OOCs depends upon the weights of codewords, the variable-weight OOCs can generate larger code size than that of constant-weight OOCs [10]. Yang introduced multimedia optical CDMA communication system employing variable-weight OOCs [19] in Manuscript received April 13, 2009; revised November 10, 2009. Date of current version July 14, 2010. The work of D. Wu was supported in part by the NSFC (No. 10961006) and by Guangxi Science Foundation (No. 0991089). The work of P. Fan was supported in part by the NSFC (No. 60772087), the 863 High-Tech Program (No. 2009AA01Z238), the 111 project (No.111-2-14), and the Sino-Swedish Int. Cooperation Program (No. 2008DFA12160). D. Wu is with the Keylab of Information Coding and Transmission, Southwest Jiaotong University, 610031, Chengdu, China, and also with the Department of Mathematics, Guangxi Normal University, 541004, Guilin, China (e-mail: [email protected]). H. Zhao is with the Department of Mathematics, Guangxi Normal University, 541004, Guilin, China (e-mail: [email protected]). P. Fan is with the Keylab of Information Coding and Transmission, Southwest Jiaotong University, 610031, Chengdu, China (e-mail: [email protected]). S. Shinohara is with the Department of Information Science, Meisei University, Hino, Tokyo, 191-8506, Japan (e-mail: [email protected]). Communicated by N. Yu, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2010.2050927
1996. In this CDMA system, the codewords of low code weight can be assigned to the low-Quality of Services (QoS) applications and high code weight codewords to high-QoS applications [10], the subscribers with different code weights will have different bit error rate (BER) performance. Hence, the variable-weight property of the OOCs enables the system to meet multiple QoS requirements. Subsequent to these developments, variable-weight OOCs have attracted much attention [5], [10], [17]–[19]. Based on the notation of [19], throughout this paper, let , , and denote the sets , and , respectively as defined below. Without loss of . generality, we assume that Definition 1: An variable-weight optical or-OOC, is a collection of thogonal code , or binary -tuples such that the following three properties hold: • Weight Distribution: Every -tuple in has a Hamming ; furthermore, there are weight contained in the set codewords of weight , i.e., indicates exactly the fraction of codewords of weight . It is clear that . • Periodic Auto-correlation: For any with Hamming weight , and any integer , ,
where the summation is carried out by treating binary symbols as reals. , • Periodic Cross-correlation: Similarly, for any , , and any integer ,
The notation
-OOC is used to denote an -OOC with the property that . The term variable-weight optical orthogonal code, or variable-weight OOC, is also used if there is no need to list the parameters. , without loss of generality, write , For each , . Let where , are integers and be the least common multiple of , and , then ,
,
. For example, suppose , and
, we have
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,
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,
, and
. In the sequel, the will be used.
notation
In [19], Yang gave upper and lower bounds on the size of variable-weight OOCs, and presented several constructions for vari, able-weight OOCs with parameters , , , some of them are optimal. In [10], Gu and Wu had constructed variable-weight OOCs by using pairwise balanced designs (PBDs) and packing designs with a partition. Some combinatorial constructions such as balanced incomplete block designs (BIBDs), difference families for variable-weight OOCs were presented in [5]. It should be noted that not all variable-weight OOCs in [19] are optimal. Further, some of the OOCs have . To the authors knowledge, no infinite classes of optimal variable-weight OOCs had been obtained except for the one in [17]. In this paper, by using cyclic packing designs, new infinite classes of optimal variable-weight OOCs are constructed. The rest of the paper is arranged as follows. In Section II, the upper bound on the size of variable-weight OOCs is packings are introduced improved, cyclic to construct variable-weight OOCs, an upper bound for packings and the number of blocks of both s is obtained, then an equivalence between ops and optimal -OOCs timal is established. In Section III, by using skew starters, two infinite are obtained for classes of optimal and . In Section IV, recursive construcs are presented, the two tions for optimal s infinite classes of optimal cyclic in Section III are then extended, and infinite classes of optimal -OOCs are obtained. Finally, a brief summary is presented in Section V. II. PRELIMINARIES In this section, the upper bound on the size of variable-weight OOCs is improved, brief mathematical background needed for the paper is given, then the equivalence between optimal cyclic packings and optimal variable-weight OOCs is established.
In [19], a -OOC is said to be optimal if the above equality holds. However, the bound is not tight , in some cases. The following is an example. Let , , , , then
. It is clear that there does not exist
a -OOC with 7 codewords since the number of codewords of weight 3 and 4 is equal. In the following, Theorem 1 will be improved. Let
Then the following result is obtained. Theorem 2: Let
Proof: Let
From Theorem 1, . So, for each , . is an integer since it is the number of codewords of weight , and hence , . Since is an integer, then . Thus . This completes the proof. For
A. Bounds
. Then
, ,
The definition of a variable-weight OOC is a generalization of the definition of OOC given in [13] and [14]. The work of Yang [19] contains lower and upper bounds on the size of variableweight OOCs to judge the goodness of the constructions, where the upper bound is given by. Theorem1 ([19]): Let
is a lowing code with 6 codewords
, , , it is easy to see that . The fol-OOC
. Then
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packing , let For a cyclic be a block in . The block orbit containing is defined to be the set of the following distinct blocks: Based on the above fact, from now on, a is said to be optimal if
-OOC
Let
Then, the following result is clear from Theorem 2. Theorem 3:
.
for . If a block orbit has distinct blocks, i.e., its set, then this block orbit wise stabilizer is equal to the identity is said to be full, otherwise short. Choose an arbitrarily fixed block from each block orbit and then call it a base block. Let denote the base blocks of a cyclic packing containing only full block orbits, that is, for any base , its setwise stabilizer . Note that in [20], block is used to denote a . a Since it is difficult to decide the upper bound for the number , the following definition is needed. of blocks in a packing is defined to be a A packing with the property that the fraction of blocks of size is , . Similarly, a is a with the property that the fraction of blocks of is , . size is the maximum number The packing number of blocks in any packings. Let
To construct optimal variable-weight OOCs, cyclic packings will be used. B. Cyclic -Packings and Variable-Weight OOCs Optimal optical orthogonal codes are closely related to some combinatorial configurations. For example, Yin [20] showed that an optimal -OOC is equivalent to an . In [6], it is proved that optimal cyclic packing -OOC is equivalent to an optimal cyclic an optimal -packing. Further information can be found in [6]. In this section, the equivalence between optimal cyclic -packings and optimal variable-weight OOCs will be established. a set of positive integers, Let be a positive integer, and each of which is greater than . If , we will omit the packing design (or packing) with order , braces. A , where is a block sizes from and index , is a pair -set and is a collection of subsets of (called blocks), each cardinality from , such that every -subset of occurs in at , then is allowed to contain most blocks of . If is the maxrepeated blocks. The packing number packing. imum number of blocks in any Consider a packing . Let be a permuta, define tion on . For any block . If , then is called an automorphism of the packing. The set of all such permutations forms a group under composition called the full automorphism group of the packing. Any of its subgroups is called an automorphism group of the packing. packing admitting a cyclic and point-regular A automorphism group is called cyclic. For a cyclic packing, the point set can be identified with , the residue ring of integers modulo . In this case, the packing has an automorphism .
The following result is obtained. Theorem 4:
.
Proof: From the definition of a -
packing, we
have
Then
Similar to the proof of Theorem 2, one can obtain the conclusion. We use base blocks in any -
to denote the maximum number of . Then it is clear that
which implies the following simple upper bound on .
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Let
, where
Applying Theorem 5 with is obtained.
,
, the following result
Lemma 1:
Then, the following result is obtained. Theorem 5: . . It is clear that is called optimal if A . , one can conGiven an optimal cyclic -sequences of length , and weight from a struct a whose nonzero bit positions are exactly base block of size indexed by the base block. It is not difficult to see that the de-OOC. Since rived (0, 1)-sequences form a cyclic , then is optimal. -OOC. For Conversely, suppose is an optimal by taking the each codeword, we construct a -subset of . This creindex set of its nonzero bit positions, where of subsets of . From the correlation propates a family forms a . Since erties of the OOC, , then is optimal. Therefore, we have the the following result. Theorem 6: An optimal is equivalent -OOC. to an optimal According to Theorem 6, in order to construct optimal -OOCs, one needs only to construct its corres. It is not easy to sponding optimal s for . When , construct optimal however, there is a feasible way. is a , for , the list of Suppose differences from is defined to be . Define . The difference leave of , denoted by , is defined to be the set of all nonzero integers in which are not covered by .A is -regular if the difference leave along with zero forms having order , which must be genan additive subgroup of . erated by integer , , suppose that , then For a . Since , the number of blocks of size is , , and hence . So, the number of then is a multiple of . The folblocks in a lowing result is obtained. Theorem 7: A necessary condition for the existence of a -regular is . Proof: Suppose blocks, then
Since
is a
-
, then the result is obtained.
with
Since the number of blocks in a tiple of
is a mul-
. Then the following result is ob-
tained. Theorem 8: If packing is optimal.
, then a -regular -
The following result is clear from Theorem 6. Corollary 1: An optimal an optimal -OOC.
is equivalent to
Example 1: Let
Then forms a . Since , then the packing -OOC in is optimal. The Section II is constructed from this packing. By virtue of Corollary 1, in order to construct optimal -OOCs, one needs only to construct its corres. sponding optimal III. CONSTRUCTIONS OF OPTIMAL CYCLIC PACKINGS In this section, two infinite classes of optimal s for and are constructed. The following concept of a skew starter is needed. Definition 2: Let be an abelian group of order . A skew starter in is a set of unordered pairs which satisfies the following three properties: • ; • ; ; • According to the definition, a skew starter in can exist only if is odd. Skew starters were used to construct optimal constant-weight optical orthogonal codes. See [1], [8], and [11] for the examples. For more details on skew starters, the interested reader may be referred to [3], [4]. The following result was stated in [1].
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Lemma 2 ([1]): There exists a skew starter in for each , is not divisible by positive integer such that 5 or is divisible by 25. There does not exist any skew starter in if . From Lemma 2, it is clear that there exists a skew starter in if . are two positive integers. If , Suppose , is isomorphic to from Chinese Reminder then Theorem in number theory. The one-to-one and onto mapping to is as follows:. from
By using the Chinese Reminder Theorem, one can get the of an optimal base blocks as follows:
One where
,
,
,
can
construct -OOC
the corresponding as follows:
,
. A. The Existence of Optimal -
s
Theorem 9: If there exists a skew starter in , then there , which is also exists a 9-regular optimal. Proof: Since a skew starter exists in , then , and, hence, is isomorphic to . is a skew starter in Suppose that , where . Define the base blocks as follows:
We compute the differences from them. For fixed , is a difference from and , denote . , . So, we need It is easy to see that only to consider for .
B. The Existence of Optimal Theorem 10: If there exists a skew starter in exists a 16-regular also optimal.
s , then there , which is
Proof: Since a skew starter exists in , then , is isomorphic to . hence, Suppose that is a skew starter in , where . Take the base blocks as follows:
Then
Now set , then does cover each element of exactly once, while any element of the additive subgroup is not covered at all. So, forms a 9-regular . and , Since then the packing is optimal from Theorem 8. This completes the proof. Example 2: Let , then skew starter in . One can get
is a
where
is the same as defined in Theorem 9.
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Now set , then does cover each element of exactly once, while any element of the additive subgroup is not covered at all. So, forms a 16-reg. ular and , Since then the packing is optimal. This completes the proof. Example 3: Let , then a skew starter in . One can get
be
cyclic difference matrix ( -CDM matrix whose each entry is an integer of such that for any two distinct rows and , the list of contains each integer of exactly once. It is easy to see that if an -CDM exists, -CDM for any positive integer . then so does an The following result was stated in [20]. For convenience, we is an odd prime, an present the construction also. When -CDM can be constructed by simply taking for and . So, we have the following. An in short) is a
an
By using the Chinese can get the following -
One
can
get
Reminder Theorem, one 16-regular and optimal packing , where
the
corresponding variable-weight -OOC as follows (see the equation at the bottom of the page). IV. RECURSIVE CONSTRUCTIONS FOR OPTIMAL S In this section, recursive constructions for optimal s are presented. More optimal s and optimal variable-weight OOCs are then obtained. In [20], a cyclic difference matrix (DM) was used to the repackings. We use the cursive constructions for cyclic definition in [7].
Lemma 3 ([20]): If is an odd prime, then there exists -CDM. The following result was stated in [7].
is odd, and , then Lemma 4 ([7]): If -CDM. there exists an Similar to the constructions in [20], the following two results are obtained. Theorem 11: Suppose that both a -regular and an optimal exist, then an opexists. Moreover, if the given timal is -regular, then so is the derived . Proof: Let given -regular pose that For each
be the family of base blocks of the , then the difference leave , where . Supis the base blocks of an optimal . , let
Since the existence of a -regular implies by Theorem 7, the difference leave of an opand an optimal have timal forms the same cardinality. So, the family . the desired optimal -
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The second part is clear according to the above construction. The proof is then complete. Theorem 12: Suppose that there exist: ; (1) a -regular -CDM; (2) an . (3) an optimal . Moreover, if Then there exists an optimal is -regular, then so is the derived the given . Proof: From Theorem 11 and condition (3), it suffices to -regular which proceeds as construct a follows. is the base blocks of the given -regular Suppose whose difference leave plus zero consists of the additive subgroup of , where . Let be an -CDM, for and . where -regular will be based Now the desired on , whose difference leave plus zero forms the subgroup of . The required base blocks can be constructed in the following way. , we have . Since there exists an For each -CDM, then the -CDM exists. For each base , where , we take block blocks
a 16-regular is clear. This completes the proof. Lemma 6: -
The following result can be obtained from Lemma 3 and Thein [20]. orem 12. Similar result was obtained for Theorem 13: Suppose that and are two odd primes, and . If both a -regular and an optimal exist, then so does . Moreover, if the given an optimal is -regular, then so is the derived . In the following, we will extend the existence results for ops in Section III by using the above contimal structions. Lemma
5:
are also optimal.
There
exist
both a 9-regular and a 16-regular . Moreover, the two packings
There exist a 9-regular and an optimal , is an integer.
Proof: If , the result comes from Lemma 5. If , the result comes from Theorem 13. This completes the proof. , is an inTheorem 14: If teger, then there exist both a 9-regular and an optimal . Proof: Write , where , are nonnegative . Then is odd and . integers, and , then , the conclusion comes from Lemma 6. If , then a skew starter in exists, and hence a 9-regIf ular and an optimal exist from , then the conclusion is true since a 9-regTheorem 9. If exist. If ular and an optimal , then the conclusion comes from Theorem 13 since a 9-regexist from ular and an optimal Lemma 6. This completes the proof. A -CDM exists from Lemma 3, a 16-regand an optimal ular exist from Lemma 5. Similar to the proof of Lemma 6, the following result is obtained. -
where the additive operation is performed in . We need only to verify that the differences arising from these blocks cover exactly once. It is a routine matter of each integer in checking and is omitted here for simplicity.
. The second part
Lemma 7: There exist a 16-regular and an optimal , is an integer.
Theorem 15: If , is an integer, then there exist both a 16-regular and an optimal . , where is an integer, and Proof: Write . Then is odd and . , then , the conclusion comes from Lemma 7. If , then a skew starter in exists, and hence a 16-regIf exist from ular and an optimal Theorem 10. If , then the conclusion is true since a 16-regexist. If ular and an optimal , the conclusion comes from Theorem 13 since a 16-regexist ular and an optimal from Lemma 7. This completes the proof. From Theorem 6, Theorems 14–15, the following results are obtained. Theorem 16: If , there exists an optimal
is an integer, then -OOC.
Theorem 17: If , then there exists an optimal
is an integer, -OOC.
Proof: Let V. CONCLUSION
Then -
forms ,
a
9-regular forms
In this paper, the upper bound on the size of variable-weight OOCs is improved. An equivalence bes and optimal tween optimal -OOCs is established. Recursive constructions for optimal s are also presented, and new infis and optimal nite classes of optimal -
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-OOCs are obtained for
and
.
ACKNOWLEDGMENT The authors would like to thank Associate Editor N. Y. Yu and the anonymous referees for their comments and suggestions that much improved the quality of this paper.
[16] M. P. Vecchi and J. A. Salehi, “Neuromorphic networks based on sparse optical orthogonal codes,” in Neural Information Processing Systems-Natural and Synthetic. New York: Amer. Inst. Phys, 1988, pp. 814–823. [17] D. Wu, P. Fan, H. Li, and U. Parampalli, “Optimal variable-weight optical orthogonal codes via cyclic difference families,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT’09), Jun. 3, 2009, pp. 448–452. [18] G. C. Yang, “Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements,” IEEE GLOBECOM ’93, vol. 1, pp. 488–492, Dec. 1993. [19] G. C. Yang, “Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements,” IEEE Trans. Commun., vol. 44, pp. 47–55, Jan. 1996. [20] J. Yin, “Some combinatorial constructions for optical orthogonal codes,” Discr. Math., vol. 185, pp. 201–219, 1998.
REFERENCES [1] K. Chen, G. Ge, and L. Zhu, “Starters and related codes,” J. Statist. Plann. Inference, vol. 86, pp. 379–395, 2000. [2] F. R. K. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: Design, analysis, and applications,” IEEE Trans. Inf. Theory, vol. 35, pp. 595–604, May 1989. [3] J. H. Dinitz, “Starters,” in The CRC Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds., Second ed. Boca Raton, FL: Chapman and Hall/CRC, 2006, pp. 622–628. [4] J. H. Dinitz and D. R. Stinson, “Room squares and related designs,” in Contemporary Design Theory, J. H. Dinitz and D. R. Stinson, Eds. New York: Wiley, 1992, pp. 137–204. [5] I. B. Djordjevic, B. Vasic, and J. Rorison, “Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs,” J. Lightw. Technol., vol. 21, pp. 1850–1856, Sep. 2003. [6] R. Fuji-Hara and Y. Miao, “Optical orthogonal codes: Their bounds and new optimal constructions,” IEEE Trans. Inf. Theory, vol. 46, pp. 2396–2406, Nov. 2000. [7] G. Ge, “On (g; 4; 1)-diffference matrices,” Discr. Math., vol. 301, pp. 164–174, 2005. [8] G. Ge and J. Yin, “Constructions for optimal (v; 4; 1) optical orthogonal codes,” IEEE Trans. Inf. Theory, vol. 47, pp. 2998–3004, Nov. 2001. [9] S. W. Golomb, Digital Communication With Space Application. Los Altos, CA: Penisula, 1982. [10] F. R. Gu and J. Wu, “Construction and performance analysis of variable-weight optical orthogonal codes for asynchronous optical CDMA systems,” J. Lightw. Technol., vol. 23, pp. 740–748, Feb. 2005. [11] S. Ma and Y. Chang, “Constructions of optimal optical orthogonal codes with weight five,” J Combin. Des., vol. 13, pp. 54–69, 2005. [12] J. L. Massey and P. Mathys, “The collision channel without feedback,” IEEE Trans. Inf. Theory, vol. 31, pp. 192–204, Mar. 1985. [13] J. A. Salehi, “Code division multiple access techniques in optical fiber networks-Part I fundamental principles,” IEEE Trans. Commun., vol. 37, pp. 824–833, Aug. 1989. [14] J. A. Salehi and C. A. Brackett, “Code division multiple access techniques in optical fiber networks-Part II systems performance analysis,” IEEE Trans. Commun., vol. 37, pp. 834–842, Aug. 1989. [15] J. A. Salehi, “Emerging optical code-division multiple-access communications systems,” IEEE Netw., vol. 3, pp. 31–39, Mar. 1989.
Dianhua Wu was born in Shandong, China, in 1966. He received the Ph.D. degree in mathematics from Suzhou University, China, in 2000. He was with Guangxi Normal University, China, as an Assistant Teacher during 1988–1991, a Lecturer during 1992–1996, and an Associate Professor from 1997 to 2000. Since 2001, he has been a full professor. His research interests include combinatorial design theory, coding theory, communication theory, and their interactions.
Hengming Zhao was born in Guangxi, China, in 1984. He received the B.S. degree in mathematics from Guangxi Normal University, China, in 2007. He is working toward the M.S. degree in mathematics with the Department of Mathematics, Guangxi Normal University.
Pingzhi Fan (SM’99) received the M.S. degree in computer science from the Southwest Jiaotong University, PRC, in 1987, and the Ph.D. degree in electronic engineering from the Hull University, U.K., in 1994. He is currently a professor and director of the Institute of Mobile Communications, Southwest Jiaotong University, PRC, and a Guest Professor of Leeds University, U.K. (1997– present), a guest professor with Shanghai Jiaotong University from (1999–present). He is the inventor of 22 patents, and the author of more than 200 research journal papers and eight books, including six books published by John Wiley and Sons, Ltd., RSP (1996), IEEE Press (2003, 2006), Springer (2004), and Nova Science (2007), respectively. His research interests include spread-spectrum and CDMA technology, information theory and coding, sequence design and applications, radio resource management, etc. Dr. Fan was a recipient of the U.K. ORS Award (1992), and the NSFC Outstanding Young Scientist Award (1998).
Satoshi Shinohara was born in Kanagawa, Japan, in 1971. He received the Ph.D. degree in Management Science and Engineering from University of Tsukuba, Japan, in 2000. He was with Meisei University as a Research Assistant (2000–2003), a Lecturer (2004–2005). From 2006, he has been an Associate Professor. His research interests include combinatorial design theory, finite geometry, coding theory, communication theory, and their interactions.
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