Relative Difference Families With Variable Block ... - Semantic Scholar

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 57, NO. 11, NOVEMBER 2011

7489

Relative Difference Families With Variable Block Sizes and Their Related OOCs Marco Buratti, Yueer Wei, Dianhua Wu, Pingzhi Fan, Senior Member, IEEE, and Minquan Cheng

Abstract—Seven infinite classes of relative difference families with variable block sizes are presented explicitly. In particular, 1)-DF with = a balanced ( 2 is explic= 3 4 5 and every coprime to 6; itly given for: (i) (ii) = 3 4 6 3 5 6 or 3 4 5 6 and every coprime to 30. As far as the authors are aware, these difference families can be viewed as the first explicit constructions of infinite classes of optimal variable-weight optical orthogonal codes with more than two weights. It is observed, however, that there are infinitely many values of for which an optimal ( 1 )-OOC exists, whatever the set of weights and the weight distribution sequence are. Index Terms—Graph decomposition, relative difference family, variable-weight optical orthogonal code.

I. INTRODUCTION PTICAL ORTHOGONAL codes (OOCs) were introduced by Salehi, as signature sequences to facilitate multiple access in optical fibre networks [43], [45]. OOCs had been found wide ranges of applications such as mobile radio, frequency-hopping spread-spectrum communications, radar, sonar, collision channel without feedback, and neuromorphic networks [24], [28], [32], [44], [46]. Most existing works on OOC’s have assumed that all codewords have the same weight and the correlation parameters are equal, see [1], [4], [5], [7], [16]–[27], [30], [31], and [56] for some of the many many examples. Some work has been also done about OOCs with constant weight but distinct correlation parameters in [33]–[36], [38], [39], [55]. 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

O

Manuscript received December 02, 2010; revised March 27, 2011; accepted July 04, 2011. Date of current version November 11, 2011. The work of D. Wu was supported in part by NSFC (No. 10961006), by Guangxi Science Foundation (No. 0991089), by the Program for Excellent Talents in Guangxi Higher Education Institutions, and by the Open Research Fund of the Key Laboratory of Information Coding and Transmission, Southwest Jiaotong University. The work of P. Fan was supported in part by NSFC (No. 60772087), by the 111 project (No. 111-2-14), and by the Sino-Swedish International Cooperation Program (No. 2008DFA12160). M. Buratti is with the Dipartimento di Matematica e Informatica, Universitá di Perugia, 06123 Perugia, Italy (e-mail: [email protected]). Y. Wei and D. Wu are with the Department of Mathematics, Guangxi Normal University, 541004 Guilin, China (e-mail: [email protected]; [email protected]). D. Wu and P. Fan are with the Keylab of Information Coding and Transmission, Southwest Jiaotong University, 610031 Chengdu, China (e-mail: [email protected]). M. Cheng is with the Institute of Policy and Planning Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan (e-mail: chengqinshi@hotmail. com). Communicated by N. Y. Yu, Associate Editor for Sequences. Digital Object Identifier 10.1109/TIT.2011.2162225

constant-weight OOCs [29]. In 1996, Yang introduced multimedia optical CDMA communication system employing variable-weight OOCs [54]. In this CDMA system, the subscribers with different code weights will have different bit error rate (BER) performance. The codewords of low code weight can be assigned to the low-QoS (Quality of Services) applications and high code weight codewords can be assigned to high-QoS requirement applications [29]. Hence, the multi-weight property of the OOCs enables the system to meet multiple QoS requirements. The interested reader may refer to [29], [50]–[54], [57] for recent results on variable-weight OOCs. be the set of all -subsets of Throughout this paper, let , the residue ring of integers modulo . We prefer to see an optical orthogonal code from the set-theoretical point of view, rather than a i.e., we will see it as a collection of subsets of collection of binary -tuples. be an ordering of a Definition 1: Let be set of integers greater than 1, let an -tuple (auto-correlation sequence) of positive integers, let be a positive integer (cross-correlation parameter), and let be an -tuple (weight distribution sequence) of positive rational numbers whose sum is 1. A optical orthogonal code is a set of subsets (codeword-sets) of with sizes (weights) from satisfying the following properties: Weight distribution property: the fraction of codeword-sets of of weight is :

Auto-correlation property: any two distinct translates of a codeword-set of weight share at most elements:

(1) Cross-correlation property: any two translates of two distinct elements: codeword-sets share at most (2) If is

for every , one simply says that -OOC. Also, speaking of an -OOC one means an -OOC where . We say that is normalized if it is written in the form with . Speaking of a balanced -OOC we mean an -OOC with , namely an OOC in which the number of codeword-sets of a given weight is a constant. an

0018-9448/$26.00 © 2011 IEEE

7490


As usual, we call list of differences of a subset of an additive group the multiset of all differences with an ordered pair of distinct elements of . More generally, the list of differences of a set of subsets of is the multiset . It is straightforward to see that conapdition (1) is equivalent to require that no element of pears in more than times for every codeword-set of weight . Also, in the case that , condition (2) is equivalent to require that the lists of differences of every two -OOC distinct codeword-sets are disjoint. Thus, an is a set of subsets of with sizes from and weight disdoes not have repeated eletribution sequence such that ments. Extending the terminology used by Yin [56] in the case is a singleton, we can also say that such a set that is a cyclic difference packing . The size of an OOC is the number of its codeword-sets and it is maximal when every other OOC with the same parameters has size not greater than . Lemma 1: If quence

is an -OOC of size with and normalized weight distribution se, then we have

Proof: It is obvious that can be partitioned into sets each of which has size and each of which has codeword-sets of weight for . Of exactly course, we have for every and hence . Thus, considering that

is an integer, we have

and the

assertion follows. An -OOC will be said optimal when its size reaches the upper bound given by the above lemma. -OOC is the The set of missing differences of an that are not covered by its list of differset of all elements of ences. An -OOC whose set of missing differences of order is said to be -regular. In particis the subgroup of ular, a 1-regular OOC is said to be perfect. The following result was stated in [52]. , then a -regular Lemma 2: If -OOC with and normalized weight distribution sequence is optimal. Using a different terminology, the existence problem for per-OOCs with a prime power and fect balanced of size at least two has been considered in [9] and completely and in solved in the special cases of [48] and [49]. The -regular OOCs can be described in terms of difference families (see [2]). The concept of a relative difference family (DF) with constant block size has been introduced in [8] as a natural extension of that of a relative difference set [42]. Subsequently, relative difference families with variable block-sizes have been considered in several papers such as [9], [11], [37]. A cyclic -DF is a set of subsets (base blocks) of with sizes from and whose differences cover exactly once, and no element of at all. If , for pointing out that has exactly

base blocks of size

for -DF. If

, we say that

is a we say

-DF. that is a balanced difference Another similar concept is that of a family where is a simple graph (see [12], [14]). This is a set of graphs (base graphs) isomorphic to with vertices in such that the list of all possible differences with an ordered pair of adjacent vertices of some base graph covers exactly once, and no element of at all. Such a difference family generates a cyclic -decomposition of , the complete -partite graph with parts of size . For general background on graph decompositions we refer to [6]. -OOC of size with It is clear that a -regular and can be viewed as a -DF. If is the nor-DF malized form of , it can be also viewed as a , that is the where is the windmill graph Wd one-point amalgamation of complete graphs (the blades of the for . windmill), of which have order The construction of a -DF can be facilitated by strong difference family (SDF for the use of a suitable short), a concept introduced in [12] as a generalization of that -SDF which corresponds to the case in which of a is the complete graph of order [10] (see also [3] and [35]). A -SDF is a collection of labelings of the vertices of with elements of such that the with and an differences ordered pair of adjacent vertices of cover all of exactly times. As shown in [12, p. 452–455], the most promising -SDF’s are those having since they often allow to get a -DF for every odd prime not smaller than the chromatic number of . In this paper, we give direct and explicit constructions for -DFs for some windmill graphs , where is the number of edges of and is any odd prime at least equal to the maximum order of the blades of . This will be realized with the implicit use of suitable or -SDFs according to whether or 1 (mod 4), respectively. Then, recursively, we construct an -DF for every integer coprime to 6 or 30 depending on the maximum order of the blades of . In this way we get, explicitly, seven infinite classes of -OOCs. In particular, optimal variable-weight we get a balanced optimal -OOC for every\break . Explicit constructions of some infinite classes of optimal -OOCs with have been also obtained in [50]–[52], [57] but, as far as the authors are aware, our main result gives the first concrete infinite classes having . On the other hand, we are going to prove, theoretically, that there are infinitely many values of for which there exists an -OOC whatever and are. We first need to recall that in [47], Wilson proved that for any graph with vertices and edges there exists a -DF for every prime (mod ) provided that . This asymptotic bound has been recently improved in [15] to , where is the degeneracy of , that is the largest minimal degree among the minimal degrees of all the subgraphs of . Let be positive integers, and gcd . Then from Dirichlet’s theorem on the distribution of primes (see [40,

BURATTI et al.: RELATIVE DIFFERENCE FAMILIES WITH VARIABLE BLOCK SIZES AND THEIR RELATED OOCS

p. 347]) we know that there exist infinitely primes such that . So, we have the following result. and any given weight disTheorem 1: For any given set tribution sequence there are infinitely many values of for which there exists an optimal -OOC. Proof: Let and let be the normalized form of . Considering that a perfect -OOC is equivalent to an -DF where is the windmill graph Wd , we have the existence -OOC as soon as is a prime such that of a perfect (mod ) and where is the number of edges of and is its degeneracy. Thus the assertion follows since there are infinitely many (mod ) in view of the Dirichlet’s theorem on primes the distribution of primes. II. DIRECT CONSTRUCTIONS In the sequel, for a fixed prime , the sets of quadratic residues will be denoted by and and of quadratic nonresidues of , respectively. Also, in the case that , we denote by a complete system of representatives for the cosets in . The first positive integer that is a quadratic of will be denoted by . Note that is a prime non-residue of with since in the opposite case we would have and hence both and are squares by definition of . It would follow that their product is also a square, that is a contradiction. We will need the following elementary facts. Lemma 3: For a fixed odd prime , we have: (mod 4); i) (mod 8); ii) (mod 12); iii) iv) (mod 10); (mod 8) ; v) (mod 8) and (mod 3) ; vi) vii) (mod 8) and (mod 3) . Proof: (i)-(iv) are very well known consequences of the Law of Quadratic Reciprocity (see, e.g., [40]). (mod 8) so that is an odd prime by Assume that so that and (ii). Thus we obviously have are squares by definition of . We also have , since is a positive integer not reaching . Hence we have . If (mod 3) we have so that 3 is is a positive integer not a square. Thus, considering that since is product reaching , we have that of the two squares 3 and . Similarly, for (mod 3) we since is product of the two squares have that 3 and . Now we we illustrate the strategy that we will apply for realizing an -DF where is a windmill graph , is the number of Wd its edges, and is an odd prime not dividing so that can in view of the Chinese Remainder be identified with Theorem. We will use an -SDF for (mod 4) -SDF for (mod 4). In the first case the and an , and we identify it with the SDF is a singleton, say

7491

of the lists of values of on each of collection blades of . In the second case the SDF the , and we identify it with the pair has size two, say of the values of of collections and on the blades. be positive integers. For , and Let define . (mod 4), . Construction I: be an -SDF and let Let be a set of subsets of such that the on is for . Considering projection of is an -SDF, it is clear that has the form that where each is a pair of elements of , say . In the case that is a nonfor each square for each , we have and hence is an -DF that is equivalent -DF. to an We point out that the above construction always succeeds for sufficiently large in view of Theorem 5.1 in [15]. We also note (mod 4) because in this case that it cannot succeed for is of the form and hence the two elements of are both squares or both non-squares by Lemma 3 (i). (mod 4), . Construction II: be an -SDF. Let Then take a set of subsets of with the projections of and on coincident with and , respectively, for . Considering that is an -SDF, we have with a quadruple of elements of for each . Assume that the ’s and the ’s can be taken in such a way that each is with . In this of the form for each and hence case we have is an -DF that is equivalent -DF. to an A.

Difference Families

Even though we do not need to split the proof of the next or 3 (mod 4), the reader theorem into the two cases of could recognize that Construction I is essentially applied using

as a Also, for the case of struction II using

-SDF in the case of (mod 4). (mod 4), we essentially apply Conwith as above and .

Theorem 2: There exists a balanced -DF for every prime . Proof: Let us show, first, that the statement is , namely that there exists a baltrue when -DF. Such a DF is given by anced with

Now assume

and identify

with

.

7492

Consider three blocks of

We have hence, setting have


of the following form:

and , we . Now note that we have where

Theorem 3: There exists a -DF for every prime . Proof: The statement is true for since a -DF is given by where we have the equation shown at the , identify with bottom of the page. Now assume , and split the proof into two cases according to the congruence class of modulo 4. 1st case: (mod 4). Consider four subsets of of the following form:

We have

where

Assume that the following condition holds: (3) In this case we have hence, setting

for

and , we have

. It follows that , i.e., is a balanced -DF. The assertion will follow if we prove that there exists a 7tuple satisfying (3) for each prime . Using Lemma 3, the reader can easily check that such a 7tuple can be taken as follows:

Assume that the following condition holds: (4) In this case, we have hence, setting we have

for every

and , is a

namely -DF. The assertion will follow if we are able to exhibit a 9tuple satisfying (4). Taking into account of Lemma 3, the reader can easily check that such a 9tuple is given by

The assertion follows. B.

Difference Families

In the proof of the next theorem the reader can recognize that we essentially apply Construction I using

as a -SDF for the case of (mod 4). Instead, for the case of (mod 4), we essentially apply Construction II using with

2nd case: (mod 4). Consider eight subsets of

of the following form:


We have

where

7493

with

taken as follows:

One can see that

is a -DF.

(mod 4). 2nd case: Consider the six subsets of

Assume that the following condition holds:

(5) In this case we have for and hence, setting , we have . It follows that

, i.e., is a -DF. The assertion will follow if we prove that there exists an satisfying (5) 11tuple for each prime . Using, again, Lemma 3, one can check that such a 11tuple can be taken as follows:

with

defined by

One can see that The next theorems will not proved in all details. The reader can use the proofs of the previous theorems as a pattern. C.

Difference Families

Theorem 4: There exists a -DF for . every prime Proof: Let us show, first, that the statement is true when , namely that there exists a -DF. It is straightforward to see that such a DF is given by where the ’s are defined as follows:

Now assume , let us identify with , and split the proof into two cases according to the residue class of modulo 4. (mod 4). 1st case: Consider the five subsets of :

is a D.

-DF. Difference Families

Theorem 5: There exists a -DF for . every prime Proof: The statement is true when , namely there -DF. Such a DF is given by exists a with

Now assume and identify Consider the following five blocks of

with :

.

7494


where the 11tuple given by

is

One can see that we get the equation shown at the bottom of the -DF. page, which is a The assertion follows. E.

where

is given by

with or according to whether we have (mod 3), respectively. One can see that

or

Difference Families

Theorem 6: There exists a every prime . with Proof: Let us identify (mod 4). 1st case: Consider the three subsets of :

with

-DF for .

is a balanced F.


-DF for Theorem 7: There exists a . every prime Proof: The assertion is true for , namely there ex-DF. Such a DF is given by ists a balanced with

taken as follows: Now assume , let us identify with , and split the proof into two cases according to the residue class of modulo 4. 1st case: (mod 4). Consider the three subsets of :

Reasoning as in the previous theorems one can see that

with is a balanced 2nd case: (mod 4). Consider the four subsets of

given by:

-DF.

with

or 3 according to whether 5 is a square or not


One can see that -DF. balanced (mod 4). 2nd case: Consider the four subsets of

is a

(mod 4). 1st case: Consider the four subsets of

7495

:

with where

given by

is given by

One can see that balanced (mod 4). 2nd case: Consider the five subsets of

is a -DF.

with the signs of and that are or according to whether or 1 (mod 3), respectively. we have One can see that

is a balanced G.


-DF for Theorem 8: There exists a . every prime , namely there exists Proof: The assertion is true for -DF. Such a DF is given by a balanced where the ’s are defined as follows:

Now assume , let us identify with , and split the proof into two cases according to the residue class of modulo 4.

with , defined by the equation shown at the bottom of the page.Note that when from Lemma 3, thus, . One can see that

is a balanced

-DF. III. RECURSIVE CONSTRUCTIONS

We recall that the chromatic number of a graph , denoted by , is the minimum positive integer for which it is possible to color the vertices of with distinct colors in such a way that any two adjacent vertices have distinct colors. As a consequence of a general theorem given in [14] the following result holds.

7496


Theorem 9: If there exists a -DF and is a positive integer whose smallest prime factor is not inferior than , then there exists a -DF. Let be a set of primes greater than and assume that -DF for every . Then applying there exists a Theorem 9 it is easily seen by induction that there exists a -DF for every integer whose prime factors all belong to . Thus, considering that the chromatic number of a windmill graph is the maximum order of its blades and taking into account all results obtained in the previous section, we can say that if is one of the following windmill graphs

and

is the number of its edges, then there exists an -DF for every coprime to 6 or 30 according to whether the maximum order of the blades is 5 or 6, respectively. In terms of OOCs, we have:

Theorem 10: For every integer coprime to 6 there exists: -OOC; a balanced 19-regular -OOC; a 22-regular a 28-regular -OOC; -OOC. a 25-regular For every integer coprime to 30 there exists: -OOC; a balanced 24-regular a balanced 28-regular -OOC; -OOC. a balanced 34-regular Each code in the above theorem is optimal in view of Lemma 2. IV. CONCLUSION In this paper, we present explicit constructions of seven new infinite classes of optimal variable-weight optical orthogonal codes. We note, in particular, that our main result, together with -OOCs [5], [41] and some others those concerning -OOCs with and about balanced (see [50]–[52], [57]) allows us to claim that for every set with and every integer coprime to 6 or not, there exists a or 30 according to whether -OOC. balanced and optimal ACKNOWLEDGMENT The authors wish to thank Associate Editor Nam Yul Yu and the anonymous referees for their comments and suggestions that much improved the quality of this paper. REFERENCES [1] R. J. R. Abel and M. Buratti, “Some progress on (v; 4; 1) difference families and optical orthogonal codes,” J. Combin. Theory, vol. 106, pp. 59–75, 2004. [2] R. J. R. Abel and M. Buratti, “Difference families,” in Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds., 2nd ed. Boca Raton, FL: Chapman & Hall/CRC, 2006, pp. 392–409. [3] K. T. Arasu and S. Sehgal, “Cyclic difference covers,” Australas. J. Combin., vol. 32, pp. 213–223, 2005. [4] S. Bitan and T. Etzion, “Constructions for optimal constant weight cyclically permutable codes and difference families,” IEEE Trans. Inf. Theory, vol. 41, no. 1, pp. 77–87, Jan. 1995.

[5] E. F. Brickell and V. Wei, “Optical orthogonal codes and cyclic block designs,” Congr. Numer., vol. 58, pp. 175–182, 1987. [6] D. Bryant and S. El-Zanati, “Graph decompositions,” in Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, Eds., 2nd ed. Boca Raton, FL: Chapman & Hall/CRC, 2006, pp. 477–486. [7] M. Buratti, “Cyclic designs with block size 4 and related optimal optical orthogonal codes,” Des. Codes Cryptogr., vol. 26, pp. 111–125, 2002. [8] M. Buratti, “Recursive constructions for difference matrices and relative difference families,” J. Combin. Des., vol. 6, pp. 165–182, 1998. [9] M. Buratti, “Pairwise balanced designs from finite fields,” Discr. Math., vol. 208/209, pp. 103–117, 1999. [10] M. Buratti, “Old and new designs via strong difference families,” J. Combin. Des., vol. 7, pp. 406–425, 1999. [11] M. Buratti, “On point-regular linear spaces,” J. Stat. Plann. Inf., vol. 94, pp. 139–146, 2001. [12] M. Buratti and L. Gionfriddo, “Strong difference families over arbitrary graphs,” J. Combin. Des., vol. 16, pp. 443–461, 2008. [13] M. Buratti, K. Momihara, and A. Pasotti, “New results on optimal (v; 4; 2; 1) optical orthogonal codes,” Des. Codes Cryptogr., vol. 58, pp. 89–109, 2011. [14] M. Buratti and A. Pasotti, “Graph decompositions with the use of difference matrices,” Bull. Inst. Combin. Appl., vol. 47, pp. 23–32, 2006. [15] M. Buratti and A. Pasotti, “Combinatorial designs and the theorem of Weil on multiplicative character sums,” Finite Fields Appl., vol. 15, pp. 332–344, 2009. [16] Y. Chang, R. Fuji-Hara, and Y. Miao, “Combinatorial constructions of optimal optical orthogonal codes with weight 4,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1283–1292, May 2003. [17] Y. Chang and L. Ji, “Optimal (4up; 5; 1) optical orthogonal codes,” J. Combin. Des., vol. 12, pp. 346–361, 2004. [18] Y. Chang and Y. Miao, “Constructions for optimal optical orthogonal codes,” Discr. Math., vol. 261, pp. 127–139, 2003. [19] K. Chen and L. Zhu, “Existence of (q; 6; 1) difference families with q a prime power,” Des. Codes Cryptogr., vol. 15, pp. 167–174, 1998. [20] K. Chen, G. Ge, and L. Zhu, “Starters and related codes,” J. Stat. Plan. Inf., vol. 86, pp. 379–395, 2000. [21] W. Chu and C. J. Colbourn, “Recursive constructions for optimal (n; 4; 2)-OOCs,” J. Combin. Des., vol. 12, pp. 333–345, 2004. [22] W. Chu and S. W. Golomb, “A new recursive construction for optical orthogonal codes,” IEEE Trans. Inf. Theory, vol. 49, no. 11, pp. 3072–3076, Nov. 2003. [23] H. Chung and P. V. Kumar, “Optical orthogonal codes-new bounds and an optimal construction,” IEEE Trans. Inf. Theory, vol. 36, no. 7, pp. 866–873, Jul. 1990. [24] F. R. K. Chung, J. A. Salehi, and V. K. Wei, “Optical orthogonal codes: Design, analysis, and applications,” IEEE Trans. Inf. Theory, vol. 35, no. 5, pp. 595–604, May 1989. [25] R. Fuji-Hara and Y. Miao, “Optical orthogonal codes: Their bounds and new optimal constructions,” IEEE Trans. Inf. Theory, vol. 46, no. 11, pp. 2396–2406, Nov. 2000. [26] R. Fuji-Hara, Y. Miao, and J. Yin, “Optimal (9v; 4; 1) optical orthogonal codes,” SIAM J. Discr. Math., vol. 14, pp. 256–266, 2001. [27] G. Ge and J. Yin, “Constructions for optimal (v; 4; 1) optical orthogonal codes,” IEEE Trans. Inf. Theory, vol. 47, no. 11, pp. 2998–3004, Nov. 2001. [28] S. W. Golomb, Digital Communication With Space Application. Los Altos, CA: Penisula, 1982. [29] 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. [30] S. Ma and Y. Chang, “A new class of optimal optical orthogonal codes with weight five,” IEEE Trans. Inf. Theory, vol. 50, no. 8, pp. 1848–1850, Aug. 2004. [31] S. Ma and Y. Chang, “Constructions of optimal optical orthogonal codes with weight five,” J. Combin. Des., vol. 13, pp. 54–69, 2005. [32] J. L. Massey and P. Mathys, “The collision channel without feedback,” IEEE Trans. Inf. Theory, vol. IT-31, no. 3, pp. 192–204, Mar. 1985. [33] M. Mishima, H. L. Fu, and S. Uruno, “Optimal conflict-avoiding codes 0 (mod 16) and weight 3,” Des. Codes Cryptogr., vol. of length n 52, pp. 275–291, 2009. [34] K. Momihara, “Necessary and sufficient conditions for tight equi-difference conflict avoiding codes of weight three,” Des. Codes Cryptogr., vol. 45, pp. 379–390, 2007. difference fam[35] K. Momihara, “On cyclic 2(k 1)-support (n; k ) ilies,” Finite Fields Appl., vol. 15, pp. 415–427, 2009.

0


[36] K. Momihara, “Strong difference families, difference covers, and their applications for relative difference families,” Des. Codes Cryptogr., vol. 51, pp. 253–273, 2009. [37] K. Momihara, “New optimal optical orthogonal codes by restrictions to subgroups,” Finite Fields Appl. to appear. [38] K. Momihara and M. Buratti, “Bounds and constructions of optimal (n; 4; 2; 1) optical orthogonal codes,” IEEE Trans. Inf. Theory, vol. 55, no. 2, pp. 514–523, Feb. 2009. [39] K. Momihara, M. Müller, J. Satoh, and M. Jimbo, “Constant weight conflict-avoiding codes,” SIAM J. Discr. Math., vol. 21, pp. 959–979, 2007. [40] M. B. Nathanson, Elementary Methods in Number Theory. New York: Springer-Verlag, 2000. [41] R. Peltesohn, “Eine Losung der beiden Heffterschen Differenzenprobleme,” Compos. Math., vol. 6, pp. 251–257, 1938. [42] A. Pott, “A survey on relative difference sets,” in Groups, Difference Sets, and the Monster. Berlin/New York: de Gruyter, 1996. [43] J. A. Salehi, “Code division multiple access techniques in optical fiber networks—Part I fundamental principles,” IEEE Trans. Commun., vol. 37, no. 8, pp. 824–833, Aug. 1989. [44] J. A. Salehi, “Emerging optical code-division multiple-access communications systems,” IEEE Network, vol. 3, pp. 31–39, Mar. 1989. [45] 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, no. 8, pp. 834–842, Aug. 1989. [46] 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: , 1988, pp. 814–823, Amer. Inst. Phys.. [47] R. M. Wilson, “Decompositions of complete graphs into subgraphs isomorphic to a given graph,” in Proc. 5th British Combinatorial Conf., C. St. J. A. Nash-Williams and J. H. Van Lint, Eds., 1975, vol. XV, pp. 647–659, Congr. Numer.. [48] D. Wu, M. Cheng, and Z. Chen, “A note on the existence of balanced (q; f3; 4g; 1) difference families,” Australas. J Combin., vol. 41, pp. 171–174, 2008. [49] D. Wu, M. Cheng, Z. Chen, and H. Luo, “The existence of balanced (v; f3; 6g; 1) difference families,” Science China Inf. Sci., vol. 53, pp. 1584–1590, 2010. [50] D. Wu, P. Fan, H. Li, and U. Parampalli, “Optimal variable-weight optical orthogonal codes via cyclic difference families,” in Proc. 2009 IEEE Int. Symp. Information Theory (ISIT’09), Jun. 28–Jul. 3 , pp. 448–452. [51] D. Wu, P. Fan, X. Wang, and M. Cheng, “A new class of optimal variable-weight OOCs based on cyclic difference families,” in Proc. 4th Int. Workshop on Signal Design and its Application in Communications (IWSDA’2009), , Oct. 19–23, 16–19, . [52] D. Wu, H. Zhao, P. Fan, and S. Shinohara, “Optimal variable-weight optical orthogonal codes via difference packings,” IEEE Trans. Inf. Theory, vol. 56, no. 8, pp. 4053–4060, Aug. 2010. [53] G. C. Yang, “Variable weight optical orthogonal codes for CDMA networks with multiple performance requirements,” Proc. IEEE GLOBECOM’93, vol. 1, pp. 488–492, Dec. 1993. [54] G. C. Yang, “Variable-weight optical orthogonal codes for CDMA networks with multiple performance requirements,” IEEE Trans. Commun., vol. 44, no. 1, pp. 47–55, Jan. 1996. [55] G. C. Yang and T. E. Fuja, “Optical orthogonal codes with unequal auto- and cross-correlation constraints,” IEEE Trans. Inf. Theory, vol. 41, no. 1, pp. 96–106, Jan. 1995.

7497

[56] J. Yin, “Some combinatorial constructions for optical orthogonal codes,” Discr. Math., vol. 185, pp. 201–219, 1998. [57] H. Zhao, D. Wu, and P. Fan, “Construction of optimal variableweight optical orthogonal codes,” J. Combin. Des, vol. 18, pp. 274–291, 2010.

Marco Buratti received the Math degree in 1985 at the University “La Sapienza”, Rome, Italy. He was an Assistant Professor at the University of L’Aquila and he is currently a Full Professor of Geometry at the University of Perugia. His area of research is Combinatorial Designs. He received the “Hall Medal” from the Institute of Combinatorics and its Applications in 1998. He is a member of the Editorial Board of the Journal of Combinatorial Designs, a council member of the Institute of Combinatorics and its Applications and a contributor of the CRC Handbook of Combinatorial Designs.

Yueer Wei was born in Guangxi, China, in 1982. She received the B.S. degree and M.S. degree in mathematics from Guangxi Normal University, China, in 2008 and 2011, respectively. Her research interests include combinatorial design theory and its applications.

Dianhua Wu was born in Shandong, China, in 1966. He received the Ph.D. degree in mathematics from Suzhou University, PRC, in 2000. He was with Guangxi Normal University as an Assistant Teacher (1988–1991), a Lecturer (1992–1996), an Associate Professor (1997–2000). From 2001, he has been a full professor. His research interests include combinatorial design theory, coding theory, communication theory, and their interactions.

Pingzhi Fan (M’93–SM’99) received his M.S. degree in computer science from the Southwest Jiaotong University, PRC, in 1987, and 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, UK (1997–), a guest professor Shanghai Jiaotong University (1999–). He was a recipient of the UK ORS Award (1992), and the NSFC Outstanding Young Scientist Award (1998). He is the inventor of 22 patents, and the author of over 200 research journal papers and 8 books, including six books published by John Wiley and Sons Ltd, RSP (1996), IEEE Press (2003, 2006) and 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.

Minquan Cheng was born in Henan, China, in 1981. He received the M.S. degree in mathematics from Guangxi Normal University, China, in 2008. He is working towards the Ph.D. degree in engineering with Institute of Policy and Planning Sciences, University of Tsukuba, Japan. His research interests include combinatorial design theory, coding theory, cryptography, and their interactions.