ISIT 2009, Seoul, Korea, June 28 - July 3, 2009
Optimal Variable-Weight Optical Orthogonal Codes via Cyclic Difference Families Dianhua Wu, Pingzhi Fan, Hengchao Li
Udaya Parampalli
Keylab of Information Coding and Transmission Southwest Jiaotong University Chengdu, 610031, China Email:
[email protected],
[email protected],
[email protected]
Department of Computer Science and Software Engineering The University of Melbourne Victoria, 3010, Australia Email:
[email protected]
integer τ, 0 < τ < n,
Abstract—Variable-weight Optical orthogonal code (OOC) was introduced by G-C Yang for multimedia optical CDMA systems with multiple quality of service (QoS) requirement. In this paper, a construction for optimal variable-weight OOCs via cyclic difference families is given. Several new constructions for cyclic difference families are also given. By using these constructions, new optimal (n, W, 1, Q)-OOCs for 2 ≤ |W | ≤ 4 are constructed.
n−1 X t=0
•
I. I NTRODUCTION Optical orthogonal codes (OOCs) were introduced by Salehi, as signature sequences to facilitate multiple access in optical fibre networks [1], [2]. OOCs are codes over symbols 0, 1 with constant weight property and can be viewed as constant weight error correcting codes [3] or balanced incomplete designs. The OOCs have found wide range of applications [4][8]. Most existing work 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. It is interesting to note that the variable-weight (or ”multi-weight”) OOCs can generate larger code size than that of constant-weight OOCs [9]. In 1996, G-C Yang introduced multimedia optical CDMA communication system employing variable-weight OOCs [10]. The variable-weight property of the OOCs enables the system to meet multiple QoS requirement. Recently variable-weight OOCs have attracted much attention [9], [10], [11]. Based on the notation of [10], throughout this paper, let W, L, and Q denote the sets {w0 , w1 , . . . , wp }, {λ0a , λ1a , . . . , λpa } and {q0 , q1 , . . . , qp }, respectively, as defined below. Definition: An (n, W, L, λc , Q) variable-weight optical orthogonal code C, or (n, W, L, λc , Q)-OOC, is a collection of binary n-tuples such that the following three properties hold: •
•
Weight Distribution: Every n-tuple in C has a Hamming weight contained in the set W ; furthermore, there are exactly qi |C| codewords of weight wi , i.e., qi indicates the fraction of codewords of weight wi . Periodic Auto-correlation: For any x = (x0 , x1 , . . . , xn−1 ) ∈ C with Hamming weight wi ∈ W , and any 978-1-4244-4313-0/09/$25.00 ©2009 IEEE
xt xt⊕τ ≤ λia ,
where the summation is carried by treating binary symbols as reals. Periodic Cross-correlation: Similarly, for any x 6= y, x = (x0 , x1 , . . . , xn−1 ) ∈ C, y = (y0 , y1 , . . . , yn−1 ) ∈ C, and any integer τ , n−1 X
xt yt⊕τ ≤ λc .
t=0
The notion (n, W, λ, Q)- OOC is used to denote a (n, W, L, λc , Q)- OOC with the property that λ0a = λ1a = . . . = λpa = λc = λ. The term variable-weight optical orthogonal code, or variable-weight OOC, is also used if there is no need to list the parameters. In [10], G-C Yang gave an upper and lower bounds on the size of variable-weight OOCs. Some constructions for variable-weight OOCs were also presented, some of them are optimal. In [9], F-R Gu and J-S Wu had constructed variableweight OOCs by using pairwise balanced designs (PBDs) and packing designs with a partition. Some combinatorial constructions were presented in [11]. The variable-weight OOCs in [10] are with |W | = 2, and not all variable-weight OOCs in [10] are optimal. Further, some of the OOCs have λa = 2. In practice, it is desirable to have OOCs with |W | more than 2 to meet different QoS constraints and λa close to 1. In this paper, we focus our attention on constructing optimal variable-weight OOCs. An explicit construction for optimal variable-weight OOCs from difference families is given. To the authors knowledge, no explicit description for constructing variable-weight OOCs from difference families is presented before. A method to construct a (v, {3, 6}, 1) difference family with v = 36t + 1, a prime number, is presented. A method to combine difference families with PBDs to produce new difference families is also given. Using these difference families, several new optimal (n, W, 1, Q)-OOCs with 2 ≤ |W | ≤ 4 are obtained.
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The paper is arranged as follows. In Section II, some necessary preliminaries concerning cyclic pairwise balanced designs needed for the paper and an upper bound on the code size are shown. In section III, main construction of variable-weight OOCs using cyclic difference families and a construction of (v, U, 1)-DFs based on (v, k, 1)-DFs and (k, U, 1)-PBD are given. Then in Section IV, a method to construct (v, {3, 6}, 1)-DFs is presented, several new optimal variable-weight OOCs are then obtained.
It is well known that (G, F) is a (v, W, 1)-DF if and only if ∆F = G \ {0}. (v, w, 1)-DFs were used to construct OOCs with constant weight w [12]-[17]. Some combinatorial constructions for variable-weight optical orthogonal codes can be found in [9], [10], [11]. III. O PTIMAL VARIABLE - WEIGHT OPTICAL ORTHOGONAL CODES AND CYCLIC DIFFERENCE FAMILIES
In this section, main construction of variable-weight OOCs using cyclic difference families is presented.
II. P RELIMINARIES In this section we briefly discuss mathematical background needed for the paper. A. Bounds The definition of a variable-weight OOC is a generalization of the definition of OOC given in [1] and [2]. The work of G-C Yang [10] contains lower and upper bounds on the size of variable-weight OOCs to judge the goodness of the constructions. Here we provide only an upper bound on the size of variable-weight OOCs given [10]. Theorem 2.1: Let λia ≥ λ(λia ∈ L). Then Φ(n, W, L, λc , Q) (n−1)(n−2)...(n−λ) , ≤ P p
A. Construction-A The first result is that, a cyclic DF family implies an OCC code. The following is an example of cyclic difference family. Example 1 Let F = {{0, 3, 16}, {0, 8, 17}, {0, 10, 12}, {0, 1, 5, 25}}. Then (Z31 , F) is a cyclic (31, {3, 4}, 1)-DF. The following lemma which is self evident. Lemma 3.1: Suppose that W = {w0 , w1 , . . . , wp }, then the necessary conditions for the existence of a cyclic (v, W, 1)-DF with ki base blocks of size wi , 0 ≤ i ≤ p, is that v=
qi wi (wi −1)(wi −2)...(wi −λ)/(λia )
ki wi (wi − 1) + 1.
i=0
i=0
where, Φ(n, W, L, λc , Q) is defined as Φ(n, W, L, λc , Q) =max{|C| : C is an (n, W, L, λc , Q)-OOC}. An (n, W, L, λc , Q)-OOC with cardinality Φ(n, W, L, λc , Q) is said to be optimal. To construct optimal variable-weight OOCs, difference families and pairwise balanced designs(PBDs) will be used. B. Cyclic Pairwise Balanced Designs For a cyclic (v, W, λ)-PBD (Zv , B), let B = {b1 , b2 , . . . , bw } be a block in B. The block orbit containing B is defined to be the set of the following distinct blocks: i B σ = B + i = {b1 + i, b2 + i, . . . , bw + i} (mod v) for i ∈ Zv . If a block orbit has v distinct blocks, i.e., its setwise stabilizer is equal to the identity {0}, then this block orbit is said to be full, otherwise short. Choose an arbitarily fixed block from each block orbit and then call it a base block. From now on, we assume that all cyclic (v, W, λ)-PBDs contain only full block orbits, that is, for any base block B ∈ B, its setwise stabilizer GB = {0}. Let F be the family of all base blocks of such a cyclic (v, W, λ)-PBD. Then the pair (Zv , F) is called a cyclic (v, W, λ) difference family ((v, W, λ)-DF in short). When W = {w}, the difference family is denoted by (v, w, λ)-DF. The term cyclic difference family is also used if there is no need to list the parameters. Suppose B is a set of an Abelian group G, define ∆B = {x − y : x, y ∈ B, x 6= y}. Suppose F = {Bi : 1 ≤ i ≤ t} is a set of subsets of G, W = {|Bi | : 1 ≤ i ≤ t}, define t S ∆F = ∆Bi . i=1
p X
Suppose that there exists a cyclic (v, W, 1)-DF with ki base blocks of size wi , 0 ≤ i ≤ p, one can construct a (0, 1)-sequence of length v and weight in W from each base block whose nonzero bit positions are exactly indexed by the base block. The resultant (0, 1)-sequences have exactly ki p P sequences of weight wi for 0 ≤ i ≤ p. Let K = ki , qi = i=0
ki /K, 0 ≤ i ≤ p. It is clear that the (0, 1)-sequences forms a (v, W, 1, Q)-OOC C with K code words. Thus, We have the following result. Theorem 3.2: Suppose that there exists a cyclic (v, W, 1)DF with ki base blocks of size wi , 0 ≤ i ≤ p, then the construction-A implies that there exists an optimal (v, W, 1, Q)-OOC C with K code words, where K=
p X
ki , and qi = ki /K, 0 ≤ i ≤ p.
i=0
Proof: Consider the code generated by the cyclic (v, W, 1)-DF as in Construction-A. The difference property implies that cross and auto correlation of the codewords are less than or equal to 1 as needed for the OCC. The optimality of the code follows from Theorem 2.1 as (v−1) (v−1) = P = K. p p P qi wi (wi −1)
i=0
ki wi (wi −1)/K
i=0
Example 2 Consider the cyclic difference family given in Example 1. Let F = {{0, 3, 16}, {0, 8, 17}, {0, 10, 12}, {0, 1, 5, 25}}. Then (Z31 , F) is a cyclic (31, {3, 4}, 1)-DF. It is easy to check that C forms an optimal (31, {3, 4}, 1, {3/4, 1/4})OOC, where
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C = {1001000000000000100000000000000, 1000000010000000010000000000000, 1000000000101000000000000000000, 1100010000000000000000000100000}. From Theorem 3.2, in order to construct optimal variableweight optical orthogonal codes, one needs only to construct its corresponding cyclic difference families. There are two ways to construct (v, U, 1)-DFs. The one is direct construction as will be shown in next section for (v, {3, 6}, 1)-DFs, the other is to use (v, k, 1)-DFs and (k, U, 1)-PBD as follows. Theorem 3.3: If there exist both a cyclic (v, k, 1)-DF with t base blocks and a (k, U, 1)-PBD with aj blocks of size uj , 0 ≤ j ≤ f , U = {u0 , u1 , . . S . , uf }, then for each 1 ≤ s ≤ t, there exists a cyclic (v, {k} U, 1)-DF with saj base blocks of size uj , 0 ≤ j ≤ f , and t − s blocks of size k. Proof: Suppose (Zv , F) is a (v, k, 1)-DF, where F = {Bi : 1 ≤ i ≤ t}, |Bi | = k, 1 ≤ i ≤ t. For each 1 ≤ j ≤ s, there exists a (k, U, 1)-PBD (Bj , Bj ). From the definition of PBDs, ∆Bj = ∆Bj . So, t t S S S S S ∆B1 ∆B2 . . . ∆Bs ∆Bi = ∆Bh = Zv \ {0}. i=s+1
h=1
This completes the proof. Note that F-R Gu and J-H Wu [9] used Singer difference sets and PBDs to construct optimal variable-weight OOCs. Theorem 3.3 is a generalization, which provides a method to combine a cyclic-DF with uniform block size k and a PBD to construct optimal variable-weight OCCs. Example 3 Let F1 = {{0, 1, 3}, {1, 2, 4}, {2, 3, 5}, {3, 4, 6}, {0, 4, 5}, {1, 5, 6}, {0, 2, 6}}. Then (Z7 , F1 ) is a (7, 3, 1)-BIBD. From [20], (Z379 , F2 ) is a cyclic (379, 7, 1)-DF with 9 base blocks, where F2 = {Bi : Bi = 221i {0, 1, 3, 24, 74, 167, 245} : 0 ≤ i ≤ 8}. Breaking block B0 by a (7, 3, 1)-BIBD. Let A = {{0, 1, 24}, {1, 3, 74}, {3, 24, 167}, {24, 74, 245}, {0, 74, 167}, {1, 167, 245}, {0, 3, 245}}. Then (Z379 , F) forms a cyclic (379, {3, 7}, 1)-DF with 7 8 SS blocks of size 3 and 8 blocks of size 7, where F = A Bi . i=1
Thus an optimal (379, {3, 7}, 1, {7/15, 8/15})-OOC is obtained. When W contains at least two distinct elements, a (v, W, λ)DF is said to be balanced if the number of base blocks of size w is constant for each w ∈ W [18]. In the remainder of this section, some known existing results on the existence of cyclic difference families, for the purpose of constructing variable-weight OCCs in next section are given. Lemma 3.4: ([19]) Let v ≡ 1 (mod 42) be a prime. If v < 261239791, v 6= 43, 127, 211 or v > 1.236597 × 1013 . Then there exists a cyclic (v, 7, 1)-DF. Let S9 = {v : v is a prime, v ≡ 1 (mod 72) < 105 , v 6= 433, 1009}, S10 = {1171, 1621, 2521, 2791, 1971, 3061, 3331, 3511,
3691, 4051, 4231, 4591, 4861, 5581, 5851, 6121, 6211, 6301, 6571, 6661, 6841, 7561, 7741}, S11 = {10781, 12211, 12541, 14081, 14411, 14741, 14851, 15401, 16061, 19031, 19141, 21011, 21341, 22111, 22441, 23321, 23761, 24091, 24971, 25411, 27611}, Lemma 3.5: ([13]) There exists a cyclic (v, w, 1)-DF for v ∈ Sw , 9 ≤ w ≤ 11. Lemma 3.6: ([21]) There exists a balanced cyclic (v, {3, 4}, 1)-DF for each prime v ≡ 1 (mod 18) and v ≥ 19. In order to construct optimal variable-weight optical orthogonal codes from Theorem 3.3, one needs the following PBDs. Lemma 3.7: ([20]) The following designs exist. (1) A (9, 3, 1)-BIBD with 12 blocks of size 3; (2) A (10, {3, 4}, 1)-PBD with 9 blocks of size 3 and 3 blocks of size 4; (3) A (11, {3, 5}, 1)-BIBD with 15 blocks of size 3 and 1 blocks of size 5. Lemma 3.8: There exists a (11, {2, 3, 4}, )-PBD with 1 block of size 2, 6 blocks of size 3, and 6 blocks of size 4. Proof: Let F = {{9, 10}, {0, 4, 8}, {1, 5, 6}, {2, 3, 7}, {0, 5, 7}, {1,3,8}, {2, 4, 6}, {0, 1, 2, 9}, {3, 4, 5, 9}, {1,3,8}, {6, 7, 8, 9}, {0, 3, 6, 10}, {1, 4, 7, 10}, {1,3,8}, {2, 5, 8, 10}}. Then (Z11 , F) is the required design. IV. C ONSTRUCTION OF CYCLIC DIFFERENCE FAMILIES In this section, a method to construct cyclic difference families defined over prime fields is presented. New optimal variable-weight OOCs are obtained. Let v be a prime and GF (v) represents prime field of order v. Let GF (v)∗ = GF (v) \ {0}, H is the multiplicative group of GF (v) and ω is a fixed primitive root in GF (v). For a fixed divisor d of v − 1, H d is the group of dth powers of H. The following result is obtained. Theorem 4.1: Let v = 36t + 1 be a prime, ε be 3rd primitive root of unity in GF (v), A1 = {b, bε, bε2 }, A2 = {1, ε, ε2 , c, cε, cε2 }, where b, c ∈ GF (v)∗ . If set L of the following elements forms a system of distinct representatives (SDR) of H 6 : L = {b(ε − 1), ε − 1, c(ε − 1), c − 1, c − ε, c − ε2 } Then the family F = {Ai ω mj : |1 ≤ j ≤ t; 1 ≤ i ≤ 2} is a balanced cyclic (v, {3, 6}, 1)-DF. If ε − 1, c(ε − 1), c − 1, c − ε and c − ε2 lie in distinct cosets of H 6 , then one can easily choose b ∈ GF (v)∗ such that L forms a SDR of H 6 . One can use this condition to construct many balanced cyclic (v, {3, 6}, 1)-DFs. We only illustrate this construction for v < 3000. Let E = {37, 73, 181, 397, 433, 541, 577, 613, 757, 829, 937, 1009, 1117, 1153, 1297, 1549, 1621, 1657, 1693, 1801, 1873, 2017, 2053, 2089, 2161, 2269, 2341, 2377, 2521, 2557, 2593, 2917, 2953}.
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Then for each v ∈ E, with the aid of a computer, such element c can be found. Theorem 4.2: For each v ∈ E, there exists a balanced cyclic (v, {3, 6}, 1)-DF. Example 4 Let v = 73, ω = 5 be a primitive element of GF(73), then ε = 8, and C06 = {1, 3, 9, 27, 8, 24, 72, 70, 64, 46, 65, 49}, C16 = {5, 15, 45, 62, 40, 47, 68, 58, 28, 11, 33, 26}, C26 = {25, 2, 6, 18, 54, 16, 48, 71, 67, 55, 19, 57}, C36 = {52, 10, 30, 17, 51, 7, 21, 63, 43, 56, 22, 66}, C46 = {41, 50, 4, 12, 36, 35, 32, 23, 69, 61, 37, 38}, C56 = {59, 31, 20, 60, 34, 29, 14, 42, 53, 13, 39, 44}. Let c = 2, b = 5, then A1 = {5, 40, 28}, A2 = {1, 8, 64, 2, 16, 55}, and F = {Ai 56j : 1 ≤ i ≤ 2, 1 ≤ j ≤ 2} = {{15, 47, 11}, {45, 68, 33}, {3, 24, 46, 6, 48, 19}, {9, 72, 65, 18, 71, 57}}. (Z73 , F) is a cyclic balanced (73, {3, 6}, 1)-DF. The corresponding optimal (73, {3, 6}, 1, { 21 , 12 })-OOC C is as follows. 0000000000010001000000000000000000000000000 000010000000000000000000000000, 0000000000000000000000000000000001000000000 001000000000000000000000010000, 0001001000000000000100001000000000000000000 000101000000000000000000000000, 0000000001000000001000000000000000000000000 000000000000001000000010000011. Now, we can use the difference families to obtain optimal variable-weight OOCs. From Theorem 3.2 and Theorem 4.2, the following result is obtained. Theorem 4.3: There exists an optimal (v, {3, 6}, 1, {1/2, 1/2})-OOC for each prime v ∈ E. In the following, one can applying the known results on cyclic difference families and PBDs in Section III with the constructions to produce various optimal variable-weight OCCs. Note that with this approach one can construct OCCs with |W | greater than or equal to 2. From Theorem 3.2, Lemmas 3.4-3.8, the following result is obtained. Theorem 4.4: The following optimal variable-weight OCCs exist. (1) (v, {3, 4}, 1, {1/2, 1/2})-OOCs for primes v ≡ 1 (mod 18) and v ≥ 19; (2) (v, {3, 7}, 1, Q)-OOCs, where v is the same as in 7s t−s Lemma 3.4, t = (v −1)/42, 1 ≤ s < t, q0 = t+6s , q1 = t+6s ; (3) (v, {3, 9}, 1, Q)-OOCs, where v ∈ S9 , t = (v − 1)/72, 12s t−s 1 ≤ s < t, q0 = t+11s , q1 = t+11s ; (4) (v, {3, 4, 10}, 1, Q)-OOCs, where v ∈ S10 , t = (v − 9s 3s t−s 1)/90, 1 ≤ s < t, q0 = t+11s , q1 = t+11s , q2 = t+11s ; 3 1 (v, {3, 4}, 1, Q)-OOCs, where q0 = 4 , q1 = 4 ; (5) (v, {3, 5, 11}, 1, Q)-OOCs, where v ∈ S11 , t = (v − 15s s t−s 1)/110, 1 ≤ s < t, q0 = t+15s , q1 = t+15s , q2 = t+15s ; 15 1 (v, {3, 5}, 1, Q)-OOCs, where q0 = 16 , q1 = 16 ; (6) (v, {2, 3, 4, 11}, 1, Q)-OOCs, where v ∈ S11 , t = (v − ss 6s 6s 1)/110, 1 ≤ s < t, q0 = t+12s , q1 = t+12s , q2 = t+12s , t−s q3 = t+12s .
V. C ONCLUSION In this paper, a construction for optimal variable-weight OOCs via cyclic difference families is given. Direct construction for cyclic (v, {3, 6}) difference families and constructions for cyclic (v, W, 1)-DFs from cyclic (v, w, 1)-DFs and pairwise balanced designs are presented. By using these constructions and the known result on cyclic difference families, several new optimal variable-weight OOCs are constructed. ACKNOWLEDGMENT The work of Dianhua Wu was supported in part by NSFC(No.10561002) and Guangxi Science Foundation (0991089); the work of Pingzhi Fan was partially supported by the 863 High-Tech R and D Program (No.2007AA01Z228), NSFC (No.60772087), and the 111 Project (No.111-2-14); the work of Hengchao Li was supported in part by Key Laboratory of Universal Wireless Communications Lab.(BUPT), MOE, P. R. China. D. Wu is also with the Department of Mathematics, Guangxi Normal University, 541004, Guilin, Guangxi, P. R. China. R EFERENCES [1] 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. [2] 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. [3] H. Chung and P. V. Kumar, ”Optical orthogonal codes-new bounds and an optimal constructions,” IEEE Tras. Inform. Theory, vol. 36, pp. 866-873, July, 1990. [4] F. R. K. Chung, J. A. Salehi and V. K. Wei, ”Optical orthogonal codes: Design, analysis, and applications,” IEEE Trans. Inform. Theory, vol. 35, pp. 595-604, May 1989. [5] S. W. Golomb, ”Digital communication with space application,” Los Altos, CA: Penisula, 1982. [6] J. L. Massey and P. Mathys, ”The collision channel without feedback,” IEEE Trans. Inform. Theory, vol. 31, pp. 192-204, Mar. 1985. [7] J. A. Salehi, ”Emerging optical code-division multiple-access communications systems,” IEEE Network, vol. 3, pp. 31-39, Mar. 1989. [8] M. P. Vecchi and J. A. Salehi, ”Neuromorphic networks based on sparse optical orthogonal codes,” in Neural Information Processing SystemsNatural and Synthetic, New York: Amer. Inst. Phys., 1988, pp. 814-823. [9] F-R. Gu and J. Wu, ”Construction and performance analysis of variableweight optical orthogonal codes for asynchronous optical CDMA systems,” J. Lightw. Technol., vol. 23, pp 740-748, Feb. 2005. [10] 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. [11] I. B. Djordjevic, B. Vasic, J. Rorison, ”Design of multiweight unipolar codes for multimedia optical CDMA applications based on pairwise balanced designs,” J. Lightwave Technol., vol. 21, pp. 1850 - 1856, Sept. 2003. [12] R. Abel and M. Buratti, ”Some progress on (v, 4, 1) difference families and optical orthogonal codes,” J. Combin. Theory (Ser. A), vol. 106, pp. 59-75, 2004. [13] M. Buratti, ”A powerful method for constructing difference families and optimal optical orthogonal codes,” J. Combin. Des., vol. 5, pp. 13-25, 1995. [14] M. Buratti, ”Cyclic designs with block size 4 and related optimal optical orthogonal codes,” Des. Codes Cryptogr., vol. 26, pp. 111-125, 2002. [15] Y. Chang, R. Fuji-Hara and Y. Miao, ”Combinatorial constructions of optimal optical orthogonal codes with weight 4,” IEEE Trans. Inform. Theory, vol. 46, pp. 2396-2406, Nov. 2000. [16] R. Fuji-Hara and Y. Miao, ”Optical orthogonal codes: Their bounds and new optimal constructions,” IEEE Trans. Inform. Theory, vol. 46, pp. 2396-2406., Nov. 2000.
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ISIT 2009, Seoul, Korea, June 28 - July 3, 2009 [17] J. Yin, ”Some combinatorial constructions for optical orthogonal codes,” Discr. Math., vol. 185, pp. 201-219, 1998. [18] M. Buratti, ”Pairwise balanced designs from finite fields,” Discr. Math., vol. 208/209, pp. 103-117, 1999. [19] K. Chen, R. Wei and L. Zhu, ”Existence of (q, 7, 1) difference with q a prime power,” J. Combin. Des., vol. 10, pp. 126-138, 2002. [20] R. C. Mullin and H-D. O. F. Gronau, ”PBDs abd GDDs: The basic,” in CRC Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz, 2nd Eds, Boca Raton, FL: CRC, 1996, pp. 231-236. [21] D. Wu, Z. Chen and M. Cheng, ”A note on the existence of balanced (q, {3, 4}, 1) difference families,” The Australasian J. Combinatorics, vol. 41, pp. , 171-174, 2008.
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