| In 1989, optical orthogonal code (OOC for short) was introduced by SaleM, and applied in optical code division multiple access (OCDMA) system. In OCDMA system, each user is assigned to an OOC as the address code. To meet multiple QoS requirements, variable-weight optical orthogonal code (VWOOC) was introduced by Yang in 1996. The VWOOC can meet multiple quality of service requirements, and have much more codes than constant-weight OOC.Let W = {w1, ..., wr} be an ordering of a set of r integers greater than 1, Λa = (λa(1), ..., λa(1)) an r-tuple of positive integers, and Q = (q1, ..., qr) an r-tuple of positive rational numbers whose sum is 1. Without loss of generality, we assume that wq < w2 < ... < wr.An (n, W,Λa, λc, Q)-OOC is a set C of subsets (codeword-sets) of Zn with sizes (weights) (3)Periodic Cross-correlation Property (?) C, C’∈C, C≠ C’, t∈ Zn,If λa(i) =λa for every i, one simply says that C is an (n, W,λa,λc, Q)-OOC. Also, speaking of an (n, W, λ, Q)-OOC one means an (n, W, λa, λc, Q)-OOC where λa = λc =λ. We say that Q is normalized if it is written in the form V = (a1/b, ...,ar/b) with gcd(a1, ..., ar) = 1. Speaking of a balanced (n, W, Λa, λc)-OOC we mean an (n, W, Λa, λc, Q)-OOC with Q = (1/r,1/r,...,1/r).Yang gave the bound of an (n, W, Λa, λc, Q)-OOC in 1996. Afterwards, the upper bound was improved by Buratti et al. Let Φ(n, W, Λa, λc, Q) =max{|C| : C is an (n, W,Λa,λc, Q)-OOC}, thenBased on the above fact, an (n, W, Λa, λc, Q)-OOC is said to be optimal if it has max-imum size. The study of (n, W, Λa, λc, Q)-OOCs is mainly on the existences of (n, W, 1, Q)-OOCs. For unequal auto-and cross-correlation constraints, some work had been done on the constructions of optimal (n, {3, 4}, Λa, 1, Q)-OOCs. As far as the author is aware of, little is known on the existence of (n, W, Λa, λc, Q)-OOCs for W = {3, 5}. This thesis focuses at-tentions on (n, {3, 5}, Λa, λc, Q)-OOCs with unequal auto-and cross-correlation constraints.Let Q = (a1/b,a2/b) be normalized, Δ12 = 6a1 + 12a2,Δ22 = 4a1 + 12a2, Δ21 =4a1 + 20a2, the upper bounds on the maximum code size of an (n, {3, 5}, Λa, 1, Q)-OOC are obtained.Theorem 1.1 Let Q=(a1/b,a2/b) be normalized, thenTheorem 1.2 Let Q = (a1/b,a2/b) be normalized, thenThis thesis discusses the existences of optimal (n, {3, 5}, Λa 1, Q)-OOCs, the following results are obtained.Theorem 1.4 For any prime p > 7, there exist a 12-regular and an optimal balanced (12p, {3,5}, (2, 1), 1)-OOC. There exist optimal balanced (12p, {3,5}, (2, 1), 1)-OOCs for p ~{3,5,7}.Theorem 1.5 Let p - 5 (rood 8) be a prime, then there exists an optimal balanced (6p, {3, 5}, (2, 1), 1)-OOC. If p _> 13, then the OOC is also 6-regular.Theorem 1.6 If p= 3 (mod 4)> 7 is a prime, then there exist a 22-regular and an optimal (22p,{3,5}, (2,1), (1/3,2/3))-OOC.Theorem 1.7 Let p= 5 (mod 8) be a prime, then there exists an optimal balanced (9p,{3,5}, (1,2),1)-OOC. If p> 29, then the OOC is also 9-regular.Theorem 1.8 If there exists a skew starter in Zv, then there exists a 18-regular balanced (18v,{3,5}, (1,2),1)-OOC.Theorem 1.9 If there exists a skew starter in Zv, then there exist a 12-regular and an optimal (12u,{3,5}, (1,2),1, (2/3, 1/3))-OOC.Theorem 1.10 If p= 3 (mod 4)> 7 is a prime, then there exist a 8-regular and an optimal balanced (8p,{3,5}, (2,2),1)-OOC.Theorem 1.11 Let p= 5 (mod 8) be a prime, then there exists an optimal balanced (8p,{3,5}, (2,2),1)-OOC. If p> 13, then the OOC is also 8-regular.Theorem 1.12 If n= 24,120 (mod 144), then there exist a 24-regular and an optimal balanced (n,{3,5}, (2,1),1)-OOC.Theorem 1.13 If n= 14,70 (mod 84)> 14, then there exist a 14-regular and an optimal (n,{3,5}, (2,1),1, (2/3, l/3))-OOC.Theorem 1.14 If n= 15,75 (mod 90)> 15, then there exists an optimal (n,{3,5}, (1,2),1,(1/3,2/3))-OOC.Theorem 1.15 If n= 14,70 (mod 84)> 14, then there exists an optimal (n,{3,5}, (2, 2),1,(1/3,2/3))-OOC.This thesis is divided into four parts. In Chapter one, we present some notations, the known results on (variable-weight) optical orthogonal codes and the main results of this the-sis. Chapter two discusses the upper bounds of Φ(n,{3,5}, Aa,1, Q). Chapter three discusses the constructions of (n,{3,5}, Aa,1, Q)-OOCs with Aa G{(2,1), (1,2), (2,2)}. Conclusions and further research problems are given in Chapter four. |
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