Bounds And Constructions For Optimal(n,{3,4},Λ_a,1,Q)-OOCs

In 1989,opticεl orthogonal code(OOC)was introduced by Salehi,and applied in opti-cal code division multiplex access(OCDMA) system as a signature sequence.In OCDMA system,each user is assigned to an OOC as the address code.To meet multiple QoS require-ments,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.In the following,the definition of a VWOOC will be given. tuple of positive ration numbers with ∑qi=1.An(n,W,Λa,λc,Q)variable-weight optical orthogonal code C,or(n,W,Λa,λc,Q)-OOC,is a collection of 0,1 n-tuples(codeword) such that the following three properties hold:(1)Weight Distribution Every codeword in C has a Hamming weight contained in the set W；furthermore,there are exactly qi|C| codewords of weghht wi,i.e.,qi indicates the ratio of codewords of weight wi；(2)Periodic Auto-correlation For any x=(x0,x1,…,Xn-1)∈C with Hamming weight wk∈W,and any integer τ,0<τ<n,(3)Periodic Cross-correlation For any x≠y,x=(xo,x1,…,xn-1)∈C,y= (y0,y1,…,yn-1)∈C,and any integer τ,0≤τ-<n, where (?) denote modulo n addition.The notion(n,W,λa,λc,Q)-OOC is used to denote an(n,W,Λa,λc,Q)-OOC with the property that λa(1)=λa(2)=…=λa(r)=λa；and the notion(n,W,λ,Q)-OOC denotes an (n,W,λa,λc,Q)-OOC with the property that λa=λc=λ.We say that Q is normalized if it is written in the form Q=(a1/b,a2/b,…,ar/b)with gcd(a1,a2,…,ar)=1.Obviously, Yang gave the upper 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 An (n, W,Λa,λc, Q)-OOC with the maximum code size for given n, W, Λa, λc and Q is called optimal. At present, some work had been done on the existence of optimal balanced (n,{3,4}, Aa, 1)-OOCs. As far as the author is aware of, no systematic results on the existence of optimalTheorem 1.1 For any prime p>5, there exists a 10-regular and optimal (10p,{3,4}, (2,Theorem 1.2 If there exists a skew starter in Zv and gcd(v,7)=1, then there exists a 14-regular and optimal (14v,{3,4}, (2,1),1, (1/3,2/3))-OOC.Theorem 1.3 If there exists a skew starter in Zv, then there exists a 12-regular and optimal (12v,{3,4}, (2,1),1, (3/4,1/4))-OOC.Theorem 1.4 For any prime p> 5, there exists a 20-regular and optimal (20p,{3,4}, (2,1),1, (1/4,3/4))-OOC. There exists an optimalTheorem 1.5 If there exists a skew starter in Zv and gcd(v,5)=1, then there existsTheorem 1.6 If there exists a skew starter in Zv and gcd(v,11)=1, then there exists an 11-regular and optimal (11v,{3,4}, (1,2),1, (1/3,2/3)-OOC.Theorem 1.7 If there exists a skew starter in Zv and gcd(v,13)=1, then there existsTheorem 1.8 For any prime p>5, there exists a 30-regular and optimal (30p,{3,4}, Theorem 1.9 For any prime p>5, there exists an 8-regular and optimal (8p,{3,4},Theorem 1.10 If there exists a skew starter in Zv and gcd(v,5)=1, then there existsTheorem 1.11 For any prime p>5, there exists a 10-regular and optimal (10p,{3,4},Theorem 1.12 For any prime p>7, there exists a 14-regular and optimal (14p,{3, {3,5,7}.Theorem 1.13 Let Q=(a1/b,a2/b) be normalized, thenTheorem 1.15 Suppose p=13 (mod 24) is a prime, then there exists a 10-regular balanced (10p,{3,4}, (2,3),1)-OOC.Theorem 1.16 For any prime p>7, there exists a 42-regular and optimal (42p,{3,Theorem 1.17 For any prime p>5, there exists an 8-regular and optimal (8p,{3,4},Theorem 1.18 If p=3 (mod 4)> 7 is a prime, then there exists a 12-regular balancedTheorem 1.19 If there exists a skew starter in Zv, then there exists a 24-regular andTheorem 1.20 If there exists a skew starter in Zv, then there exists a 9-regular andTheorem 1.21 If there exists a skew starter in Zv, then there exists a 9-regular and optimalThis thesis is divided into five parts. In Chapter one, we present some notations, the known results on optical orthogonal codes and the main results of this thesis. Chapter two discusses the constructions of optimal Chapter three discusses the upper bounds of The constructions of optimal four. Conclusions and further research problems are given in Chapter five.