Bounds And Constructions For Optimal (n,﹛3,4,5﹜, Λ_a,1,Q)-OOCs

In 1989,optical orthogonal code(00C)was introduced by Salehi,to set optical code division multiple access(OCDMA)system as a signature sequence.OOC(constant-weight) can not meet multiple quality of service(QoS)requirements,thus,variable-weight optical or-thogonal code(VWOOC)was introduced by Yang in 1996.In multimedia OCDMA system, the subscribers with different code weights will have different bit error rate(BER)perfor-mance.The codewords of low code weight can be assigned to the low-QoS applications and high code weight codewords can be assigned to high-QoS requirement applications.There-fore,VWOOC can meet multiple quality of service requirements.The definition of VWOOC will be given below.Let n, λc be positive integers,W={w1,w2,…,wr}be a set of positive integers, Λα=(λα(1)),λα(2),...,λα(r))be an r-tuple of positive integers,and Q=(q1,q2,...,qr)be an r-tuple of positive ration numbers with variable-weight optical orthogonal code C,or(n,W,Λα,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 weight wi,i.e.,gi indicates the ratio of codewords of weight wi；(2)Periodic Auto-correlation:For any x=(xo,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=(x0,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,Λα,λc,Q)-OOC is used to denote an(n,W,Λα,λc,Q)-OOC with the property that λα(1)=λα(2)=...=λα(r))=λα；and the notion(n,W,Λ,λ,Q)-OOC denotes an(n,W,λα,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, By a balanced(n,W,Λα,λc,Q)-OOC we mean an(n,W,Λα,λc,Q)-OOC with Q=(1/r,1/r,....,1/r), namely an OOC in which the number of blocks of a given size is a constant.Let Q be normalized,Φ(n,W,Λα,λc,Q)=max{|C|:C be an(n,W,Λα,λc,Q)-OOC},an upper bound on the codeword size of variable-weight OOCs was given by Buratti et al asAn(n,W,Λα,λc,Q)-OOC with the maximum code size for given n,W,Λα,λc and Q is called optimal.At present,some people focus attentions on the existence of optimal(n,W,Λα,λc,Q)-OOCs. Few people discuss the existence of optimal(n,W,Λα,λc,Q)-OOCs with W={3,4},{3,5}, and Λα≠(1,1).In this thesis,we focus our attentions on the optimal(n,W,Λα,1,Q)-OOCs with W={3,4,5},and Λα=(1,2,1),(1,1,2),(1,2,2). The upper bounds of Φ(n,W,Λα,1,Q)with Λα=(1,2,1),(1,1,2),(1,2,2)are given below:Theorem 1.1 If Q=(a1/b,a2/b,a3/b)is normalized,then where△121=6a1+8a2+20a3.Theorem 1.2 If Q=(a1/b,a2/b,a3/b)is normalized,then where△112=6a1+12a2+12a3.Theorem 1.3 If Q=(a1/b,a2/b,a3/b)is normalized,then where△122=6a1+8a2+12a3.By using skew starter and quadratic residue,optimal variable-weight OOCs are obtained below.Theorem 1.4 For any prime p≥5,and p≠17,there exist a 17-regular and an op-timal balanced(17p,{3,4,5},(1,2,1),1)-OOC.There exists an optimal balanced(17p,{3,4, 5},(1,2,1),1)-OOC for p=17.Theorem 1.5 For any prime p≥5,there exist a 27-regular and an optimal(27p,{3,4, 5},(1,2,1),1,(1/4,1/4,2/4))-OOC.Theorem 1.6 For any prime p≥5,and p≠7,there exist a 21-regular and an op-timal(21p,{3,4,5},(1,2,1),1,(1/4,2/4,1/4))-OOC.There exists an optimal(21p,{3,4,5}, (1,2,1),1,(1/4,2/4,1/4))-OOC for p=7.Theorem 1.7 If there exists a skew starter in Zv and gcd(v,5)=1,then there exist a 20-regular and an optimal(20v,{3,4,5},(1,2,1),1,(2/4,1/4,1/4))-OOC.Theorem 1.8 If there exists a skew starter in Zv and gcd(v,5)=1,then there exist a 15-regular and an optimal balanced(15v,{3,4,5},(1,1,2),1)-OOC.Theorem 1.9 For any prime p≥5,and p≠7,there exist a 21-regular and an op-timal(21p,{3,4,5},(1,1,2),1,(1/4,1/4,2/4))-OOC.There exists an optimal(21p,{3,4,5}, (1,1,2),1,(1/4,1/4,2/4))-OOC for p=7.Theorem 1.10 For any prime p≥5,and p≠7,there exist a 21-regular and an op-timal(21p,{3,4,5},(1,1,2),1,(1/4,2/4,1/4))-OOC.There exists an optimal(21p,{3,4,5}, (1,1,2),1,(1/4,2/4,1/4))-OOC for p=7.Theorem 1.11 If there exists a skew starter in Zv,then there exist a 18-regular and an optimal(18v,{3,4,5},(1,1,2),1,(2/4,1/4,1/4))-OOC.Theorem 1.12 For any prime p≥5,and p≠13,there exist a 13-regular and an op-timal balanced(13p,{3,4,5},(1,2,2),1)-OOC.There exists an optimal balanced(13p,{3,4, 5},(1,2,2),1)-OOC for p=13.Theorem 1.13 For any prime p≥5,and p≠19,there exist a 19-regular and an op-timal(19p,{3,4,5},(1,2,2),1,(1／4,1／4,2/4))-OOC.There exists an optimal(19p,{3,4,5}, (1,2,2),1,(1/4,1/4,2/4))-OOC for p=19.Theorem 1.14 For any prime p≥5,and p≠17,there exist a 17-regular and an op-timal(17p,{3,4,5},(1,2,2),1,(1/4,2/4,1/4))-OOC.There exists an optimal(17p,{3,4,5}, (1,2,2),1,(1/4,2/4,1/4))-OOC for p=17.Theorem 1.15 If there exists a skew starter in Zv,then there exist a 16-regular and an optimal(16v,{3,4,5},(1,2,2),1,(2/4,1/4,1/4))-OOC.The 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 thesis. Chapter two discusses the upper bounds of optimal(n,{3,4,5},Aa,1,Q)-OOCs. The existence of optimal(n,{3,4,5},Λα,1,Q)-OOCs with Λα∈{(1,2,1),(1,1,2),(1,2,2)} are presented in Chapter three. Conclusions and further research problems are given in Chapter four.