The Combinatorial Constructions For OOCs With Correlations Two And Two-Dimensional Variable-Weight OOCs

In1989, one-dimensional constant-weight optical orthogonal code (1-D CWOOC) was introduced by Salehi, and applied in optical code division multiple access (OCDMA) system as a signature sequence.1-D CWOOC can not meet multiple quality of service (QoS) requirements, one-dimensional variable-weight optical orthogonal code (1-D VWOOC) was introduced by Yang in1996. In multimedia OCDMA system, the different codeword weights of OOC have different bit-error rates (BER). Therefore, the1-D VWOOC can meet multiple quality of service requirements. Below, the definition of1-D VWOOC will be given,of positive integers, and Q=(q1,q2,...,qr) be an r-tuple of positive ration numbers, re-spectively as defined below. A (v, W, Λa, λc, Q) variable-weight optical orthogonal code C, or (v, W, Λa, λc, Q)-OOC, is a collection of binary v-tuples (codeword) such that the following three properties hold:(1) Weight Distribution:Every v-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 ratio of codewords of weight wi, and∑qi=1.(2) Periodic Auto-correlation:For any x=(x0,X1,...,xv-1)∈C with Hamming weight wk∈W, and any integer τ,0<τ<v,(3) Periodic Cross-correlation:For any x≠y, x=(x0,x1,...,xv-1)∈C, y=(y0,y1,...,yv-1)∈C, and any integer τ,0≤τ<v,where (?) denote modulo v addition. The notion (v, W, λ, λc, Q)-OOC is used to denote a (v,W,Λa, λc,Q)-OOC with the property that λa(1)=λa(2)=...=λa(r)=λ. The notion (v, W, λ,Q)-OOC is used to denote a (v, W, Λa, λc, Q)-OOC with the property that 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, b=∑ai.A (v,w, λ)-OOC (constant-weight OOC) is a special1D-VWOOC. One-dimensional optical orthogonal code C is said to be optimal if it has maximum code size.With the rapid development of the society and economy, people need high speed, large capacity, different bit error rate optical code division multiple access. Two-dimensional constant-weight optical orthogonal code (2-D CWOOC) was introduced by Yang in2001, it has large capacity, and can not meet multiple quality of service (QoS) requirements. To solve the problem, two-dimensional variable-weight optical orthogonal code was introduced. Below, the definition of2D-VWOOC will be given.An (uxv,W, Λa, λc, Q) variable-weight optical orthogonal code C, or (uxv, W, Λa, λc, Q)-OOC, is a collection of binary v-matrix (codeword) such that the following three properties hold:(1) Weight Distribution:The ratio of codewords of C with Hamming weight wk is qk,1≤k≤r;(2) Periodic Auto-correlation:For0<τ≤v-1, and matrix X in C with Hamming weight Wk, where(3) Periodic Cross-correlation:For0≤τ≤v-1, and any two distinct matrices X, Y in C, wherewhere (?) denote modulo v addition. The notion (u×v, W, λ, Q)-OOC is used to denote an (u×v, W, Λa, λc, Q)-OOC with the property that λa(1)=λa(2)=...=λa(r)=λc=λ. An (u×v, w, λ)-OOC (constant-weight OOC) is a special2D-VWOOC. Two dimensional optical orthogonal code C is said to be optimal if it has maximum code size. Many results had been done on optimal(v,W,1,Q)一OOCs and optimal(u×v,k,1)-OOCs,while not so much had been done on optimal(v,W,2,Q)-OOCs and (u×v,W,Q)-OOCs.By using Strict Cyclic Packing Design,Rotational Steiner Quadruple Systems,Strict Cyclic Balanced Design,Good Quadruple System,Quadratic Residue,Balanced Incomplete Block Design and Group Divisible Design,the following results are obtained:Theorem1.1where Φ(v,W,1,Q)=max{|C|:C is a(v,W,1,Q)-OOC}.Theorem1.2Suppose that there exists an RoSQS(u+1),u≡1(mod6).and h is an integer.Then(1)there exists an optimal(u,{3,4},2(2)there exists an optimal(u,{3,4},2,Theorem1.3Suppose p=12t+7is a prime, h is an integer.Then(1)there exists an optimal(2)there exists an optimal1≤h<(2t+1)(3t+1).Theorem1.4Suppose u≡7,31(mod60)is a prime,then there exists an optimalTheorem1.6Suppose that there exists an RoSQS(u+1),u≡1(mod6),and h is an integer.Then(1)there exists an optimal(2u,{3,4},2,(10/u+7,u-3/u+7))-OOC(2)there exists an optimal(2u,{3,4},2Theorem1.5Suppose n≥3is odd,h is an integer.Then(1)there exists an optimal(2n-1,{3,4},(2)there exists an optimal(2n-1,{3,4}-Theorem1.7Suppose p=12t+7is a prime,h is an integer.Then(1)there exists an optimal (2)there exists an optimal (2p,{3,4},2,(10(p-1)+24h/(p+7)(p-1)+18h,(p-1)(p-3)+6h/(p+7)(p-1)+18h))-OOC,for each1<h<(p-1)(p-3)/(?).Theorem1.8Suppose u≡31,43(mod60)is a prime,then there exists an optimal (2u,{3,4},2,(1/2,1/2))-OOC.Theorem1.9(1)there exists an optimal(2)there exists an optimal(2n+1-2,{3,4},2,(10(2n-1-1)+12h/(2n++)(2n-1-1)+9b,(2n-1-1)(2n-4)-3h/(2n+6)(2n-1-1)+9b))-OOC,Theorem1.10There exists an optimal (4×v,{3,5},1,(2/3,1/3))-OOC for each positive integer v whose prime factors are all congruent1modulo4.Theorem1.11There exists an optimal (4×v,{3,4,5},1,(2/5,1/5,2/5))-OOC for each positive integer u whose prime factors are all congruent1modulo4.Theorem1.12There exists an optimal (5×v,{3,5},1,(5/6,1/6))-OOC for each positive integer v whose prime factors are all congruent3modulo4.Theorem1.13If there exists a g-regular2-SCP(k,1；v),an optimal2-SCP(W,Q,1；u×v)and a(W,Q,1)-GDD(uk),then there exists an optimal(u×v,W,1,Q)-OOC.Theorem1.14There exists an optimal(3×V,{3,4},1,(4/5,1/5))-OOC for each pos-itive integer v≡6(mod12).Theorem1.15If there exists a g-regular2-SCP(W,1,Q；v),an optimal2-SCP(W,1,Q； u×g)and a(u,k,1)-BIBD,a(ki,1)-MGDD(wik)for any wi∈W,then there exists an optimal (u×v,W,1,Q)-OOC.Theorem1.16Suppose u≡1(mod36)is a prime power.Then there exists an optimal2-SCP({3,4},1,(1/2,1/2)；u×v)for each positive integers v whose prime factors are all congruent to1modulo18.Theorem1.17Suppose u≡1(mod24)is a prime.Then there exists an optimal2-SCP({3,4},1,(2/3,1/3)；u×v)for each positive integers v whose prime factors are all congruent to1modulo24.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 thesis.Chapter two discusses the existence of optimal(v,{3,4},2,Q)-OOCs.Chapter three focuses on the existence of(u×v,W,1,Q)-OOCs.Further research problems are given in Chapter four.