| In 1989,optical orthogonal code(OOC)was introduced by Salehi,as a special sequence,optical orthogonal code is widely used in Optical Code Division Multiple Access(OCDMA)system.The bit error rate(BEQ)of OOC depends on the weight of the optical orthogonal code,the constant weight orthogonal code can not satisfy the multiple quality of service(QoS),thus,variable weight optical orthogonal code(VWOOC)was introduced by Yang in 1996.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.Therefore,VWOOC can meet multiple quality of service requirements.Let u,?c be positive integers,K = ?k1,k2,…,ks} be a set of positive integers,?a =(?a(1),?a(2),…,?a(s))be a-tuple of positive integers,and R=(r1,r2,…,rs)be an s-tupleof positive rational numbers with ?i=1s ri= 1.An(u,K,?a,?c,R)-OOCs is a set F of sub-sets(codeword-sets)of Zu with sizes(weights)from K,auto-correlation vector ?a,cross-correlation parameter ?c,codeword weight distribution vector R.The notion(u,K,?a,?c,R)-OOC is used to denote an(u,K,?a,?c,R)-OOC with the property that ?a(1)=?a(2)=...=?a(s=)?a,and(u,K,?,R)-OOC denotes an(u,K,?a,?c,R)-OOC with the property that?a =?c=?,We say that R is normalized if it is written in the form R =(h1/h,h2/h,…,hs/h)with gcd(h1,h2,…,hs)= 1.Obviously,h = ?i=1s hi.By a balanced(u,K,?a,?c)-OOC wemean an(u,K,?a,?c,R)-OOC with R=(1/h,1/h,…,1/h).Let R be normalized,?(u,K,?a,?c,R)=max{|F|:F is an(u,K,?a,?C,R)-OOC},an upper bound on the codeword size of variable-weight OOCs was given by Buratti et al below.An(u,K,?a,?,R)-OOC with the maximum code size for given u,K,?a,?c and R is called optimal.Given a subset A of Zu,define the list of differences of ?A by A = {a-b:a,b ? A,a? b}as a multiset,and the support of ?A.Let C be a set of subsets of Zu,define AC = ?A?C ?A.The difference leave of an(u,K,?a,1,R)-OOC C,denoted by DL(C),is the set of all elements of Z,which are not covered by AC.We say that C is m-regular if DL(C)is the subgroup of Zu of order m.Obviously m divisible by u.At present,some people focus attentions on the existence of optimal(u,K,1,R)-OOCs.Some work had been done for optimal(u,K,?a,1,R)-OOCs with K E {{3,4},{3,5}},and?a ?(1,1),In this thesis,we also focus our attentions on the optimal(u.K,?a,1,R)-OOCs with K = {3,4},{3,5},and ?a =(1,2),(1,3),(2,1),(2,2).The following further results are obtained:Theorem 1.1 Suppose q = 3(mod 4)is a prime,then there exists an optimal(19q,{3,4,?(1,2),1,(l/5,4/5))-OOC.If q>7,then the OOC is also 19-regular.Theorem 1.2 Suppose q = 3(mod 4)is a prime,then there exists an optimal(16q,{3,4},(1,2),1,(4/5,1/5))-OOC.If q>7,then the OOC is also 16-regular.Theorem 1.3 Suppose q= 3(mod 4)is a prime,then there exists an optimal[18q,{3,4},(1,2),1,(2/5,3/5))-OOC.lf q? 7,then the OOC is also 18-regular.Theorem 1.4 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(17q,{3,4},(1,2),1,(3/5,2/5))-OOC.If q ? 7,then the OOC is also 17-regular.Theorem 1.5 Suppose q = 5(mod 24)is a prime,then there exists an optimal(12q,{3,4},(1,3),1,(1/4,3/4))-OOC.If q ? 29,then the OOC is also 12-regular.Theorem 1.6 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(12q,{3,4},(1,3),1,(3/4,1/4))-OOC.If q ? 7,then the OOC is also 12-regular.Theorem 1.7 Suppose q = 3(mod 4)is a prime,then there exists an optimal(15q {3,4},(1,3),1,(1/5,4/5))-OOC.lf q ? 7,then the OOC is also 15-regular.Theorem 1.8 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(15q,{3,4},(1,3),1,(4/5,1/5))-OOC.If q ? 7,then the OOC is also 15-regular.Theorem 1.9 Suppose q = 3(mod 4)is a prime,then there exists an optimal(15q,{3,4},(1,3),1,(2/5,3/5))-OOC.If q ? 7,then the OOC is also 15-regular.Theorem 1.10 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(15q,{3,4},(1,3),1,(3/5,2/5))-OOC.If q ?7,then the OOC is also 15-regular.Theorem 1.11 Suppose q ?3(mod 4)is a prime,then there exists an optimal(26q,{3,4},(2,1),1,(1/5,4/5))-OOC.If q ? 7,then the OOC is also 26-regular.Theorem 1.12 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(14g,{3,4},(2,1),1,(4/5,1/5))-OOC.If q?7,then the OOC is also 14-regular.Theorem 1.13 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(22q,?3,4?(2,1),1,(2/5,3/5))-OOC.If q ? 7,then the OOC is also 22-regular.Theorem 1.14 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(18q,{3,4},(2,1),1,(3/5,2/5))-OOC.If q ? 7,then the OOC is also 18-regular.Theorem 1.15 Suppose q ?3(mod 4)is a prime,then there exists an optimal(18q,{3,4},2,1,(1/5,4/5))-OOC.If q ? 7,then the OOC is also 18-regular.Theorem 1.16 Suppose q ? 5(mod 8)is a prime,then there exists an optimal and 12-regular(12q,?3,4},2,1,(4/5,1/5))-OOC.Theorem 1.17 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(16q,{3,4},2,1,(2/5,3/5))-OOC.If q ? 7,then the OOC is also 16-regular.Theorem 1.18 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(14q,{3,4},2,1,(3/5,2/5))-OOC,If q ? 7,then the OOC is also 14-regular.Theorem 1.19 Suppose q? 11,19(mod 24)is a prime,then there exists an optimal(21q,{3,5},(1,2),1,(1/4,3/4))-OOC.If q ? 11,then the OOC is also 21-regular.Theorem 1.20 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(15g,{3,5},(1,2),1,(3/4,1/4))-OOC.If q ? 7,then the OOC is also 15-regular.Theorem 1.21 Suppose q ? 3(mod 4)is a prime,then there exists an optimal(32q,{3,5},(2,1),1,(1/4,3/4))-OOC.If q ? 7,then the OOC is also 32-regular.Theorem 1.22 Suppose q ?3(mod 4)is a prime,then there exists an optimal(16q,{3,5},(2,1),1,(3/4,1/4))-OOC.If q? 7,then the OOC is also 16-regular.The thesis is divided into four 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 existence of optimal(u,{3,4},?a,1,R)-OOCs with ?a =(1,2),(1,3),(2,1),(2,2),R =(1/4,3/4),(3/4,1/4),(1/5,4/5),(4/5,1/5),(2/5,3/5),(3/5,2/5).Chapter three,discusses the existence of optimal(u,{3,5},?a,1,R)-OOCs with ?a =(1,2),(2,1),R =(1/4,3/4),(3/4,1/4).Conclusions and further research problems are given in Chapter four. |
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