The Combinatorial Constructions For Optical Orthogonal Codes

In 1989,optical orthogonal code was introduced by Salehi,as signature sequences to facilitate optical code division multiple access(OCDMA)system.Variable-weight optical orthogonal code was introduced by Yang for multimedia OCDMA systems with multiple quality of service(QOS)requirement.Let W,and Q denote the sets{w0,w1,…,wp},{λa0,λa1,…λap} and {qo,q1,…,qp}, respectively as defined below.Without loss of generality,we assume that w0<w1<…< wp.An(v,W,L,λc,Q)variable-weight optical orthogonal code C,or(v,W,L,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 fraction of codewords of weight wi, and (?)qi=1.(2)Periodic Auto-correlation:For any x=(x0,x1,…,xv-1)∈C with Hamming weight wi∈W,and any integerτ.0<τ<v,(3)Periodic Cross-correlation:Similarly,for any x=(x0,x1,…,xv-1)∈C,y (y0,y1,…yv-1)∈C,x≠y,and any integerτ, where"(?)" denote modulo v addition.The notion(v,W,λ,Q)-OOC is used to denote a (v,W,L,λc,Q)-OOC with the property thatλa0=λa1=…λap=λc=λ.If W={w}, a(v,W,L,λc,Q)-OOC is a(v,W,λa,λc)-OOC.Similarly,The notion(v,w.λ)-OOC is used to denote a(v,W,λa,λc)-OOC with the property thatλa=λc=λ.LetΦ(v,W,L,λ,Q)=max{│C│:C is a(v,W,L,λ,Q)-OOC}.For the upper of variable-weight OOCs.the following result is given by Yang.Letλai≥λ(λai∈L),thenΦ(v,W,L,λ,Q) For each qi∈Q,without loss of generality,write qi=bi/ai,where ai,bi are integers and gcd(ai,bi)=1,0≤i≤p.Let f(Q)=lcm(a0,a1,...ap),and fi(Q)=f(Q)qi,then (?)fi(Q)=f(Q).In this paper,the upper bound on the size of variable--weight OOCs is improved as follows.Theorem 1.0 Letλai≥A,0≤i≤p,thenΦ(v,W L,λ,Q)By using Additive Sequence of Permutations,Quadruple Systems,Perfect Bases,Dif-ference Families,Perfect Difference Families and Skew Starters,the following results are obtained.Theorem 1.1 Suppose m,n,t are non-negative integers and not all equal to zero, then there exists an optimal(3×21m×45t×49n,4,1)-OOC.Theorem 1.2 Suppose m,n,t are non-negative integers,then there exists an optimal (99×21m×45t×49n,4,1)-OOC.Theorem 1.3 suppose a(g,K,1)-PDF exists,where K={k1,k2,…kn}. Let sl be the number of the blocks of size kl,1≤l≤n,s=s1+s2+…+sn and M=max{4,k1,k2,…,kn}.For any positive integer u whose prime factors are all congruent to 1 modulo 4 and no less than M,and(u,g+2)=1,then(1)If kl≠4,1≤l≤n,then there exists an optimal((g+2)u,{4,k1,k2,…,kn},1, {1/(4s+1),4s1/(4s+1),4s2/(4s+1),…,4sn/(4s+1)})-OOC;(2)If there exists l∈[1,n],such that kl=4,and gcd(4sl+1,4s1,…,4sl-1, 4sl+1,…,4sn)=1,then there exists an optimal((g+2)u,{4,k1,…,kl-1,kl+1,…,kn}, 1,{(4sl+1)/(4s+1),4s1/(4s+1),…,4sl-1/(4s+1),4sl+1/(4s+1),…,4sn/(4s+1)})-OOC.Theorem 1.4 Suppose a(g,K,1)-PDF exists,where K={k1,k2,…,kn). Let sl be the number of the blocks of size kl,1≤l≤n,s=s1+s2+…+sn and M=max{3,k1,k2,…,kn}.For ally positive integer u whose prime factors are all congruent to 1 modulo 2 and no less than M,and(u,g+2)=1:then(1)If kl≠3,1≤l≤n,then there exists an optimal((g+2)u,{3,k1,k2,…,kn},1, {1/(2s+1),2s1/(2s+1),2s2/(2s+1),…,2sn/(2s+1)})-OOC;(2)If there exists l∈[1,n],such that kl=3,and gcd(2sl+1,2s1,…,2sl-1,2sl+1,…,2sn)=1,then there exists an optimal((g+2)u,{3,k1,…kl-1,kl+1,…,kn},1,{(2sl+ 1)/(2s+1),2sl/(2s+1),…,2sl-1/(2s+1),2sl+1/(2s+1),…,2sn/(2s+1)})-OOC.Theorem 1.5 Suppose a cyclic(g,K,1)-DF exists,where K={k1,k2,…,kn}. Let sl be the number of the blocks of size kl,1≤l≤n,s=s1+s2+…+sn and M=max{3,k1,k2,…,kn).For any positive integer u whose prime factors are all congruent to 1 modulo 6 and no less than M,and (u,g)=1,then(1)If kl≠3,1≤l≤n,then there exists an optimal(gu,{3,k1,k2,…,kn),1,{1/(6s+ 1),6s1/(6s+1),6s2/(6s+1),…,6sn/(6s+1)})-OOC;(2)If there exists l∈[1,n],such that kl=3,and gcd(6sl+1,6s1,…,6sl-1,6sl+1,…, 6sn)=1,then there exists an optimal(gu,{3,k1,…,kl-1,kl+1,…,kn),1,{(6sl+1)/(6s+ 1),6s1/(6s+1),…,6sl-1/(6s+1),6sl+1/(6s+1),…,6sn/(6s+1)})-OOC.Theorem 1.6 Suppose a(g,K,1)-PDF exists,where K={k1,k2,…,kn}.Let sl be the number of the blocks of size kl,1≤l≤n,s=s1+s2+…+Sn and M=max{4,k1,k2,…,kn}.For any positive integer u whose prime factors are all congruent to 1 modulo 6 and no less than M,and(u,g)=1,then(1)If kl≠4,1≤l≤n,then there exists an optimal((g+1)u,{4,k1,k2,…,kn},1, {1/(6s+1),6s1/(6s+1),6s2/(6s+1),…,6Sn/(6s+1)})-OOC;(2)If there exists l∈[1,n],such that kl=4,and gcd(6sl+1,6s1,…,6sl-1,6sl+1,…, 6sn)=1,then there exists an optimal((g+1)u,{4,k1,…,kl-1,kl+1,…,kn},1,{(6st+ 1)/(6s+1),6s1/(6s+1),…,6sl-1/(6s+1),6sl+1/(6s+1),…,6sn/(6s+1)})-OOC.Theorem 1.7 If v≡15(mod 30)is an integer,and v＞15,then there exists an optimal(v,{3,4},1,{1/3,2/3}-OOC.Theorem 1.8 If v≡30,150 (mod 180)is an integer,and v>30,then there exists an optima1(v,{3,4},1,{1/3,2/3})-OOC.Theorem 1.9 If v≡12 (mod 24)is an integer,and v>12,then there exists an optimal(v,{3,4},1,{2/3,1/3})-OOC.Theorem 1.10 If v≡24,120(mod 144)is an integer,and v>24,then there exists an optimal(v,{3,4},1,{2/3,1/3})-OOC.The paper is divided into five parts. In Chapter one,we present some notations,the known:results on(variable-weight)optical orthogonal codes and the main results of t his paper.Chapter two discusses the existence of optimal(v,4.1)一OOCs.Chapter three gives some general constructions of optimal variable-weight OOCs. The existences of optimal (v,{3,4},1,Q)-OOCs are treated in Chapter four,where Q∈{{1/3,2/3},{2/3,1/3}}.Fur-ther research problems are given in Chapter five.