The Existence Of Variable-Weight 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)requirements.Let W,L,and Q denote the sets{ω0,ω1,...,ωp),{λα0,λα1,...,λαp}and{q0,q1,...,qp), respectively as defined below.Without loss of generality,we assume thatω0<ω1<...< 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 weightωi,i.e.,qi indicates the fraction of codewords of weightωi,and∑i=0pqi=1.(2)Periodic Auto-correlation:For any x=(x0,X1,...,xv-1)∈C with Hamming weightωi∈W,and any integerτ,0<τ<v,(3)Periodic Cross-correlation:Similarly,for any x≠y,x=(x0,x1,...,xv-1)∈C, y=(y0,y1,...,yv-1)∈C,and any integerτ, where"(?)"denote modulo v addition.The notation(v,W,λ,Q)-OOC is used to denote a (v,W,L,λc,Q)-OOC with the property thatλα0=λα1=...=λαp=λc=λ.LetΦ(c,W,L,λ,Q)=max{|C|:C is a(v,W,L,λ,Q)-OOC},Yang gave an upper bound on the codeword size of variable-weight OOCs as follow.Letλαi≥λ(λαi∈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)=qif(Q),then∑i=0pfi(Q)=f(Q).Recently,the upper bound was improved by Jiang.Letλαi≥λ(λαi:∈L), thenΦ(v,W,L,λ,Q)≤f(Q)(?)Based on the above fact,from now on,a(v,W,L,λ,Q)-OOC is said to be optimal if the above equality holds.The study of(v,W,L,λ,Q)-OOCs is mainly on the existences of (v,W,1,Q)-OOCs for |W|=2.As far as the author is aware,little is known on the existence of(v,W,1,Q)-OOCs for |W|≥3.In this thesis,(v,W,1,Q)-OOCs for |W|=3 is considered, and the following results are obtained.Theorem 1.4 There exists an optimal(19v,{3,4,5},1,{1/3,1/3,1/3})-OOC for any positive integer v whose prime factors are all congruent to 3 modulo 4 and no less than 7.Theorem 1.5 There exists an optimal(22v,{3,4,5},1,{1/2,1/4,1/4})-OOC for any positive integer v whose prime factors are all congruent to 3 modulo 4 and no less than 7.Theorem 1.6 There exists an optimal(28v,{3,4,5},1,{2/5,2/5,1/5})-OOC for any positive integer v whose prime factors are all congruent to 3 modulo 4 and no less than 7.Theorem 1.7 There exists an optimal(25v,{3,4,5},1,{3/5,1/5,1/5})-OOC for any positive integer v whose prime factors are all congruent to 3 modulo 4 and no less than 7.Theorem 1.8 There exists an optimal(24v,{3,4,6},1,{1/3,1/3,1/3})-OOC for any positive integer v whose prime factors are all congruent to 3 modulo 4 and no less than 7.Theorem 1.9 There exists an optimal(51v,{3,4,6},1,{12/17,1/17,4/17})-OOC for any positive integer v whose prime factors are all congruent to 1 modulo 4 and no less than 5.Theorem 1.10 There exists an optimal(45v,{3,4,6},1,{8/13,1/13,4/13})-OOC for any positive integer v whose prime factors are all congruent t0 1 modulo 4 and no less than 5.Theorem 1.11 There exists an optimal(47v,{3,4,5},1,{8/17,5/17,4/13})-OOC for any positive integer v whose prime factors are all congruent to 1 modulo 4.Theorem 1.12 There exists an optimal(41v,{3,4,5},1,{4/13,5/13,4/13})-OOC for any positive integer v whose prime factors are all congruent to 1 modulo 4.Letλα=max{λαi:0≤i≤p),to the author's knowledge,very little is known on optimal(v,W,L,λ,Q)-OOC forλα≠λ.Even more,the upper bound for the codeword size of(v,{3,4},L,1,Q)-OOC is not decided till now,where L={{1,2},{2,2},{2,1}).This thesis gives the upper bounds of(v,{3,4},L,1,Q)-OOCs,and discusses the existences of optimal(v,{3,4},L,1,(1/2,1/2})-OOCs,then gets the results as follow.Theorem 1.13 Suppose q=10t±1 is a prime such that(?)is a primitive root of Zq,then there exist an optimal(g,{3,4},{1,2},1,{1/2,1/2})一OOC.Theorem 1.14 Suppose q=5(mod 8)is a prime,then there exists an optimal (7q,{3,4},{1,2},1,{1/2,1/2})-OOC.Theorem 1.15 Suppose q三1(mod 6)is a prime,then there exists an optimal (2q,{3,4},{2,2},1,{1/2,1/2})-OOC.Theorem 1.16 Suppose v=5km,here gcd(m,30)=1,and k>1,then there exists an optimal(8v,{3,4},{2,1},1,{1/2,1/2})-OOC.The paper 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 paper.Chapter two discysses the existences of optimal(v,W,1,Q)-OOCs,where |W|=3. The existences of(v,{3,4},L,1,{1/2,1/2})-OOCs are treated in Chapter three,where L∈{{1,2},{2,2},{2,1}}.Further research problems are given in Chapter four.