New Results Of Two-Dimensional Variable-weight Optical Orthogonal Codes

In 1989.one-dimensional constant-weight optical orthogonal code（1D CWOOC）was introduced by Salchi and applied in optical code division multiple access（OCDMA）system as a signature sequence.As 1D CWOOC can not meet multiple quality of service（QoS）requirements.Yang introduced one-dimensional variable-weight optical orthogonal code（1D VWOOC）in 1996.With the rapid development of the society and the requirement for different forms of information are improved,people need OCDMA with high speed,large ca-pacity and different bit error rate.In order to expand the capacity of OOC,two-dimensional constant-weight optical orthogonal code（2D CWOOC）was introduced by Yang in 1997,similar to 1D CWOOC,2D CWOOC can not meet multiple quality of service requirements too.To solve the problem,two-dimensional variable-weight optical orthogonal code（2D VWOOC）was introduced.The definition of 2D VWOOC will be given bellow.Let W={ω₁,...,ω_r} be an ordering of a set of r integers greater than 1,∧a=（λa（1）,...,λa（r））an r-tuple of positive integers,and Q=（q1,...,qr）an r-tuple of positive rational numbers whose sum is 1.Without loss of generality,we assume that ω₁<ω₂<...<ω_r.An（u × v,W,∧a,λc,Q）variable-weight optical orthogonal code or（u × v,W,∧a,λc,Q）-OOC C,is a collection of（0,1）u × v matrices（codewords）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:For 0<τ<v-1,and matrix X in C with Hamming weight wk,where（?）（?）（3）Periodic Cross-correlation:For 0 ≤ τ<v-1 and any two distinct matrices X,Y in C,where where（?）denote modulo v addition.The notion（u×v,W,λa,λc,Q）-OOC is used to denote an（u×v,W,∧a,λc,Q）-OOC with the property that.λa（1）= λa（2）=...= λa（i）= λa.Also,speaking of an（u×v,W,λ,Q）-OOC one means an（u×v,W,λa,λc,Q）-OOC where λa = λc = λ.We say that Q is normalized if it is ω_ritten in the form Q =（a1/b,a2/b,...ar/b）with ged（a1,a2,...,ar）= 1.Obviously,b =Σr i=1 ai,An（u×v,w,λ）-OOC（constant-weight,OOC）is an（u×v,{w}.λ,（1））-OOC with W = {w},Q =（1）.An OOC is said to be optimal if it has maximum code size.Many results had been done on optimal（v,W,1,Q）-OOCs,while not so much had been done on optimal 2D VWOOCs.In this thesis,the following results are obtained:Theorem 1.1:There exist a 1-regular and an optimal（6×v,{3,4,6},1,（5/7,1/7,1/7））-OOC for each positive integer r whose prime factors p are all congruent 3 modulo 4.and p≥ 11.Theorem 1.2:There exist a 1-regular and an optimal（5×v,{3,4,5},1,（1/4.2/4,1/4））-OOC for each positive integer v whose prime factors p are all eongruent 5 modulo 8,and P≥53.Theorem 1.3:There exist a 1-regular and an optimal（6×v,{3,4,5},1,（2/11,6/11,3/11））-OOC for each positive integer v whose prime factors p are all congruent 5 modulo 8,and p ≥ 53.Theorem 1.4:There exist a 1-regular and an optimal（6×v,{3,4},1,（14/19,5/19））-OOC for each positive integer v whose prime factors p are all congruent 5 modulo 8,and P ≥ 29.Theorem 1.5:There exist a 1-regular and an optimal（6×v,{3,4},1,（14/19,5/19））-OOC for each positive integer r whose prime factors p are all congruent 5 modulo 8,and p ≥ 53.Theorem 1.6:There exist a 1-regular and an optimal（6×v,{3,4},1,（14/19,5/19））-OOC for each positive integer v whose prime factors p are all congruent,7 modulo 12.and（?）（?）p≥31.Theorem 1.7:There exist a 1-regular and an optimal（6 × v,{3,4},1,（14/19,5/19））-OOC for each positive integer v whose prime factors p arc all congruent 7 modulo 12,and P ≥ 19.Theorem 1.8:There exist a 1-regular and an optimal（5 × v,{3,4},1,（23/24,1/24））-OOC for each ppsitive integer v whose prime factors p are all congruent 7 modulo 12,and p≥31.The thesis is divided into four parts.In Chapter one,some notations and the main results are presented.In Chapter two,we give the construction of optimal（u × v,W,1,Q）-OOCs with W = {3,4,5},{3,4,6}.In Chapter three,new optimal（u × v,{3,4},1,Q）-OOCs are constructed.Conclusions and further research problems are presented in Chapter four.