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={?₁,...,?_r} be an ordering of a set of r integers greater than 1,?a=??a?1?,...,?a?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 ?₁<?₂<...<?_r.An?u × v,W,?a,?c,Q?variable-weight optical orthogonal code or?u × v,W,?a,?c,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<?<v-1,and matrix X in C with Hamming weight wk,where???????3?Periodic Cross-correlation:For 0 ? ?<v-1 and any two distinct matrices X,Y in C,where where???denote modulo v addition.The notion?u×v,W,?a,?c,Q?-OOC is used to denote an?u×v,W,?a,?c,Q?-OOC with the property that.?a?1?= ?a?2?=...= ?a?i?= ?a.Also,speaking of an?u×v,W,?,Q?-OOC one means an?u×v,W,?a,?c,Q?-OOC where ?a = ?c = ?.We say that Q is normalized if it is ?_ritten in the form Q =?a1/b,a2/b,...ar/b?with ged?a1,a2,...,ar?= 1.Obviously,b =?r i=1 ai,An?u×v,w,??-OOC?constant-weight,OOC?is an?u×v,{w}.?,?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.