Further Research On Two-dimensional Variable-weight Optical Orthogonal Codes

In 1989,one-dimensional constant-weight optical orthogonal code?1D CWOOC?was introduced by Salehi and applied in optical code divi,sion 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 = {w1,...,wr} be an ordering of a set of r integers great.er than 1.?a =(?_a¹,…,?_a^r)an r-tupie 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 w1<w2<...<wrAn?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;where???denote modulo v addition,whore???denote modulo v addition.?2?Periodic Auto-correlation:For 0???v-1,and matrix X in C with Hamming weight w_k,where?3?Periodic Cross-correlation:For 0???v-1,and any two distinct matrices X,Y in C,where where ?denote mudulo v addition.where ? denote modulo v addition.If ?_a¹=?_a²=...= ?_a^r=?_a,we use the notion?u×v,W.?a,?r,Q?-OOC to denote an?u×v,W,?a,?c,Q?-OOC.Also,an?u×v,W,?,Q?-OOC means an?u × v,W,?a,?c,Q?-OOC whene ?_a = ?_c = ?.If Q =?a1/b,a2/b,...,ai/b?with ged?a1,a2,...,ai?...1.b=?r i=1 ai,we say that Q is normalized.Obviously,an?u×v,w,??-OOC?const ant-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 boon 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:Let v be a positive integer,if each prime factors p of v are all satisfy p ? 7?mod 12?,then there exist a 1-regular and an optimal?3×v,{3,4},1,?5/7,2/7??-OOC,p?31.Theorem 1.2:Let v be a positive integer,if each prime factors p of v are all satisfy p ? 7?mod 12?,then there exist a 1-regular and an optimal?3 × v,{3,4},1,?7/8,1/8??-OOC,p ? 31.Theorem 1.3:Let v be a positive integer,if each prime factors p of v are all satisfy p ? 5?mod 8?,then there exist a 1-regular and an optimal?7 × v,{3,5},1,?16/21,5/21??-OOC,p ? 29.Theorem 1.4:Let v be a positive integer,if each prime factors p of v are all satisfy p? 1?mod 4?,then there exist a 1-regular and an optimal?6 × v,{3,5},1:?2/5,3/5??-OOCTheorem 1.5:Let v be a positive integer,if each prime factors p of v are all satisfy p ? 1?mod 4?,then there exist a 1-regular and an optimal?5×v,{3,5},1,?5/6,1/6??-OOC.Theorem 1.6:Let v be a positive integer,if each prime factors p of v arc all satisfy p ? 1?mod 4?,then there exist a 1-regular and an optimal?7×v,{3,5},1,?14/17,3/17??-OOC.Theorem 1.7:If there is a skew starter in Zv,then there exist a 1-regular and optimal?7 × v,{3,5},1,?13/14,1/14??-OOC.Theorem 1.8:If there is a skew st.arter in Zv,then there exist a 1-regular and optimal?8 × v,{3,5},1,?18/19,1/19??-OOC.Theorem 1.9:If there is a skew starter in Zv,t hen t here exist,a 1-regular and optimal?9 × v,{3,5},1,?17/20.3/20??-OOC.Theorem 1.10:Let v be a positive integer,if each prime factors p of v are all satisfy p ? 1?mod 4?,then there exist a 1-regular and an optiml?5 × v,{3,4,5},1,?8/11,1/11,2/11??-OOC.Theorem 1.11:If there is a skew starter in Zv,then there exist a 1-regular and an optimal?7 × v,{3,4,5},1,?9/12,2/12,1/12??-OOC.Theorem 1.12:If there is a skew starter in Zv,then there exist a 1-regular and an optimal?8 × v,{3,4,5},1,?14/17,2/17,1/17??-OOC.This thesis is divided into five parts.In Chapter one,some notations and the main results of this thesis are presented.The construction of optimal?u × v,{3,4},1,Q?-OOCs,?u x v,{3,5},1,Q?-OOCs,and?u × v,{3,4,5},1,Q?-OOCs are given in Chapters 2-4 respec-tively.Conclusions and further research problems are presented in Chapter five.