二维变重量光正交码的组合构造

发布时间：2018-05-14 16:14

本文选题：二维光正交码 + 变重量　；参考：《广西师范大学》2017年硕士论文

【摘要】：1989 年 Salehi 提出了一维常重量光正交码(One-Dimensional Constant-Weight Op-tical Orthogonal Code,1D CWOOC)的概念,它作为一种签名序列被应用于光码分多址(OCDMA)系统.由于一维常重量光正交码不能满足多种服务质量(QoS)需求,Yang于1996 年引入了一维变重量光正交码(One-Dimensional Variable-Weight Optical Orthogonal Code,1D VWOOC)用于OCDMA系统.随着社会的高速发展,人们对不同类型信息的需求逐渐提高,这就要求产生高速率、大容量、不同误码率的OCDMA系统.为了给光正交码扩容,Yang于1997年提出了二维常重量光正交码(Two-Dimensional Constant-Weight Optical Orthogonal Code,2D CWOOC),但类似于一维常重量光正交码,二维常重量光正交码也只能满足单一质量的服务需求.为了解决这一问题,Yang于2001年引入二维变重量光正交码(Two-Dimensional Variable-Weight Optical Orthogonal Code,2D VWOOC).下面给出二维变重量光正交码的定义.设W ={w1,w2,...,wr}为正整数集合,Λa =(λa(1),λa(2),...,λa(r))为正整数数组,Q =(q1,q2,...,qr)为正有理数数组且(?).不失一般性,我们假设w1w2...wr.二维(u×v,W,Λa,λc,Q)变重量光正交码或(u×v,W,Λa,λc,Q)-OOC C,是一簇u×v的(0,1)矩阵(码字),并且满足以下三个性质:(1)码字重量分布:C中的码字所具有的汉明重量均在集合W中,且C恰有qi|C|个重量为wi的码字,1≤i≤r,即qi为重量等于wi的码字占总码字个数的百分比,因而(?).(2)周期自相关性:对任意矩阵X∈C.其汉明重量wk∈W,整数τ，0τv-1,(3)周期互相关性:对任意两个不同矩阵X,Y∈C,整数τ,0≤τ＜v-1,上述符号(?)表示对v取模运算.若λa(1)=λa(2)=...=λa(r)=λa,我们将(u×v,W,Λa,λc,Q)-OOC 记为(u×v,W,Λa,λc,Q)-OOC.若λa=λc=λ.则记为(u×v,W,Λa,λc,Q)-OOC.若 Q =(a1/b·a2/b,...,ar/b)且gcd(a1,a2,...,ar)= 1:则称Q是标准的,显然,(?).若W = {w},则Q =(1).所以，，常重量的(u×v,w,λ)-OOC可以看作是(u×v,{w},λ,(1))-OOC.对于光正交码,当它的码字个数达到最大值时称其为最优的.而对于最优(u×v,W,1,Q)-OOC的构造己有一些成果,但就作者目前所知对于最优二维变重量光正交码的存在性结果不多,本文将做继续研究并且得到以下主要结果.定理1.1 如果在Zv上存在斜Starter,则存在1-正则且最优(6×v,{3,4},1,(4/5,1/5))-OOC.定理1.2 如果在Zv上存在斜Starter,则存在1-正则(6×v,{3,4},1,(2/3,1/3))-OOC.定理1.3 如果在Zv上存在斜Starter,则存在1-正则(9×v,{3,4},1,(7/8,1/8))-OOC.定理1.4 设v为正整数且v的每个质因子p≡1(mod 4),则存在1-正则且最优(3×v,{3,4},1,(4/5,1/5))-OOC.定理1.5 设v为正整数且v的每个质因子p≡1(mod 4),则存在1-正则且最优(6×v,{3,4},1,(6/7,1/7))-OOC.定理1.6 设v为正整数且v的每个质因子p≡1(mod 4).则存在1-正则且最优(6×v,{3,4},1,(10/11,1/11))-OOC.定理1.7 设v为正整数且v的每个质因子p≡1(mod 4).则存在1-正则且最优(6×v,{3,4},1,(22/23,1/23))-OOC.定理1.8 设v为正整数且v的每个质因子p≡1(mod 6),则存在1-正则且最优(6×v,{3,4},1,(1/2,1/2))-OOC.定理1.9设v为正整数且u的每个质因子p≡1(mod 6),则存在1-正则且最优(4 × v,{3,4},1,(2/5,3/5))-OOC.定理1.10设v为正整数且v的每个质因子p≡1(mod 6),则存在1-正则且最优(4× v,{3,4},1,(6/7,1/7))-OOC.定理1.11设v为正整数且u的每个质因子p≡1(mod 6),则存在1-正则且最优(4 × v,{3,4},1,(10/13,3/13))-OOC.定理1.12设u为正整数且v的每个质因子p≡1(mod 6),则存在1-正则且最优(5 × v,{3,4},1,(3/4,1/4))-OOC.定理1.13设v为正整数且v的每个质因子p≡1(mod 6),则存在1-正则且最优(5 × v,{3,4},1,(19/22,3/22))-OOC.定理1.14设v为正整数,v的每个质因子p≡7(mod 12)且p31,则存在1-正则且最优(4 × v,{3,4},1,(6/11,5/11))-OOC.定理1.15设v为正整数且u的每个质因子p ≡ 1(mod 4),则存在1-正则且最优(5 × v,{3,4,5},1,(3/5,1/5,1/5))-OOC.定理1.16如果在Zv上存在斜Starter,则存在1-正则且最优(7 × {3,4,5},1,(7/11,3/11,1/11))-OOC.本文共分四章:第一章介绍本文相关概念及本文的主要结果,第二章给出最优(u×u,{3,4}.1,Q)-OOCs的构造,第三章给出最优(u × v,{3,4,5},1,Q)-OOCs的构造,第四章是小结及可进一步研究的问题.
[Abstract]:In 1989, Salehi proposed the concept of one dimensional One-Dimensional Constant-Weight Op-tical Orthogonal Code (1D CWOOC), which was used as a signature sequence in the optical code division multiple access (OCDMA) system. One-Dimensional Variable-Weight Optical Orthogonal Code (1D VWOOC) is used for OCDMA systems. With the rapid development of the society, the demand for different types of information is gradually improved. This requires the production of high speed, large capacity, and different bit error rate OCDMA systems. In order to extend the optical orthogonal code, Yang is proposed in 1997. Two dimensional constant weight optical orthogonal codes (Two-Dimensional Constant-Weight Optical Orthogonal Code, 2D CWOOC) are given, but similar to one dimensional constant weight optical orthogonal codes, two dimensional constant weight optical orthogonal codes can only meet single quality service requirements. In order to solve this problem, Yang introduced a two-dimensional variable weight optical orthogonal code (Two-Dimensional) in 2001. Variable-Weight Optical Orthogonal Code, 2D VWOOC). Below the definition of two-dimensional variable weight optical orthogonal codes. Set W ={w1, W2, and wr} are positive integer sets, and a = (lambda a (1), lambda a (2)) as positive integer array. The weight optical orthogonal code or (U * V, W, a, C, Q) -OOC C, which is a cluster of U * V (0,1) matrix (codeword), and satisfies the following three properties: (1) the code word weight distribution: the Hamming weight of the codeword in the C is all in the set, and 1 is less than equal or equal to the number of total codewords. 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Obviously, it is = (1). So, the constant weight is considered to be an optical orthogonal code, when its number of codes reaches the maximum value, it is called optimal. The structure of the optimal (U * V, W, 1, Q) -OOC has some achievements, but there is not much existence of the existence of the optimal two-dimensional variable weight optical orthogonal codes known by the author. This paper will continue to study and obtain the following main results. Theorem 1.1 there is a 1- regular and optimal (6 x V, {3,4}, 1, (4/5,1/5)) -O if there is a skew Starter on Zv. OC. theorem 1.2 if there is a skew Starter on Zv, there is a 1- regular (6 x V, {3,4}, 1, (2/3,1/3)) -OOC. theorem 1.3 if there is a skew Starter in Zv, then there exists a 1- regular (9 * V, 1, 1) theorem 1.4 for every qualitative factor 1. 1.5 set V as positive integer and each qualitative factor of V (MOD 4), there is a 1- regular and optimal (6 x V, {3,4}, 1, (6/7,1/7)) -OOC. theorem 1.6 to set V as positive integer and V for every qualitative factor p 1 (4). 1- regular and optimal (6 * V, {3,4}, 1, (22/23,1/23)) -OOC. theorem 1.8 set V as positive integer and every qualitative factor p of v 1 (MOD 6), then there exists 1- regular and optimal (6 * V, {3,4}, 1,) 1.9 for every qualitative factor 1 (6). Positive integers and every qualitative factor of V (MOD 6), there is a 1- regular and optimal (4 * V, {3,4}, 1, (6/7,1/7)) -OOC. theorem 1.11 to set V as a positive integer and u for every qualitative factor p 1 (MOD 6), then there exists a regular and optimal (4 x, 1, 1) theorem 1.12 with a positive integer and every qualitative factor 1 (6), and there is a regular regularity. And the optimal (5 x V, {3,4}, 1, (3/4,1/4)) -OOC. theorem 1.13 set V as a positive integer and every qualitative factor of V P 1 (MOD 6), then there is a 1- regular and optimal (5 * V, {3,4}, 1, (19/22,3/22)) theorem 1.14. V is a positive integer and every qualitative factor of u p 1 (MOD 4), there is a 1- regular and optimal (5 x V, {3,4,5}, 1, (3/5,1/5,1/5)) -OOC. theorem 1.16 if there is a Starter Starter on Zv, then there are 1- canonical and optimal (7 x 1, (1)) altogether four chapters: the first chapter introduces the relevant concepts and the main results of this article, first chapter The two chapter gives the structure of the optimal (U * u, {3,4}.1, Q) -OOCs, and the third chapter gives the construction of the optimal (U * V, {3,4,5}, 1, Q) -OOCs. The fourth chapter is a summary and further research.

【学位授予单位】：广西师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157.4

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