二维变重量光正交码的新结果

发布时间：2019-05-28 17:54

【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.为了满足多种服务质量(QoS)的需求,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)的概念.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 ={ω_1,ω_2,...,ω_r}为正整数集合,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ω_1ω_2...ω_r.二维(u × v,∧a,λc,Q)变重量光正交码,或(u × v,W,∧a,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∑r i=1 qi=1.(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 ≤ τv-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,∧a,λc,Q)-OOC 记为(u ×v,W,λa,λc,Q)-OOC.若λa=λc= λ,则记为(u × v,W,λ,Q)-OOC.若Q =(a1/b,a2/b,...,ar/b)且gcd(a,a2,...,ar)= 1,则称Q是标准的,显然,b =∑r i=1 ai.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的构造已有一些成果,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设v为正整数,v的每个质因子≡ 3(mod 4)且p ≥ 11,则存在1-正则且最优(6 × v,{3.4.6},1,(5/7.1/7.1/7))-OOC.定理1.2设v为正整数,uw的每个质因子p ≡ 5(mod 8)且p ≥ 53,则存在1-正则且最优(5 × v,{3.4.5}.1,(1/4.2/4.1/4))-OOC.定理1.3设v为正整数,v的每个质因子p三5(mod 8)且p ≥ 53,则存在1-正则且最优(6 × v,{3,,4,5},1,(2/11,,6/11,3/11))-OOC.定理1.4设v为正整数,,v的每个质因子p ≡ 5(mod 8)且p ≥ 29,则存在1-正则且最优(6 × v,{3,4},1,(14/19,5/19))-OOC.定理1.5设v为正整数,v的每个质因子p≡5(mod 8)且p≥53,则存在1-正则且最优(6 × v，{3,4},1,(10/17,7/17))-OOC.定理1.6设v为正整数,v的每个质因子p三7(mod 12)且p ≥ 31,则存在1-正则-且最优(4 × v,{3,4},1,(14/15,1/15))-OOC.定理1.7设v为正整数,v的每个质因子p三7(mod 12)且p≥ 19,则存在1-正则且最优(4 × v,{3,4},1,(2/9,7/9))-OOC.定理1.8设v为正整数,v的每个质因子p三7(mod 12)且p ≥ 31,则存在1-正则且最优(5 × v,{3,4},1,(23/24,1/24))-OOC.本文共分为四章:第一章介绍与本文有关的概念及本文的主要结果,第二章给出W={3,4,5},{3,4,6}的最优(u × v,,W,1,Q)-OOCs 的构造,第三章给出最优(u × v,{3,4},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 constant-weight optical orthogonal code (1D CWOOC), which is applied to the optical code division multiple access (OCDMA) system as a signature sequence. In order to meet the needs of a variety of quality of service (QoS), Yang introduced the concept of one-dimensional variable-weight optical orthogonal code (1D VWOOC) in 1996. With the rapid development of the society, the demand for different types of information is gradually improved. In order to expand the optical orthogonal code, Yang proposed two-dimensional constant-weight optical orthogonal code (2D CWOOC) in 1997, but it is similar to one-dimensional constant-weight optical orthogonal code. The definition of a two-dimensional variable-weight optical orthogonal code is given below. Let W = {1 _ 1,2 _ 2,... , __ r} is a positive integer set, is a positive integer array, Q = (q1, q2...). (qr) is a positive rational number array without loss of generality. A two-dimensional (u, v, a, c, q) variable weight optical orthogonal code, or (u, v, w, a, c, q)-ooc c, is a (0,1) matrix (code word) of a cluster u, v, and satisfies the following three properties: (1) the codeword weight distribution: (1) the codeword in c has a hamming weight in the set w and c is qi. | c | a code word with a weight of wi,1, i, r, i. e., the number of code words with a weight equal to wi as a percentage of the total number of codewords, and thus, r i = 1 qi = 1. (2) Periodic self-correlation: for any matrix X-C, its Hamming weight, wk, W, integer number,0, v-1, (? ) (? ) (3) Periodic cross-correlation: for any two different matrices X, Y, C, integer number,0, and v-1, (? ). The above-mentioned symbol (?) represents a modulo operation for v. If the symbol (1) = (a (2) =... = (a (r) = (a), we will (u, v, W, a, c, Q)-OOC as (u, v, W, a, c, Q)-OOC. If the ratio of (u, v, W, a, c, Q)-OOC is recorded as (u, v, W, HCO3, Q)-OOC. If Q = (a1/ b, a2/ b,... , ar/ b) and gcd (a, a2,... (ar) = 1, then Q is standard and it is clear that b = {r i = 1 ai. If W = {w}, Q = (1). Therefore, the constant weight (u, v, w,1)-OOC can be considered as (u, v, {w},1, (1))-OOC. For optical orthogonal codes, it is said to be optimal when its number of codewords reaches the maximum. However, for the best (u, v, W,1, Q)-OOC, there are some results, but for the best (u, v, W,1, Q)-OOC, we will continue to study and get the following main results. Theorem 1.1.1 Let v be a positive integer, v for each mass factor of 3 (mod 4) and p = 11, there is 1-regular and optimal (6% v, {3.4.6},1, (5/ 7.1/ 7.1/7)) There are 1-regular and optimal (5, v, {3.4.5}.1, (1/ 4.2/ 4.1/4))-OOC. Theorem 1.3 is set to v is a positive integer, v for each mass factor p-3 5 (mod 8) and p-equal to 53, there are 1-regular and optimal (6, v, {3,,4,5},1, (2/11,6/11,3/11))-OOC. The theorem 1.4 is set to a positive integer, and each of the quality factors p-5 (mod 8) and p-{29} of v are 1-regular and optimal (6, v, {3,4},1, (14/19,5/19))-OOC. The theorem 1.5 is a positive integer, and each of the qualitative factors of v is p-5 (mod 8) and p = 53, there is 1-regular and optimal (6} v, {3,4}, The structure of W,1, Q)-OOCs and the third chapter give the optimal (u, v, {3,4},1, Q)-OOCs. The fourth chapter is the summary and the further study.
【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

【相似文献】