光正交签名码及其相关设计的组合构作

发布时间：2018-05-21 07:22

  本文选题：光正交签名码 + 填充设计　； 参考：《北京交通大学》2015年博士论文

【摘要】：1994年,Kitayama基于光纤通信技术在医学图像、数字视频广播以及超计算机可视化图像传输方面的应用,提出了一种新型的、采用空间扩频技术实现的光码分多址并行图像传输系统.与此同时,光正交签名码作为码分多址并行图像传输系统的首选光地址码,受到了信息论领域、组合设计领域等众多学者的关注. 设k,m,n,λa和λc为正整数.参数为(m,n,k,λa,λc)的光正交签名码l是一族自相关系数为λa,互相关系数为λc,且Hamming重量为k的m×n(0,1)-矩阵(码字),简记为(m,n,k,λa,λc)-OSPC一个光正交签名码的码字容量即为其所包含码字的个数,它决定了码分多址并行图像传输系统能够同时承载的最大用户量.令Θ(m,n,k,λa,λc)表示所有(m,n,k,λa,λc)-OOSPC中码字容量的最大值,则称码字容量为Θ(m,n,k,λa,λc)的(m,n,k,λa,λc)-OOSPC是最优的(或最大的).当λa=λc=λ时,记号(m,n,k,λa,λc)-OOSPC与Θ(m,n,k,λa,λc)分别简写为(m,n,k,λa,λc)-OOSPC与Θ(m,n,k,λ) 本文主要围绕如下的两个问题展开深入讨论. (1)如何确定Θ(m,n,k,λa,λc)的精确值? (2)如何构作码字容量为Θ(m,n,k,λa,λc)的最优(m,n,k,λa,λc)-OOSPC 文章结构组织如下. 第1章简要介绍光正交签名码的研究背景、意义以及现状. 第2章探讨最优(m,n,k,λa,λc)-OOSPC码字容量的计算问题.利用有限群理论中群作用的思想给出λα=k或k-1情形下,Θ(m,n,k,λa,k-1)值的计算公式. 第3章从组合设计理论的角度出发,剖析光正交签名码的组合结构,指出光正交签名码与一类特殊填充设计之间的等价关系.进一步地,借助组合设计理论中的两类辅助设计：差阵与可分组设计,给出(m,n,k,λa,λc)-OOSPC的一系列递归构作.除此之外,还将以(3,n,4,1)-OOSPC为例,介绍两种直接构作光正交签名码的方法,同时给出(3,n,4,1)-OOSPC的一些无穷类. 第4与第5章将利用第3章给出的诸多构作方法去解决最优(m,n,4,1)-OOSPC和最优(m,n,3,1)-OOSPC的构作问题.其中,第4章主要是拓展(m,n,4,1)-OOSPC的已有结果.最终围绕三类参数：(1)gcd(m,18)=3且n三0(mod12),(2)mn三8,16(mod24)且gcd(m,n,2):2,及(3)mn三0(mod24)且gcd(m,n,6)=2,获得最优(m,n,4,1)-OOSPC的若干无穷类.第5章则是在第3章所给构作方法的基础上,针对某些特殊参数下的(m,n,3,1)-OOSPC,创新出新的构作方法,从而使得最优(m,n,3,1)-OOSPC的构作问题得以彻底解决. 基于差阵在递归构作光正交签名码时的应用,第6章集中探讨了有限交换群G上(G,4,λ)差阵(即(G,4,λ)-DM)的存在性问题.最终针对λ=1且G为非循环交换群,以及λ1为奇数且G为交换群两种情形,证明了(G,4,λ)-DM存在的充要条件是G没有循环的Sylow2-子群.除此之外,还在第6章的末尾指出,对任意偶数λ≥2和任意有限交换群G,(G,4,λ)-DM总是存在的.
[Abstract]:Based on the application of optical fiber communication technology in medical image, digital video broadcasting and visual image transmission by supercomputer in 1994, Kitayama proposed a new optical code division multiple access (OCDMA) parallel image transmission system using spatial spread spectrum technology. At the same time, optical orthogonal signature codes, as the first choice of optical address codes for code division multiple access (CDMA) parallel image transmission systems, have attracted much attention from many scholars in the field of information theory, combinatorial design and so on. Let k n, 位 a and 位 c be positive integers. The optical orthogonal signature code l with the parameters of Hamming, 位 _ a, 位 _ c is a family of autocorrelation coefficient 位 _ a, and the correlation number is 位 _ c, and the Hamming weight k is m 脳 n ~ (0) ~ (1) ~ (-1) matrix. The codeword capacity of an optical orthogonal signature code is the number of the code words contained in the optical orthogonal signature code (位 _ (a), 位 _ (a) 位 _ (c), and the number of the code words contained in the code is the same as the number of the code words contained in the optical orthogonal signature code (位 _ a, 位 _ (a), 位 _ c)-OSPC). It determines the maximum number of users that the code division multiple access (CDMA) parallel image transmission system can carry at the same time. In this paper, we make mmmnnnnk, 位 _ a, 位 _ c denote the maximum capacity of all codewords in c)-OOSPC, then we say that the code-word capacity is mmnnk, 位 _ (a, 位 _ c) is the best (or the largest) of which the codeword capacity is mnnnnk, 位 _ (a, 位 _ c), 位 _ (a), 位 _ (a, 位 _ c) is the best (or the largest). When 位 _ a = 位 _ c = 位 _ c, the notation c)-OOSPC, 位 _ a, 位 _ c)-OOSPC and ~ (?) are abbreviated as "mechnik, 位 _ a, 位 _ c), respectively). 位 _ a, 位 _ (c)-OOSPC) and ~ () ~ = 位, 位 _ _ _ This paper focuses on the following two issues are discussed in depth. (1) how to determine the exact value of themnnhk, 位 _ a, 位 _ c)? (2) how to construct the optimal c)-OOSPC of a codeword with a capacity of -, 位 _ (a, 位 _ c) (= The structure of the article is as follows. Chapter 1 briefly introduces the research background, significance and current situation of optical orthogonal signature codes. In chapter 2, we discuss the calculation of the optimal c)-OOSPC codeword capacity. By using the idea of group action in the finite group theory, the formulas for calculating the value of 位 伪 ~ (n) k or k ~ (-1) in the case of 位 _ 伪 _ k or k ~ (-1) are given. In chapter 3, the combinatorial structure of optical orthogonal signature codes is analyzed from the point of view of combinatorial design theory, and the equivalent relationship between optical orthogonal signature codes and a class of special fill designs is pointed out. Furthermore, with the aid of two kinds of auxiliary design in combinatorial design theory: difference matrix and grouping design, a series of recursive constructions of c)-OOSPC are given. In addition, two methods of directly constructing optical orthogonal signature codes are introduced, and some infinite classes of OOSPC are given. In the fourth and fifth chapters, we will solve the construction problems of the optimal MNU 4OOSPC and the OOSPC by using the construction methods given in Chapter 3. The results show that the structure of the OOSPC is better than that of the OOSPC, and that of the OOSPC is better than that of the OOSPC. Among them, chapter 4 is mainly to expand the existing results of OOSPC. Finally, around three parameters: 1 / 1 / g / d / m ~ (18) / 3 and n ~ 30 / 0 / d ~ (12) / ~ (2) mn ~ (3 / 8) / ~ (16) ~ (24) and 3 / 3 / mn ~ (30) mod24) and / or gcdlum / mn ~ (6) / 2), some infinite classes of OOSPC are obtained. In chapter 5, on the basis of the construction method given in chapter 3, a new construction method is innovated for some special parameters. Based on the application of differential matrix in the recursive construction of optical orthogonal signature codes, in chapter 6, we focus on the existence of the differential matrices (i.e., G ~ (4), 位 ~ (-DM) over a finite commutative group G. In this paper, we prove that G is a noncyclic commutative group and 位 1 is odd and G is a commutative group. The necessary and sufficient condition is that G does not have a cyclic Sylow2-subgroup. In addition, at the end of Chapter 6, it is pointed out that for any even number 位 鈮，

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