Topics in Algebraic Design Theory

代数设计理论专题

• 批准号: 0400411
• 负责人: Qing Xiang
• 金额: $ 11万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0400411&HistoricalAwards=false
• 关键词: Topics; Algebraic; Design; Theory

The project addresses several problems of central importance in the areas of designs, codes and association schemes. The emphasis will be on the use of algebraic and number theoretic methods in the above areas. Topics of the proposed research include Smith normal forms of subspace-inclusion matrices (q-analogues of the subset-inclusion matrices); connections between difference sets and maximal arcs in projective spaces; strongly regular graphs, two-weight codes and association schemes. Number theoretic tools, such as Gauss and Jacobi sums, have been used with success by the PI and others in all of the afforementioned topics. It is expected that deeper p-adic number theory and representation theory of the general linear group will be necessary in the further investigations of these topics. Combinatorial designs first arose in recreational mathematics problems, such as Kirkman's 15 schoolgirls problem, and later in the design of statistical experiments. They found many applications in coding theory, finite geometry, cryptography, computer science, and electrical engineering. Designs and codes are intimately related. Many efficient error-correcting codes are constructed from designs. These codes are used nowadays in our daily life, for example, in CD players, high-speed modems, and cellular phones. This proposal investigates several problems in the interface of designs and codes, and gives summer support for a graduate student who will study and work in the areas of design theory and coding theory related to the above mentioned applications.

该项目解决了设计、规范和联合方案领域的几个核心问题。重点将放在使用代数和数论方法在上述领域。拟议的研究课题包括史密斯正规形式的子空间包含矩阵（q-类似物的子集包含矩阵）;之间的连接差集和极大弧在射影空间;强正则图，两个重量码和关联计划。数论工具，如高斯和雅可比和，已被PI和其他人成功地用于所有上述主题。我们期望在这些课题的进一步研究中，更深入的p进数理论和一般线性群的表示理论将是必要的。组合设计首先出现在娱乐数学问题中，如柯克曼的15个女学生问题，后来出现在统计实验的设计中。他们发现许多应用在编码理论，有限几何，密码学，计算机科学和电气工程。设计和代码是密切相关的。许多有效的纠错码都是从设计中构造出来的。这些代码现在在我们的日常生活中使用，例如，在CD播放器，高速调制解调器和蜂窝电话中。该提案调查了设计和代码接口中的几个问题，并为将在与上述应用相关的设计理论和编码理论领域学习和工作的研究生提供夏季支持。