Topics in Algebraic Design Theory

该项目解决了设计、规范和关联方案领域的几个核心问题。重点将放在上述领域中代数和数论方法的使用上。建议的研究主题包括子空间包含矩阵的Smith范式（子集包含矩阵的q-类似物）；射影空间中差集与极大弧之间的联系强正则图，二权码和关联方案。数论工具，如高斯和雅可比和，已经被PI和其他人成功地用于上述所有主题。在这些课题的进一步研究中，需要更深入的p进数论和一般线性群的表示理论。组合设计首先出现在娱乐性的数学问题中，比如柯克曼的15个女学生问题，后来出现在统计实验的设计中。他们在编码理论、有限几何、密码学、计算机科学和电子工程中找到了许多应用。设计和规范是密切相关的。许多有效的纠错码都是从设计中构造出来的。这些代码如今在我们的日常生活中使用，例如CD播放机、高速调制解调器和移动电话。本提案研究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暑期支持。

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.