Invariants of incidence matrices, difference sets and strongly regular graphs

关联矩阵、差分集和强正则图的不变量

• 批准号: 0701049
• 负责人: Qing Xiang
• 金额: --
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701049&HistoricalAwards=false
• 关键词: Invariants; incidence; matrices; difference; sets

This grant will support the PI and his collaborators to work on invariants of incidence matrices, difference sets, strongly regular graphs and the related error-correcting codes. Incidence matrices arise whenever one considers relations between two finite sets. Invariants of incidence matrices, such as p-ranks and Smith normal forms, can reveal a great deal of information of the underlying incidence structures. The PI plans to continue his research on various geometric incidence matrices, such as the subspace-inclusion matrices and their symplectic analogues. The Smith normal form results will have interesting geometric applications, for example, in classifying ovoids in three-dimensional projective space over a finite field of even order. Other topics of research include constructions and nonexistence proofs of difference sets and strongly regular graphs. The PI's investigation will make use of techniques from several different areas of mathematics, including representation theory, p-adic number theory and combinatorics.Incidence matrices play an important role in many parts of discrete mathematics, including the theory of error-correcting codes. Most of the incidence matrices considered in this proposal can be used to generate low density parity check codes. Efficient error-correcting codes are used nowadays in our daily life, for example, in CD players, high speed modems, and cellular phones. Cyclic difference sets are the same objects as binary sequences with two-level periodic autocorrelation functions. Such sequences have many applications in radar, spread-spectrum communications and cryptography. This project will concentrate on the study of invariants of incidence matrices and the existence questions of difference sets.

这笔赠款将支持 PI 及其合作者研究关联矩阵、差异集、强正则图和相关纠错码的不变量。每当考虑两个有限集之间的关系时，就会出现关联矩阵。关联矩阵的不变量，例如 p 秩和 Smith 范式，可以揭示潜在关联结构的大量信息。 PI 计划继续研究各种几何关联矩阵，例如子空间包含矩阵及其辛类似矩阵。史密斯范式结果将具有有趣的几何应用，例如，在偶数阶有限域上的三维射影空间中对卵形进行分类。其他研究主题包括差异集和强正则图的构造和不存在证明。 PI 的调查将利用多个不同数学领域的技术，包括表示论、p 进数论和组合数学。关联矩阵在离散数学的许多部分中发挥着重要作用，包括纠错码理论。本提案中考虑的大多数关联矩阵可用于生成低密度奇偶校验码。如今，高效的纠错码已广泛应用于我们的日常生活中，例如 CD 播放器、高速调制解调器和蜂窝电话。循环差分集与具有两级周期性自相关函数的二进制序列是相同的对象。此类序列在雷达、扩频通信和密码学中有许多应用。该项目将集中研究关联矩阵的不变量和差异集的存在性问题。