Modular Ranks of Incidence Matrices and Related Topics

关联矩阵的模块化排序及相关主题

• 批准号: 1001557
• 负责人: Qing Xiang
• 金额: $ 17.54万
• 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: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001557&HistoricalAwards=false
• 关键词: Modular; Ranks; Incidence; Matrices; Related

The principal investigator (PI) and his collaborators are interested in a diverse set of problems concerning incidence matrices, additive combinatorics, difference sets and Hadamard matrices. Incidence matrices arise whenever one considers relations between two finite sets. Many questions in finite geometry, combinatorics and representation theory of finite groups reduce to the computation of ranks of certain incidence matrices over finite fields. The PI and his collaborators have had some successes in the past few years investigating the modular ranks and the Smith normal forms of classes of incidence matrices arising from finite geometry and combinatorics. The PI intends to continue his work in this direction, aiming at solving several open problems in this area. Secondly, the PI will continue his recent work on Snevily's conjecture on Latin transversals of addition tables of finite abelian groups and apply the exterior algebra method to other problems in additive combinatorics. In the third part, the PI intends to study relative difference sets and further explore their connections to Hadamard matrices and mutually unbiased bases.Incidence matrices are basic mathematical objects which are frequently encountered in various branches of mathematics, computer science and engineering. Many of the incidence matrices considered in this proposal can be used to generate efficient error-correcting codes, which are used nowadays in our daily life, for example, in CD players, high speed modems, and cellular phones. Difference sets and Hadamard matrices are important objects in combinatorial designs theory, which have found many applications in radar, spread-spectrum communications and cryptography.

首席研究员 (PI) 和他的合作者对涉及关联矩阵、加法组合、差分集和哈达玛矩阵的各种问题感兴趣。每当考虑两个有限集之间的关系时，就会出现关联矩阵。 有限几何、组合学和有限群表示论中的许多问题都归结为有限域上某些关联矩阵的秩的计算。在过去的几年里，PI 和他的合作者在研究由有限几何和组合学产生的关联矩阵类的模秩和 Smith 范式方面取得了一些成功。 PI 打算继续朝这个方向工作，旨在解决该领域的几个未决问题。其次，PI将继续他最近关于Snevilly关于有限交换群加法表的拉丁横截的猜想的工作，并将外代数方法应用于加法组合学中的其他问题。在第三部分中，PI打算研究相对差集，并进一步探讨它们与Hadamard矩阵和互无偏基的联系。关联矩阵是在数学、计算机科学和工程学的各个分支中经常遇到的基本数学对象。本提案中考虑的许多关联矩阵可用于生成高效的纠错码，这些代码在当今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。差分集和 Hadamard 矩阵是组合设计理论中的重要对象，在雷达、扩频通信和密码学中有许多应用。