Modular Ranks of Incidence Matrices and Related Topics

首席研究员(PI)和他的合作者对一系列关于关联矩阵、可加组合、差集和Hadamard矩阵的问题感兴趣。只要考虑两个有限集之间的关系，就会产生关联矩阵。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计算。在过去的几年里，PI和他的合作者在研究有限几何和组合数学中产生的关联矩阵的模秩和Smith范式方面取得了一些成功。国际和平协会打算继续朝这一方向工作，旨在解决这一领域的几个未决问题。其次，PI将继续他最近关于有限交换群的加法表的拉丁横截的Snevly猜想的工作，并将外代数方法应用于加性组合学中的其他问题。在第三部分中，PI打算研究相对差集，并进一步探索它们与Hadamard矩阵和互无偏基的关系。该方案中考虑的许多关联矩阵可以用来生成有效的纠错码，这些纠错码如今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。差集和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着广泛的应用。

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.