Invariants of incidence matrices, difference sets and strongly regular graphs

这笔拨款将支持PI和他的合作者研究关联矩阵、差集、强正则图和相关纠错码的不变量。只要考虑两个有限集之间的关系，就会产生关联矩阵。关联矩阵的不变量，如p-秩和Smith正规形，可以揭示关联结构的大量信息。PI计划继续他对各种几何关联矩阵的研究，如子空间包含矩阵及其辛类似物。Smith范式的结果将在几何上有意义的应用，例如，在偶数阶有限域上的三维射影空间中的卵形分类。其他研究主题包括差集和强正则图的构造和不存在证明。PI的研究将利用几个不同数学领域的技术，包括表示论、p-进数论和组合学。概率矩阵在离散数学的许多部分都发挥着重要作用，包括纠错码理论。该方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，高效纠错码在我们的日常生活中被使用，例如，在CD播放机、高速调制解调器和移动电话中。循环差集与具有两级周期自相关函数的二元序列具有相同的对象。这种序列在雷达、扩频通信和密码学中有着广泛的应用。本课题将集中研究关联矩阵的不变量和差集的存在性问题。

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.