New constructions of partial difference sets

这次合作的总体目标是开发强大的新的偏差分集结构，偏差分集是具有特殊对称性和正则性的数学结构。这些结构在编码理论、密码学、图论和有限几何中以各种形式出现，并通过相应信号的有利物理行为在数字通信中具有重要应用。这项新合作的参与者是Jonathan Jedwab博士（不列颠哥伦比亚省西蒙弗雷泽大学），James Davis博士（弗吉尼亚州里士满大学）和John Polhill博士（宾夕法尼亚州布鲁姆斯堡大学）。具体的目标是：(1)理解群的性质和它所包含的偏差分集之间的关系；(2)找到偏差分集的新的递归结构，以便从较小的例子中构造更大的例子；(3)发现偏差分集和其他数学构型之间的新联系。此次合作的成果，将作为后续联盟国际合作基金提案的基础，将包括：(a)更深入地了解离散数学的这些基本结构存在的时间和原因，(b)改进计算搜索偏差分集的算法，(c)在数字通信中更有效和更安全的信号设计的新可能性，涉及降低功耗或更小的辐射发射。(d)一个初步的偏差分集数据库，在未来将得到显著的增强。这些成果将加强加拿大在组合设计和编码理论及其在实际数字通信中的有效和安全应用方面的研究领导地位。

The overall goal of this collaboration is to develop powerful new constructions of partial difference sets, which are mathematical structures with special symmetry and regularity properties. These structures occur in various guises in coding theory, cryptography, graph theory, and finite geometry, and have important applications in digital communications via the advantageous physical behaviour of the corresponding signals. The participants in this new collaboration are Dr. Jonathan Jedwab (Simon Fraser University, BC), Dr. James Davis (University of Richmond, VA) and Dr. John Polhill (Bloomsburg University, PA). The specific objectives are to:(1) Understand the relationship between the properties of a group and the partial difference sets it contains,(2) Find new recursive constructions of partial difference sets so that larger examples can be constructed from smaller examples, and(3) Discover new connections between partial difference sets and other mathematical configurations.The outcomes of the collaboration, to be used as the foundation for a subsequent Alliance International Collaboration Grant proposal, will include:(a) a deeper understanding of when and why these fundamental structures of discrete mathematics exist,(b) improved algorithms for searching for partial difference sets computationally,(c) new possibilities for more efficient and safer signal design in digital communications, involving reduced power consumption or smaller radiated emissions, and(d) a preliminary database of partial difference sets, to be significantly augmented in future. These outcomes will strengthen Canada's position as a research leader in the theory of combinatorial designs and coding, and in their efficient and safe application in practical digital communications.