Association Schemes and Configurations in Real and Complex Space

该项目研究了通信、信息论、网络和纯数学中众多主题的各个领域的基础组合对象。该团队由首席研究员（PI）和一名博士生组成，将研究真实复杂空间中的关联方案和配置。 该项目的动机是纠错码理论、密码学、量子信息论、结理论、极值网络、有限几何以及球面和射影码的应用。关联方案和相关工具在这些和其他领域发挥着基础作用，例如指导我们了解大多数数字通信设备中使用的二进制纠错块码的最著名的效率界限。随着数字技术规模和复杂性的增长，我们看到对此类代数工具的需求不断增加，这些工具可以从数据中有效地提取重要的结构信息。该项目更广泛的影响包括培训高素质人才以及向当地学校的学生和教师进行推广。过去 50 年来协会计划的进展是惊人的，该理论处理的新应用的种类逐年增加。然而，一些基本问题（例如关于彗星关联方案）仍未解决，直到最近才出现一些新想法。该项目探索了从代数几何到代数拓扑的强大数学工具，以打造更强大、更通用的关联方案理论，该方案将能够解决现有的开放性问题（理论和应用）以及可能在这种通用组合语言中提出的未来挑战。要解决的具体问题包括：在彗形方案的第一本原幂等列集上消失的多项式理想的有效描述；具有少量角度的线系统的构造（通过有限几何和设计理论）和边界；关联方案和代码的效率参数的界限；确定加法完全正则纠错码的结构。所使用的数学工具主要由代数组合学界开发，包括线性代数和环理论技术，以及多项式环和离散同伦中的零维理想。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project investigates combinatorial objects fundamental to various areas of communications, information theory, networks, and numerous topics in pure mathematics. The team, consisting of the Principal Investigator (PI) and a PhD student will study association schemes and configurations in real and complex space. The project is motivated by applications in the theory of error-correcting codes, cryptography, quantum information theory, knot theory, extremal networks, finite geometry and spherical and projective codes. Association schemes and related tools play a fundamental role in these and other areas, for example guiding us to the best known efficiency bounds for binary error-correcting block codes used in most digital communications devices. As digital technologies grow in scale and complexity, we see increasing need for algebraic tools of this sort that extract important structural information efficiently from data. The project's broader impacts include training of highly qualified personnel and outreach to students and teachers in local schools.Progress on association schemes over the past 50 years has been phenomenal, and the variety of new applications that the theory handles has increased with each passing decade. Yet some fundamental problems, regarding cometric association schemes for example, remain unresolved with few new ideas emerging until recently. This project explores powerful mathematical tools, ranging from algebraic geometry to algebraic topology, to forge a stronger and more versatile theory of association schemes that will be equipped to attack existing open questions - both theoretical and applied - as well as future challenges that are likely to be framed in this general combinatorial language. Specific problems to be attacked include: efficient description of the ideal of polynomials vanishing on the set of columns of the first primitive idempotent of a cometric scheme; constructions (via finite geometry and design theory) and bounds for systems of lines with few angles; bounds on efficiency parameters of association schemes and codes; determining the structure of additive completely regular error-correcting codes. The mathematical tools to be employed have mainly been developed by the algebraic combinatorics community, including linear-algebraic and ring-theoretic techniques, as well as zero-dimensional ideals in polynomial rings and discrete homotopy.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.