Combinatorial designs in quantum information theory and digital communications

如果没有卫星通信、手机、便携式音乐播放器、闪存驱动器、GPS（全球定位系统）导航和点播电影等数字通信技术，现代社会将无法辨认。这些技术变得无处不在、被广泛接受和高度可靠的一个关键原因是，嵌入式数学结构和算法提供了对用户不可见的关键支持。在这些技术中嵌入数学的要求来自于物理约束的组合，例如，希望有效地使用能量将信息从一个设备发送到另一个设备，或者即使外部人员可以监视交换，也要在双方之间安全地传递信息，或者即使原始信号被噪声破坏，也要正确地恢复传输的信息。这些物理约束的组合对应于在多个约束下排列物体的数学问题;这些问题通过组合设计来解决。** 本提案的短期目标涉及使用组合设计来解决数字通信中三个众所周知的挑战性问题：*（1）构建大量复杂的等角线。(2)约束最大数量的复杂互无偏基。* (3)构造具有大渐近价值因子的二元序列族。*寻求这些问题的完整解决方案是非常雄心勃勃的：几十年来，每个问题都抵制了众多数学家，物理学家和工程师的相当大的努力。问题1和2出现在量子信息理论中，这是一个动态的研究领域，处理传统物理学不再适用的非常小尺度的信息处理。解决这些问题将有利于量子版本的密码学（人们希望让对手难以阅读信息）和量子版本的纠错码（相反，人们希望让其他人即使在存在噪声的情况下也能轻松阅读信息）。组合设计在解决这些问题中的重要性尚未确定，这是本提案的一个主要创新方面。问题3出现在传统（非量子）数字通信和统计力学中。它的解决方案将允许设计更有效的雷达和无线通信，并创造可申请专利的技术。我们的目标不仅是解决这些具体问题，但在这样做，以创建新的工具和方法来分析与数字通信相关的广泛的组合结构类。这链接到我的研究计划的长期目标：结合联合收割机组合，代数，分析和计算技术，从根本上改变经典和新兴问题的研究是重要的实际和理论的数字通信。

Modern society would be unrecognizable without digital communications technologies such as satellite communication, cell phones, portable music players, flash drives, GPS (global positioning system) navigation, and movies on demand. A key reason these technologies have become ubiquitous, widely accepted, and highly dependable is the critical support provided by embedded mathematical structures and algorithms that remain invisible to the user.******The requirement for embedded mathematics in these technologies derives from a combination of physical constraints, for example the desire to send information from one device to another using energy efficiently, or to pass information securely between two parties even though outsiders can monitor the exchange, or to recover transmitted information correctly despite corruption of the original signal by noise. These combinations of physical constraints correspond to mathematical problems of arranging objects subject to multiple constraints; such problems are solved by combinatorial designs. ******The short-term objectives of this proposal involve the use of combinatorial designs to solve three notoriously challenging problems of digital communications:***(1) Construct large sets of complex equiangular lines.***(2) Constrain the largest number of complex mutually unbiased bases.***(3) Construct families of binary sequences with large asymptotic merit factor.***Seeking a complete solution to these problems is highly ambitious: each has resisted considerable effort over decades by numerous mathematicians, physicists, and engineers.******Problems 1 and 2 arise in quantum information theory, a dynamic research area dealing with the processing of information at very small scales where conventional physics no longer applies. Solving these problems would benefit the quantum version of cryptography (where one wishes to make messages difficult for an adversary to read) and the quantum version of error-correcting codes (where, on the contrary, one wishes to make messages easy for others to read even in the presence of noise). The importance of combinatorial designs in solving these problems has not yet been established, and is a major innovative aspect of this proposal.******Problem 3 arises in conventional (non-quantum) digital communications and in statistical mechanics. Its solution would allow the design of more efficient radar and wireless communications, and the creation of patentable technology.******The aim is not only to solve these specific problems, but in doing so to create new tools and methods for analyzing a wide class of combinatorial structures related to digital communications. This links to the long-term goal of my research program: to combine combinatorial, algebraic, analytical, and computational techniques to transform radically the study of classical and emerging problems that are important in practical and theoretical digital communications.