CRII: AF: RUI: New Frontiers in Fundamental Error-Correcting Codes

本项目研究现代应用中纠错码的基本内容。纠错码是保护信息免受各种计算环境中噪声的不利影响的数学对象。例如，插入/删除校正码有助于保护DNA存储或磁存储存储的信息的完整性，量子纠错码保护量子态的微妙纠缠，这将量子计算机与经典计算机区分开来。纠错码也是理论上的基本对象，它与算法、计算复杂性和密码学等领域有着意想不到的联系，这些领域的应用对抗噪声能力没有明显的需求。纠错码有着悠久而丰富的历史。然而，对噪声弹性的新需求以及与计算机科学和数学中其他主题的新联系激发了新的问题，并重新引起了人们对旧问题的兴趣。本项目考虑这些新的和更新的问题。该项目还包括一个教育部分，其中包括本科生研究指导和本科生课程开发。该项目解决了关于纠错码的三个基本问题：第一个问题考虑纠正插入和删除错误的代码：代码可以保护信息免受插入和删除错误的限制是什么？该项目还利用新发现的与插入/删除代码的算法连接来找到用于估计序列相似性的异常快速的算法，这是计算生物学中的一个中心问题。第二个问题考虑列表可解码码，这在理论和实践中都很重要：什么样的代码可以实现列表解码的理论最优？第三个问题考虑量子纠错码：量子体系结构的物理约束如何限制我们保持纠缠量子态的能力？该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估的支持。

This project studies fundamental contexts of error-correcting codes in modern applications. Error correcting codes are mathematical objects that protect information from the adverse effects of noise in a variety of computational contexts. For example, insertion/deletion-correcting codes help protect the integrity of information stored with DNA storage or magnetic storage, and quantum error-correcting codes protect the delicate entanglement of quantum states that distinguishes quantum computers from classical computers. Error correcting codes are also theoretically fundamental objects that find unexpected connections to areas like algorithms, computational complexity, and cryptography, where the applications have no apparent need for noise-resilience. Error-correcting codes have a long and rich history. Nevertheless, new demands for noise resilience and new connections to other topics in computer science and mathematics have inspired new questions and revived interest in old ones. This project considers these new and renewed questions. This project also has an educational component, which includes undergraduate research mentorship and undergraduate course development.This project tackles three fundamental questions about error-correcting codes: The first question considers codes that correct insertion and deletion errors: what are the limits of how well codes can protect information from insertions and deletions errors? This project also leverages a newly discovered algorithmic connection with insertion/deletion codes to find exceptionally fast algorithms for estimating the similarity of sequences, a central question in computational biology. The second question considers list-decodable codes, which are important in both theory and practice: what kinds of codes can achieve the theoretical optimum for list-decoding? The third question considers quantum error-correcting codes: how do physical constraints of quantum architectures limit our ability to preserve entangled quantum states?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.