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

CRII：AF：RUI：基本纠错码的新领域

• 批准号: 2347371
• 负责人: Ray Li
• 金额: $ 17.43万
• 依托单位: Santa Clara University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-04-01 至 2026-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2347371&HistoricalAwards=false
• 关键词: CRII; AF; RUI; New; Frontiers

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.

这个项目研究了纠错码在现代应用中的基本情况。纠错码是一种数学对象，可在各种计算环境中保护信息免受噪声的不利影响。例如，插入/删除纠错码有助于保护存储在DNA存储或磁存储中的信息的完整性，而量子纠错码保护量子态的微妙纠缠，这是量子计算机与经典计算机的区别。纠错码在理论上也是基本的对象，可以找到与算法、计算复杂性和密码学等领域的意想不到的联系，在这些领域，应用程序显然不需要抗噪。纠错码有着悠久而丰富的历史。然而，对噪音恢复能力的新需求，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，并重新唤起了人们对旧问题的兴趣。这个项目考虑了这些新的和新的问题。这个项目还有一个教育部分，包括本科生研究指导和本科生课程开发。这个项目解决了关于纠错码的三个基本问题：第一个问题考虑了纠正插入和删除错误的代码：代码保护信息免受插入和删除错误的限制是多少？该项目还利用新发现的与插入/删除代码的算法联系来寻找异常快速的算法来估计序列的相似性，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译码，这在理论和实践上都很重要：什么样的码可以达到列表译码的理论最优？第三个问题是量子纠错码：量子体系结构的物理约束如何限制我们保存纠缠量子态的能力？这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。