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.

该项目研究现代应用中纠错码的基本背景。纠错码是一种数学对象，可以保护信息免受各种计算环境中噪声的不利影响。例如，插入/删除校正码有助于保护 DNA 存储或磁存储所存储信息的完整性，而量子纠错码则可以保护量子态的微妙纠缠，这正是量子计算机与经典计算机的区别。理论上，纠错码也是基本对象，可以与算法、计算复杂性和密码学等领域发现意想不到的联系，而这些领域的应用程序对抗噪性没有明显的需求。纠错码有着悠久而丰富的历史。尽管如此，对抗噪能力的新要求以及与计算机科学和数学其他主题的新联系激发了新问题并重新燃起了对旧问题的兴趣。该项目考虑这些新的和更新的问题。该项目还具有教育部分，其中包括本科生研究指导和本科生课程开发。该项目解决有关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：代码的好坏有哪些限制可以保护信息免受插入和删除错误的影响吗？该项目还利用新发现的与插入/删除代码的算法连接来寻找异常快速的算法来估计序列的相似性，这是计算生物学的一个核心问题。第二个问题考虑了列表解码码，这在理论和实践中都很重要：什么样的码可以实现列表解码的理论最佳值？第三个问题考虑量子纠错码：量子结构的物理约束如何限制我们保存纠缠量子态的能力？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进行评估，被认为值得支持标准。