Collaborative Research: Evaluation Codes, Duals, and Applications

合作研究：评估代码、对偶和应用

• 批准号: 2401558
• 负责人: Hiram Lopez Valdez
• 金额: $ 24.7万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-10-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401558&HistoricalAwards=false
• 关键词: Collaborative; Research; Evaluation; Codes; Duals

Coding theory supports reliable, robust communication even while using imperfect channels, which distort messages. It has evolved since its emergence in the 1940s to support various applications ranging from secure communications to data storage. Bespoke codes and customized error correction or erasure recovery algorithms satisfy ever-changing technological needs. Evaluation codes employ tools and underlying structures from algebraic geometry and commutative algebra to provide flexible constructions that address various scenarios. They build on the heavily utilized Reed-Solomon and Reed-Muller codes which depend on polynomial structures. This project focuses on designing codes and algorithms which recover erased data or correct errors using less information than traditional methods, building frameworks for use in practical settings, and employing them in quantum error correction. The impact of this proposed research expands beyond the immediate scientific applications to serve as a platform for student and postdoctoral training and efforts to diversify the discipline. The PIs have a history of providing research engagement opportunities and amplifying them via other outreach such as Cleveland State University's STEM Peer Teachers and Association of Latin Professionals for America as well as Virginia Tech's Broadening Engagement and Participation in Undergraduate Research, SWIMM: Supporting Women in Mathematics through Mentoring, and Virginia's Commonwealth Cyber Initiative. Evaluation codes are a large family of error-correcting codes, encompassing algebraic geometry codes and polynomial codes, such as Reed-Solomon and Reed-Muller codes. This project advances their utility by investigating their duals, focusing on the hull of a code, meaning the intersection of the code and its dual. The hull plays a role in the complexity of several algorithms in code-based cryptography, protection against side-channel and fault injection attacks, and quantum error correction. Tools from algebraic geometry and commutative algebra will be employed to determine duals and hulls of evaluation codes, including multivariate Goppa codes and codes from curves, and provide explicit constructions for codes with controlled duals. Objectives include designing evaluation codes and algorithms which utilize less information in erasure recovery (via linear exact repair) and error correction (via fractional decoding); extending the framework for polar coding via evaluation codes to channels with memory; and constructing new codes for quantum error correction. The proposed research serves as an ideal training ground for students and postdoctoral researchers due to multiple points of entry and the opportunity for computation, allowing for increased opportunities to diversify the discipline.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.

即使使用不完美的通道（会扭曲消息），编码理论也支持可靠、稳健的通信。自 20 世纪 40 年代出现以来，它不断发展，支持从安全通信到数据存储等各种应用。定制代码和定制纠错或擦除恢复算法满足不断变化的技术需求。评估代码采用代数几何和交换代数的工具和基础结构来提供解决各种场景的灵活结构。它们建立在大量使用的 Reed-Solomon 和 Reed-Muller 码的基础上，这些码依赖于多项式结构。该项目的重点是设计代码和算法，使用比传统方法更少的信息来恢复已擦除的数据或纠正错误，构建在实际环境中使用的框架，并将其用于量子纠错。这项拟议研究的影响超出了直接的科学应用范围，成为学生和博士后培训以及学科多元化努力的平台。 PI 长期以来一直提供研究参与机会，并通过其他外展活动扩大这些机会，例如克利夫兰州立大学的 STEM 同行教师和美国拉丁专业人士协会，以及弗吉尼亚理工大学扩大本科生研究的参与和参与、SWIMM：通过辅导支持女性数学研究以及弗吉尼亚联邦网络倡议。评估码是一大类纠错码，包括代数几何码和多项式码，例如里德-所罗门码和里德-穆勒码。该项目通过研究它们的对偶来提高它们的实用性，重点关注代码的外壳，即代码与其对偶的交集。外壳在基于代码的密码学、防止侧信道和故障注入攻击以及量子纠错等多种算法的复杂性中发挥着重要作用。代数几何和交换代数的工具将用于确定评估代码的对偶和外壳，包括多元 Goppa 代码和曲线代码，并为具有受控对偶的代码提供显式构造。目标包括设计在擦除恢复（通过线性精确修复）和纠错（通过分数解码）中使用较少信息的评估代码和算法；通过评估代码将极性编码框架扩展到具有记忆的通道；并构建用于量子纠错的新代码。由于多个切入点和计算机会，拟议的研究成为学生和博士后研究人员的理想培训基地，从而增加了学科多样化的机会。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。