Collaborative Research: Evaluation Codes, Duals, and Applications

编码理论支持可靠、健壮的通信，即使在使用不完美的信道时，也会扭曲信息。自20世纪40年代出现以来，它已经发展到支持从安全通信到数据存储的各种应用。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。计算代码使用代数几何和交换代数中的工具和底层结构来提供处理各种场景的灵活结构。它们建立在依赖于多项式结构的大量使用的Reed-Solomon和Reed-Muller码的基础上。该项目侧重于设计代码和算法，使用比传统方法更少的信息来恢复被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，作为学生和博士后培训的平台，并努力使学科多样化。pi有提供研究参与机会的历史，并通过其他外展活动扩大这些机会，如克利夫兰州立大学的STEM同行教师和美洲拉丁专业人员协会，以及弗吉尼亚理工大学的扩大参与和参与本科研究，SWIMM：通过指导支持女性数学，以及弗吉尼亚的联邦网络倡议。求值码是一大类纠错码，包括代数几何码和多项式码，如Reed-Solomon码和Reed-Muller码。该项目通过研究它们的双重性来提高它们的实用性，重点关注代码的外壳，即代码与其双重性的交集。在基于代码的加密中，外壳在几种算法的复杂性中发挥作用，防止侧信道和故障注入攻击，以及量子纠错。代数几何和交换代数的工具将被用来确定计算码的对偶和壳，包括多元Goppa码和曲线码，并提供具有控制对偶的码的显式结构。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；通过求值码将极性编码框架扩展到具有内存的信道；构建量子纠错的新代码。由于有多个切入点和计算机会，拟议的研究可以作为学生和博士后研究人员的理想训练场，从而增加了学科多样化的机会。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。

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.