Collaborative Research: Evaluation Codes, Duals, and Applications

编码理论支持可靠、稳健的通信，即使在使用不完美的信道时也是如此，这会扭曲消息。自20世纪40年代出现以来，它已经发展为支持从安全通信到数据存储的各种应用。定制的代码和定制的纠错或擦除恢复算法可以满足不断变化的技术需求。评估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灵活的结构，以应对各种情况。它们建立在广泛使用的依赖于多项式结构的里德-所罗门和里德-穆勒码的基础上。本项目致力于设计恢复擦除数据或纠正错误的代码和算法，使用比传统方法更少的信息，构建用于实际环境的框架，并将其应用于量子纠错。这项拟议研究的影响超越了直接的科学应用，成为学生和博士后培训的平台，并努力使该学科多样化。私人投资机构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，例如克利夫兰州立大学的STEM同行教师和美国拉丁专业人员协会，以及弗吉尼亚理工大学扩大对本科生研究的接触和参与，游泳：通过导师支持女性数学，以及弗吉尼亚州的联邦网络倡议。评估码是纠错码的一个大家族，包括代数几何码和多项式码，如里德-所罗门和里德-穆勒码。这个项目通过研究它们的对偶来提高它们的实用性，重点放在代码的外壳上，这意味着代码及其对偶的交集。外壳在基于代码的密码学中的几种算法的复杂性中扮演着重要角色，保护免受旁路和故障注入攻击，以及量子纠错。利用代数几何和交换代数的工具来确定评价码的对偶和壳，包括多元Goppa码和曲线上的码，并给出具有受控对偶码的显式构造。目标包括设计在擦除恢复(通过线性精确修复)和纠错(通过分数译码)中利用较少信息的评估码和算法；通过评估码将极坐标编码的框架扩展到有记忆的信道；以及构造用于量子纠错的新码。这项拟议的研究为学生和博士后研究人员提供了一个理想的培训场地，因为它有多个切入点和计算机会，允许更多的机会使学科多样化。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。

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.