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.

编码理论即使使用不完美的渠道，也支持可靠，稳健的通信，这会扭曲消息。自1940年代出现以来，它已经发展起来，以支持从安全通信到数据存储的各种应用程序。定制代码和自定义的误差校正或擦除恢复算法满足了不断变化的技术需求。评估代码采用代数几何形状和交换代数的工具和基础结构来提供灵活的结构来解决各种情况。它们建立在依赖多项式结构的大量利用的芦苇 - 溶剂和芦苇 - 毛刺代码上。该项目着重于设计代码和算法，这些代码和算法使用传统方法恢复删除的数据或更改错误，在实用设置中使用的构建框架，并将其用于量子错误校正。这项拟议研究的影响扩大了直接的科学应用，可以作为学生和博士后培训的平台以及使学科多样化的努力。 PI具有提供研究参与机会的历史，并通过其他外展活动（例如克利夫兰州立大学的STEM同伴教师和拉丁专业人员协会）以及弗吉尼亚理工大学的扩大参与和参与本科研究的参与，游泳：通过辅导和弗吉尼亚州的英联邦网络网络支持妇女在数学领域，为妇女提供支持的妇女。评估代码是一个大型误差校正代码，包括代数几何代码和多项式代码，例如Reed-Solomon和Reed-Muller代码。该项目通过调查双重功能，重点关注代码的船体，这意味着代码及其双重的交集，从而提高了他们的效用。船体在基于代码的密码学，防止侧通道和断层注射攻击以及量子误差校正中的多种算法的复杂性中起作用。代数几何形状和交换代数的工具将用于确定评估代码的双重和船体，包括多元GOPPA代码和曲线的代码，并为具有受控二元组的代码提供明确的构造。目标包括设计评估代码和算法，这些算法在擦除恢复（通过线性精确修复）和误差校正（通过分数解码）中使用的信息较少；通过评估代码将极地编码的框架扩展到具有内存的通道；并构建新的代码进行量子误差校正。拟议的研究是由于多个入学点和计算机会，为学生和博士后研究人员提供了理想的培训理由，从而增加了更多机会多样化该学科。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估来通过评估来获得支持的。