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Codes from Curves: Structure, Decoding, and Modern Applications

曲线编码：结构、解码和现代应用

基本信息

批准号：
1855136
负责人：
Gretchen Matthews
金额：
$ 18万
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：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855136&HistoricalAwards=false
关键词：
Codes Curves Structure Decoding Modern

项目摘要

This award supports research into algebraic coding theory. The large volume of data generated today motivates the need for distributed storage systems which can provide long-term storage of data with highly reliable retrieval and availability of information to users. A distributed storage system consists of a network of storage nodes. A stored file may be retrieved by accessing these nodes. In such a system, individual nodes may be unreliable; for instance, they may become unavailable due to routine maintenance. For this reason, how information is stored across the network is highly relevant. The goal is to create a storage system that allows for recovery of lost data while limiting both network traffic and the number of disks accessed. This project addresses the problem by appealing to the underlying structure of a family of error-correcting codes defined using algebraic geometry. The codes have robust structure that yields powerful local properties and potential to balance storage overhead, reliability, network traffic, and repair bandwidth. This project considers the use of codes obtained via algebraic geometric constructions in highly relevant applications. The following problems will be studied: coding for distributed storage via algebraic geometric and combinatorial points of view, with the goals of minimizing node access and repair bandwidth; and explicit construction of locally correctable codes, quantum codes, and secret sharing schemes using algebraic geometric tools. Code structure will be explored, and decoding codes which are directs sum may be handled by decoding or repair within its factors. Full use will be made of the automorphism group of the code as well as natural nested structures inherited from the underlying algebraic geometry. Much of the work in this proposal is focused on algebraic geometric constructions utilized in situations where limiting network traffic in the decoding process is desirable; the aim is to accomplish this by capitalizing on the natural underlying structure of algebraic geometric codes rather than appealing exclusively to locality. In doing so, the performance guaranteed by the code construction is maintained (meaning code parameters do not suffer from local requirements) while making the codes more amenable to current applications.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.

该奖项支持代数编码理论的研究。如今产生的大量数据激发了对分布式存储系统的需求，该系统可以提供长期的数据存储，并为用户提供高度可靠的检索和信息可用性。分布式存储系统由存储节点网络组成。可以通过访问这些节点来检索存储的文件。在这样的系统中，单个节点可能是不可靠的；例如，它们可能由于日常维护而变得不可用。因此，信息如何在网络上存储是高度相关的。目标是创建一个存储系统，允许恢复丢失的数据，同时限制网络流量和访问的磁盘数量。该项目通过利用代数几何定义的一系列纠错码的底层结构来解决这个问题。这些代码具有强大的结构，可产生强大的本地属性和平衡存储开销、可靠性、网络流量和修复带宽的潜力。该项目考虑在高度相关的应用中使用通过代数几何构造获得的代码。将研究以下问题：通过代数几何和组合的观点对分布式存储进行编码，以最小化节点访问和修复带宽为目标；以及使用代数几何工具显式构造局部可校正代码、量子代码和秘密共享方案。将探索代码结构，并且可以通过在其因子内解码或修复来处理直接和的解码代码。将充分利用代码的自同构群以及从底层代数几何继承的自然嵌套结构。该提案中的大部分工作都集中在需要在解码过程中限制网络流量的情况下使用的代数几何构造；目的是通过利用代数几何代码的自然基础结构而不是仅仅诉诸局部性来实现这一目标。这样做，可以保持代码结构所保证的性能（意味着代码参数不受当地要求的影响），同时使代码更适合当前的应用。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。