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.

该奖项支持对代数编码理论的研究。当今生成的大量数据激发了分布式存储系统的需求，这些存储系统可以为用户提供高度可靠的检索和信息可用性的长期存储。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，单个节点可能不可靠。例如，由于常规维护，它们可能无法获得。因此，如何在整个网络中存储信息非常相关。目标是创建一个存储系统，该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。该项目通过吸引使用代数几何定义的错误校正代码家族的基础结构来解决该问题。这些代码具有强大的结构，可产生强大的本地属性，并具有平衡存储开销，可靠性，网络流量和维修带宽的潜力。该项目考虑通过在高度相关应用中通过代数几何结构获得的代码。将研究以下问题：通过代数几何和组合观点编码分布式存储的编码，其目标是最大程度地减少节点访问和维修带宽；并使用代数几何工具明确构建本地可更正的代码，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素内解码或维修来处理直接总和的解码代码。完全使用代码的自动形态组以及从基础代数几何形状继承的天然嵌套结构。该提案中的大部分工作都集中在代数几何结构上，这些几何结构是在解码过程中限制网络流量的情况下；目的是通过利用代数几何代码的自然潜在结构来实现这一目标，而不是仅吸引当地。在此过程中，通过代码构建所保证的性能（意味着代码参数不会遭受本地要求），同时使代码更适合当前应用程序。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优点和更广泛的审查标准通过评估来进行评估的。