Codes from Curves: Structure, Decoding, and Modern Applications

该奖项支持对代数编码理论的研究。今天生成的大量数据激发了对分布式存储系统的需求，该分布式存储系统可以为用户提供具有高度可靠的信息检索和可用性的数据的长期存储。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，信息如何在网络中存储非常重要。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。这个项目解决了这个问题，呼吁使用代数几何定义的纠错码的家庭的底层结构。代码具有强大的结构，产生强大的本地属性和潜力，以平衡存储开销，可靠性，网络流量和修复带宽。 这个项目考虑了在高度相关的应用程序中使用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的角度来看，分布式存储的编码，以最小化节点访问和修复带宽的目标;和明确的本地可纠正的代码，量子码，和秘密共享计划的建设使用代数几何工具。对码的结构进行了探讨，对于直接和的译码码，可以在其因子内进行译码或修复。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。在这个建议中的大部分工作都集中在代数几何结构中使用的情况下，限制网络流量的解码过程是可取的，目的是实现这一目标，利用代数几何代码的自然底层结构，而不是专门呼吁地方。这样，代码结构所保证的性能得以保持（这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。

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.