Codes from Curves: Structure, Decoding, and Modern Applications

该奖项支持对代数编码理论的研究。今天产生的大量数据促使对分布式存储系统的需求，这种存储系统可以提供长期的数据存储，并向用户提供高度可靠的信息检索和可用性。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，单个节点可能不可靠；例如，由于例行维护，它们可能变得不可用。因此，信息在网络中的存储方式非常重要。我们的目标是创建一种存储系统，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。这个项目通过求助于使用代数几何定义的纠错码族的底层结构来解决这个问题。这些代码具有健壮的结构，可产生强大的本地属性，并具有平衡存储开销、可靠性、网络流量和修复带宽的潜力。这个项目考虑通过代数几何构造获得的代码在高度相关的应用中的使用。将研究以下问题：以最小化节点访问和修复带宽为目标的基于代数几何和组合观点的分布式存储编码；以及使用代数几何工具显式构造局部可校正码、量子码和秘密共享方案。探讨了码的结构，对于直接和的译码码，可以通过译码或在其因子范围内进行修正来处理。将充分利用代码的自同构组以及从基础代数几何继承的自然嵌套结构。该方案中的大部分工作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结构；其目的是通过利用代数几何码的自然底层结构来实现这一点，而不是仅限于局部性。在这样做的同时，代码构建保证的性能得到了保持(意味着代码参数不受当地要求的影响)，同时使代码更符合当前的应用程序。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。

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.