Topics in algebraic geometry codes

代数几何代码主题

• 批准号: 1403062
• 负责人: Gretchen Matthews
• 金额: $ 21万
• 依托单位: Clemson University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1403062&HistoricalAwards=false
• 关键词: Topics; algebraic; geometry; codes

Error control is essential in all digital communications or storage, ranging from bank transactions to internet commerce, from CD or DVD to cloud computing, from cellphone conversations to outer space explorations, among many other applications. Many practical error correcting codes are constructed via algebraic curves over finite fields. These codes have been extensively studied from both theoretical and practical point of views. However, many questions still remain open, particularly on code structures, code constructions and decoding complexity. This project will strive to study structural properties and decoding algorithms for codes from algebraic geometry with the aim of making these codes more amenable to applications. The project is multi-disciplinary lying at the crossroads of mathematics, computer science,and electronic engineering. It bridges pure mathematics, particularly discrete mathematics and algebraic geometry, with practical applications in digital communications. Any new result or any good algorithm for codes could be used to improve communication capability in practice. Algebraic geometry (AG) codes have a tremendous amount of algebraic structure. Exploiting this algebraic structure enables construction of codes, efficient encoding, and efficient decoding. While many advances in decoding algorithms for AG codes has been made in the last decade, most of these apply only to one-point AG codes. Multipoint codes can have much better parameters than comparable one-point codes. This project will study how to realize this advantage in term of encoding and decoding algorithms. Another important issue relates to explicit constructions of bases for AG codes, especially those from curves in higher dimensional spaces (function field towers). This project studies both list decoding beyond half distance and fast unique decoding below half distance. Techniques include power series and Grobner bases. For many AG codes, even if decoding is only up to half distance, their error control capability is much higher than Reed-Solomon codes which are widely used in practice. To make AG codes more suitable for practical implementations, the project aims to reduce the decoding complexity and memory requirements via power series representations.

错误控制在所有数字通信或存储中至关重要，从银行交易到互联网商务，从CD或DVD到云计算，从手机对话到外层空间探索以及许多其他应用程序。许多实际错误纠正代码是通过有限场上的代数曲线构建的。从理论和实际观点的观点中，这些代码已经进行了广泛的研究。但是，许多问题仍然保持开放，尤其是在代码结构，代码构造和解码复杂性上。该项目将努力研究代数几何形状的代码的结构属性和解码算法，以使这些代码更适合应用程序。该项目是多学科的，位于数学，计算机科学和电子工程的十字路口。它桥接了纯数学，尤其是离散的数学和代数几何形状，并在数字通信中使用了实际应用。任何新的结果或代码的任何好算法都可以用来提高实践中的沟通能力。代数几何（AG）代码具有大量代数结构。利用这种代数结构可以构建代码，有效的编码和有效的解码。尽管在过去十年中，已经在解码AG代码的算法方面取得了许多进步，但其中大多数仅适用于单点AG代码。多点代码可以比可比的单点代码具有更好的参数。该项目将研究如何在编码和解码算法方面实现这一优势。另一个重要的问题是与AG代码的碱基构造有关，尤其是在较高维空间（功能场塔）中的曲线的构造。该项目研究列出了一个超过半距离的解码，又列出了一半以下的快速独特解码。技术包括Power Series和Grobner基础。对于许多AG代码，即使解码仅高于一半距离，它们的错误控制能力也比在实践中广泛使用的Reed-Solomon代码高得多。为了使AG代码更适合实际实施，该项目旨在通过电源序列表示来减少解码的复杂性和内存要求。