Topics in algebraic geometry codes

错误控制在所有数字通信或存储中都是必不可少的，从银行交易到互联网商务，从CD或DVD到云计算，从手机通话到外太空探索，以及许多其他应用。许多实用的纠错码都是利用有限域上的代数曲线构造的。人们从理论和实践的角度对这些规范进行了广泛的研究。然而，许多问题仍然是开放的，特别是在代码结构，代码构造和解码的复杂性。本计画将致力于从代数几何学研究编码的结构特性与解码演算法，以使这些编码更易于应用。该项目是多学科躺在数学，计算机科学和电子工程的十字路口。它将纯数学，特别是离散数学和代数几何与数字通信中的实际应用联系起来。任何新的结果或任何好的编码算法都可以用于实际通信中提高通信性能。代数几何（AG）码具有大量的代数结构。利用这种代数结构能够构造代码、高效编码和高效解码。虽然在过去的十年中，在AG码的解码算法方面已经取得了许多进展，但这些进展中的大多数仅适用于单点AG码。多点编码的参数要比单点编码好得多。本计画将研究如何在编码与解码演算法上实现此优点。另一个重要的问题涉及到明确的建设基地的AG码，特别是那些从曲线在高维空间（功能领域塔）。本项目研究了半距离以上的列表译码和半距离以下的快速唯一译码。技巧包括幂级数和Grobner基。对于许多AG码，即使译码距离只有一半，它们的差错控制能力也比实际中广泛使用的Reed-Solomon码高得多。为了使AG码更适合于实际实现，该项目旨在通过幂级数表示来降低解码复杂度和内存需求。

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.