Topics in algebraic geometry codes

差错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，从手机通话到外层空间探索，以及许多其他应用。许多实用的纠错码都是通过有限域上的代数曲线来构造的。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，许多问题仍然悬而未决，特别是在代码结构、代码构造和译码复杂性方面。这个项目将致力于研究代数几何中码的结构性质和译码算法，目的是使这些码更适合于应用。该项目是一个多学科的项目，位于数学、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在实践中用于提高通信能力。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，AG码的译码算法取得了许多进展，但大多数都只适用于一点AG码。多点编码可以具有比可比的单点编码更好的参数。本项目将从编解码算法方面研究如何实现这一优势。另一个重要的问题是AG码的基的显式构造，特别是高维空间(函数域塔)中曲线的基的显式构造。本课题既研究了半距离以上的列表译码，又研究了半距离以下的快速唯一译码。技巧包括幂函数级数和Grobner基。对于许多AG码来说，即使译码距离只有一半，它们的差错控制能力也远远高于实际中广泛使用的里德-所罗门码。为了使AG码更适合于实际实现，该项目旨在通过幂函数级数表示来降低译码复杂度和存储需求。

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.