Topics in algebraic geometry codes

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，错误控制在所有数字通信或存储中都是必不可少的。许多实际的纠错码是用有限域上的代数曲线构造的。从理论和实践的角度对这些规范进行了广泛的研究。然而，许多问题仍然没有解决，特别是在代码结构、代码结构和解码复杂性方面。本项目将努力研究代数几何代码的结构特性和解码算法，目的是使这些代码更适合应用。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。在实际应用中，任何新的结果或好的编码算法都可以用来提高通信能力。代数几何码具有大量的代数结构。利用这种代数结构可以构造代码、高效编码和高效解码。虽然在过去十年中，AG码的解码算法取得了许多进展，但其中大多数只适用于一点AG码。多点代码可以有比单点代码更好的参数。本课题将研究如何在编解码算法上实现这一优势。另一个重要的问题涉及到AG码的基的显式构造，特别是那些来自高维空间（函数场塔）的曲线。本课题研究了半距离以外的列表解码和半距离以下的快速唯一解码。技术包括幂级数和Grobner基。对于许多AG码来说，即使译码距离只有一半，其误差控制能力也远远高于实际应用中广泛使用的Reed-Solomon码。为了使AG码更适合实际实现，该项目旨在通过幂级数表示来降低解码复杂性和内存需求。

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.