CIF: Small: List Decoding for Algebraic Geometry Codes: Theoretical Analysis, Efficient Algorithms, Practical Implementation

CIF：小：代数几何编码的解码：理论分析，高效算法，实际实现错误控制编码确保了在嘈杂环境中传输数据的可靠性，因此是通信系统的关键组成部分。通过给数据增加一点冗余，并使用复杂的数学算法进行编码和解码，系统中的错误可以减少到任意低的阈值。该项目涉及代数几何（AG）代码，这是一个庞大而强大的代码族，其中包括Reed-Solomon （RS）代码，这是当今商业产品中使用的标准代码。Reed-Solomon码的标准解码算法是Berlekamp-Massey算法，它解码到球体填充界。在20世纪80年代和90年代，AG码被发现比RS码产生更好的纠错性能，并开发了有效的一般化Berlekamp-Massey算法。在20世纪90年代，苏丹发现了一种有效解码超出球体填充界的方法，称为列表解码。本课题从理论、算法和应用三个方面对AG码的列表解码进行了深入的研究。有强有力的证据表明，苏丹的方法，当用于高速率AG码时，可以比当前的分析预测更好地执行，因此主要的重点是改进列表解码的理论基础，以发现其对AG码的最大能力。研究人员还通过结合经典的Berlekamp-Massey型方法和几位研究人员的最新创新，提高了当前算法的效率，并使其适合硬件实现。研究人员与工业界和学术界的硬件和通信工程师合作，确定AG代码的特殊属性将特别有利的应用。

CIF:Small:Decoding of Algebraic Geometry Codes:Theoretical Analysis, Efficient Algorithms, Practical ImplementationError control coding ensures the reliability of data transmitted in a noisy environment and is therefore a critical component of communications systems. By adding a bit of redundancy to data, and using sophisticated mathematical algorithms to encode and decode, errors in the system can be reduced to an arbitrarily low threshold. This project concerns algebraic geometry (AG) codes, a large and powerful family of codes that includes Reed-Solomon (RS) codes, which are the standard code used in commercial products today. The standard decoding algorithm for Reed-Solomon codes is the Berlekamp-Massey algorithm, which decodes up to the sphere-packing bound. In the 1980's and 1990's, AG codes were discovered that yielded better error correction performance than RS codes, and efficient algorithms generalizing Berlekamp-Massey were developed. A method for efficiently decoding beyond the sphere-packing bound, called list decoding, was discovered in the 1990's by Sudan.This project advances the theoretical, algorithmic and applied understanding of list decoding for AG codes. There is strong evidence that Sudan's method, when used on high-rate AG codes, can perform much better than current analysis predicts, so a primary focus is improving the theoretical underpinnings of list decoding to discover its maxim capabilities for AG codes. The investigators also improve the efficiency of current algorithms and tailor them to hardware implementation by combining classical Berlekamp-Massey type methods and recent innovations due to several researchers. The investigators work with hardware and communication engineers in industry and in academia to identify applications where the special properties of AG codes will be particularly advantageous.