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

CIF：Small：代数几何编码的译码：理论分析、高效算法、实际实现差错控制编码确保了在噪声环境中传输的数据的可靠性，因此是通信系统的重要组成部分。通过向数据添加一点冗余，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)码，这是一个庞大而强大的码族，其中包括里德-所罗门(RS)码，这是当今商业产品中使用的标准码。里德-所罗门码的标准译码算法是Berlekamp-Massey算法，其译码精度可达球填充极限。在20世纪80年代的S和90年代的S中，AG码被发现比RS码具有更好的纠错性能，并发展了推广Berlekamp-Massey的有效算法。列表译码是Sandan在1990年代的S中发现的一种超越球面包装界限的高效译码方法，对AG码的列表译码在理论上、算法上和应用上都有了新的认识。有强有力的证据表明，苏丹的方法在高码率AG码上的性能比当前分析预测的要好得多，因此一个主要的重点是改进列表译码的理论基础，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类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，并将其定制为硬件实现。研究人员与工业和学术界的硬件和通信工程师合作，以确定AG代码的特殊特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.