Error-correcting codes are a method for representing data, so that one can recover the original information even if some parts of it were corrupted. The basic idea, which dates back to the revolutionary work of Shannon and Hamming about a century ago, is to encode the data into a redundant form, so that the original information can be decoded from the redundant encoding even in the presence of some noise or corruption. One prominent family of error-correcting codes are Reed-Solomon Codes which encode the data using evaluations of low-degree polynomials. Nearly six decades after they were introduced, Reed-Solomon Codes, as well as some related families of polynomial-based codes, continue to be widely studied, both from a theoretical perspective and from the point of view of applications. Besides their obvious use in communication, error-correcting codes such as Reed-Solomon Codes are also useful for various applications in theoretical computer science. These applications often require the ability to cope with many errors, much more than what is possible information-theoretically. List-decodable codes are a special class of error-correcting codes that enable correction from more errors than is traditionally possible by allowing a small list of candidate decodings. These codes have turned out to be extremely useful in various applications across theoretical computer science and coding theory. In recent years, there have been significant advances in list decoding of Reed-Solomon Codes and related families of polynomial-based codes. This includes efficient list decoding of such codes up to the information-theoretic capacity, with optimal list-size, and using fast nearly-linear time, and even sublinear-time, algorithms. In this book, we survey these developments.

翻译：纠错码是一种数据表示方法，使得即使在部分数据损坏的情况下仍能恢复原始信息。其基本思想可追溯至约一个世纪前香农和汉明的开创性工作，即将数据编码为冗余形式，使得即使在存在噪声或损坏的情况下，仍能从冗余编码中解码出原始信息。里德-所罗门码是纠错码中的重要家族，它通过低次多项式求值对数据进行编码。在问世近六十年后，里德-所罗门码以及一些相关的基于多项式的码族，无论是从理论角度还是应用视角，仍然被广泛研究。除了在通信领域的直接应用外，诸如里德-所罗门码等纠错码在理论计算机科学的多个应用中也具有重要价值。这些应用通常需要处理远超信息论理论上限的巨量错误。列表可解码码是一类特殊的纠错码，它通过允许输出少量候选解码结果，能够纠正比传统方法更多的错误。这类码在理论计算机科学和编码理论的诸多应用中展现出极大效用。近年来，里德-所罗门码及相关多项式码族的列表解码研究取得了显著进展。这包括：实现达到信息论容量的高效列表解码，具有最优列表大小，采用近乎线性的快速算法，甚至亚线性时间算法。本书将系统梳理这些研究进展。