The classical family of Reed-Solomon codes consist of evaluations of polynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at $n$ distinct field elements. These are arguably the most widely used and studied codes, as they have both erasure and error-correction capabilities, among many others nice properties. In this survey we study closely related codes, folded Reed-Solomon codes, which are the first constructive codes to achieve the list decoding capacity. We then study two more codes which also have this feature, \textit{multiplicity codes} and \textit{derivative codes}. Our focus for the most part are the list decoding algorithms of these codes, though we also look into the local decodability of multiplicity codes.

中文翻译：

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超越 Guruswami-Sudan（和 Parvaresh-Vardy）半径：折叠 Reed-Solomon、多重性和衍生代码

Reed-Solomon 码的经典系列包括在有限域 $\mathbb{F}_q$ 上的多项式的评估，其次数小于 $k$，在 $n$ 不同的域元素处。这些可以说是最广泛使用和研究的代码，因为它们具有擦除和纠错能力，以及许多其他不错的特性。在本次调查中，我们研究了密切相关的代码，折叠 Reed-Solomon 代码，这是第一个实现列表解码能力的建设性代码。然后我们研究另外两个也有这个特征的代码，\textit{multiplicity 代码}和\textit{衍生代码}。我们主要关注这些代码的列表解码算法，尽管我们也研究了多重代码的局部可解码性。