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Beyond the Guruswami-Sudan (and Parvaresh-Vardy) Radii: Folded Reed-Solomon, Multiplicity and Derivative Codes
arXiv - CS - Discrete Mathematics Pub Date : 2020-03-11 , DOI: arxiv-2003.05400
Neophytos Charalambides

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:

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。
更新日期:2020-03-12
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