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The classical family of Reed-Solomon codes consist of evaluations ofpolynomials over the finite field $\mathbb{F}_q$ of degree less than $k$, at$n$ distinct field elements. These are arguably the most widely used andstudied codes, as they have both erasure and error-correction capabilities,among many others nice properties. In this survey we study closely relatedcodes, folded Reed-Solomon codes, which are the first constructive codes toachieve the list decoding capacity. We then study two more codes which alsohave 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.

Beyond the Guruswami-Sudan (and Parvaresh-Vardy) Radii: Folded Reed-Solomon, Multiplicity and Derivative Codes