  Understanding the limits of list-decoding and list-recovery of Reed-Solomon
(RS) codes is of prime interest in coding theory and has attracted a lot of
attention in recent decades. However, the best possible parameters for these
problems are still unknown, and in this paper, we take a step in this
direction. We show the existence of RS codes that are list-decodable or
list-recoverable beyond the Johnson radius for \emph{any} rate, with a
polynomial field size in the block length. In particular, we show that for any
$\epsilon\in (0,1)$ there exist RS codes that are list-decodable from radius
$1-\epsilon$ and rate less than $\frac{\epsilon}{2-\epsilon}$, with constant
list size. We deduce our results by extending and strengthening a recent result
of Ferber, Kwan, and Sauermann on puncturing codes with large minimum distance
and by utilizing the underlying code's linearity.
