List-Decoding and List-Recovery of Reed–Solomon Codes Beyond the Johnson Radius for Every Rate

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
every
rate, with a polynomial field size in the block length. In particular, we show that for every
$\epsilon \in (0,1)$
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.