  List-decodability of Reed-Solomon codes has received a lot of attention, but
the best-possible dependence between the parameters is still not
well-understood. In this work, we focus on the case where the list-decoding
radius is of the form $r=1-\varepsilon$ for $\varepsilon$ tending to zero. Our
main result states that there exist Reed-Solomon codes with rate
$\Omega(\varepsilon)$ which are $(1-\varepsilon,
O(1/\varepsilon))$-list-decodable, meaning that any Hamming ball of radius
$1-\varepsilon$ contains at most $O(1/\varepsilon)$ codewords. This trade-off
between rate and list-decoding radius is best-possible for any code with list
size less than exponential in the block length.
  By achieving this trade-off between rate and list-decoding radius we improve
a recent result of Guo, Li, Shangguan, Tamo, and Wootters, and resolve the main
motivating question of their work. Moreover, while their result requires the
field to be exponentially large in the block length, we only need the field
size to be polynomially large (and in fact, almost-linear suffices). We deduce
our main result from a more general theorem, in which we prove good
list-decodability properties of random puncturings of any given code with very
large distance.
