  List-decodability of Reed-Solomon codes has received a lot of attention by
different researchers, 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 $(1-\varepsilon,
O(1/\varepsilon))$-list-decodable Reed-Solomon codes with rate
$\Omega(\varepsilon)$ (which is best-possible for any code which is
list-decodable with radius $1-\varepsilon$ and list size less than exponential
in the block length). This improves a recent result of Guo, Li, Shangguan,
Tamo, and Wootters, and resolves the main motivating question of their work. We
deduce our main result from a more general theorem, in which we prove good
list-decodability properties of random puncturings of a given code with very
large distance.
