  This paper shows that, with high probability, randomly punctured Reed-Solomon
codes over fields of polynomial size achieve the list decoding capacity. More
specifically, we prove that for any $\epsilon>0$ and $R\in (0,1)$, with high
probability, randomly punctured Reed-Solomon codes of block length $n$ and rate
$R$ are $\left(1-R-\epsilon, O({1}/{\epsilon})\right)$ list decodable over
alphabets of size at least $2^{\mathrm{poly}(1/\epsilon)}n^2$. This extends the
recent breakthrough of Brakensiek, Gopi, and Makam (STOC 2023) that randomly
punctured Reed-Solomon codes over fields of exponential size attain the
generalized Singleton bound of Shangguan and Tamo (STOC 2020).
