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

Randomly Punctured Reed-Solomon Codes Achieve the List Decoding Capacity over Polynomial-Size Alphabets