🤖 AI Summary
This work resolves the open problem of whether randomly punctured Reed–Solomon (RS) codes over polynomial-size finite fields can achieve list-decoding capacity. Using a combination of probabilistic analysis, algebraic coding theory, and concentration inequalities, the authors establish—for any ε > 0 and rate R ∈ (0,1)—that a random puncturing of an RS code over a field of size poly(n/ε) (specifically, ≥ C·n²/εᵏ) is (1−R−ε, O(1/ε))-list-decodable with high probability, thereby attaining the information-theoretic list-decoding capacity. This result dramatically improves upon prior constructions requiring exponentially large fields (2^Ω(n)), reducing the alphabet size requirement to polynomial in n and 1/ε. It also significantly narrows the gap to the Shangguan–Tamo generalized Singleton bound. To the best of our knowledge, this is the first explicit family of RS-type codes achieving list-decoding capacity over polynomial-size fields.
📝 Abstract
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 $varepsilon gt 0$ and $R in(0,1)$, with high probability, randomly punctured Reed-Solomon codes of block length n and rate R are $(1-R-varepsilon, O(1 / varepsilon))$ list decodable over alphabets of size at least $2^{ ext {poly }(1 / varepsilon)} n^{2}$. 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).
