Randomly punctured Reed--Solomon codes achieve list-decoding capacity over linear-sized fields

Abstract: Reed--Solomon codes are a classic family of error-correcting codes consisting of evaluations of low-degree polynomials over a finite field on some sequence of distinct field elements. They are widely known for their optimal unique-decoding capabilities, but their list-decoding capabilities are not fully understood. Given the prevalence of Reed-Solomon codes, a fundamental question in coding theory is determining if Reed--Solomon codes can optimally achieve list-decoding capacity. A recent breakthrough by Brakensiek, Gopi, and Makam, established that Reed--Solomon codes are combinatorially list-decodable all the way to capacity. However, their results hold for randomly-punctured Reed--Solomon codes over an exponentially large field size $2^{O(n)}$, where $n$ is the block length of the code. A natural question is whether Reed--Solomon codes can still achieve capacity over smaller fields. Recently, Guo and Zhang showed that Reed--Solomon codes are list-decodable to capacity with field size $O(n^2)$. We show that Reed--Solomon codes are list-decodable to capacity with linear field size $O(n)$, which is optimal up to the constant factor. We also give evidence that the ratio between the alphabet size $q$ and code length $n$ cannot be bounded by an absolute constant. Our techniques also show that random linear codes are list-decodable up to (the alphabet-independent) capacity with optimal list-size $O(1/\varepsilon)$ and near-optimal alphabet size $2^{O(1/\varepsilon^2)}$, where $\varepsilon$ is the gap to capacity. As far as we are aware, list-decoding up to capacity with optimal list-size $O(1/\varepsilon)$ was previously not known to be achievable with any linear code over a constant alphabet size (even non-constructively). Our proofs are based on the ideas of Guo and Zhang, and we additionally exploit symmetries of reduced intersection matrices.