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.

翻译：里德-所罗门码是一类经典的纠错码，由有限域上低次多项式在若干不同域元素处的求值构成。这类码因其最优唯一译码能力而广为人知，但其列表译码能力尚未完全明确。鉴于里德-所罗门码的普遍应用，编码理论中的一个基本问题是：里德-所罗门码能否最优地达到列表译码容量？Brakensiek、Gopi和Makam近期的一项突破性工作表明，里德-所罗门码在组合意义上可全程达到容量的列表可译性。然而，他们的结果仅适用于在指数级大域大小$2^{O(n)}$上的随机打点里德-所罗门码，其中$n$为码的码长。一个自然的问题是：里德-所罗门码是否仍能在更小的域上达到容量？近期，Guo和Zhang证明了域大小为$O(n^2)$时里德-所罗门码可达到列表译码容量。我们证明当域大小为线性$O(n)$时（该常数因子已达最优），里德-所罗门码同样可达到列表译码容量。同时我们给出证据表明，字母表大小$q$与码长$n$的比值无法由绝对常数限定。我们的技术还表明，随机线性码可在与字母表无关的容量下达到列表可译性，其最优列表大小为$O(1/\varepsilon)$，字母表大小近乎最优为$2^{O(1/\varepsilon^2)}$，其中$\varepsilon$为与容量的差距。据我们所知，此前即便对恒定字母表大小（甚至非构造性情形）的任意线性码，也未知能否以最优列表大小$O(1/\varepsilon)$达到容量下的列表译码。我们的证明基于Guo和Zhang的思想，并进一步利用了约化交集矩阵的对称性。