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纠错码，由有限域上低次多项式在若干互异域元素序列上的求值构成。其最优唯一解码能力广为人知，但列表解码性能尚未完全阐明。鉴于里德-所罗门码的广泛应用，编码理论中的一个基本问题是：里德-所罗门码能否最优地实现列表解码容量？Brakensiek、Gopi与Makam的最新突破表明，里德-所罗门码在组合意义下可完全达到列表解码容量。然而，该结论仅适用于指数级大域大小$2^{O(n)}$上的随机打孔里德-所罗门码（其中$n$为码块长度）。一个自然的问题是：里德-所罗门码在更小域上是否仍能达到容量？近期，Guo与Zhang证明了域大小为$O(n^2)$时里德-所罗门码可列表解码至容量。我们进一步证明：当域大小与码长呈线性关系$O(n)$时，里德-所罗门码仍可列表解码至容量，该结果在常数因子意义上最优。同时，我们给出证据表明字母表大小$q$与码长$n$的比值无法被绝对常数界定。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实现列表解码，且最优列表大小$O(1/\varepsilon)$与近最优字母表大小$2^{O(1/\varepsilon^2)}$（其中$\varepsilon$为容量间隙）。据我们所知，此前即便仅考虑存在性（非构造性），亦未有任何常数字母表大小的线性码能被证明可实现最优列表大小$O(1/\varepsilon)$的容量级列表解码。本文证明基于Guo与Zhang的思想，并额外利用了简化交集矩阵的对称性。