In this paper, we prove that with high probability, random Reed-Solomon codes approach the half-Singleton bound - the optimal rate versus error tradeoff for linear insdel codes - with linear-sized alphabets. More precisely, we prove that, for any $\epsilon>0$ and positive integers $n$ and $k$, with high probability, random Reed--Solomon codes of length $n$ and dimension $k$ can correct $(1-\varepsilon)n-2k+1$ adversarial insdel errors over alphabets of size $n+2^{\mathsf{poly}(1/\varepsilon)}k$. This significantly improves upon the alphabet size demonstrated in the work of Con, Shpilka, and Tamo (IEEE TIT, 2023), who showed the existence of Reed--Solomon codes with exponential alphabet size $\widetilde O\left(\binom{n}{2k-1}^2\right)$ precisely achieving the half-Singleton bound. Our methods are inspired by recent works on list-decoding Reed-Solomon codes. Brakensiek-Gopi-Makam (STOC 2023) showed that random Reed-Solomon codes are list-decodable up to capacity with exponential-sized alphabets, and Guo-Zhang (FOCS 2023) and Alrabiah-Guruswami-Li (STOC 2024) improved the alphabet-size to linear. We achieve a similar alphabet-size reduction by similarly establishing strong bounds on the probability that certain random rectangular matrices are full rank. To accomplish this in our insdel context, our proof combines the random matrix techniques from list-decoding with structural properties of Longest Common Subsequences.

翻译：本文证明，在高概率下，随机Reed-Solomon码能够逼近半Singleton界——这是线性插入删除码在码率与纠错能力之间的最优折衷——且仅需线性规模的字母表。具体而言，我们证明对于任意 $\epsilon>0$ 及正整数 $n$ 和 $k$，在高概率下，长度为 $n$、维度为 $k$ 的随机Reed-Solomon码能够在规模为 $n+2^{\mathsf{poly}(1/\varepsilon)}k$ 的字母表上纠正 $(1-\varepsilon)n-2k+1$ 个对抗性插入删除错误。这一结果显著改进了Con、Shpilka和Tamo（IEEE TIT, 2023）工作中所展示的字母表规模，他们证明了存在字母表规模为指数级 $\widetilde O\left(\binom{n}{2k-1}^2\right)$ 的Reed-Solomon码能够精确达到半Singleton界。我们的方法受到近期关于Reed-Solomon码列表解码研究的启发。Brakensiek-Gopi-Makam（STOC 2023）证明了随机Reed-Solomon码在指数级字母表规模下可实现达到容量的列表解码，而Guo-Zhang（FOCS 2023）和Alrabiah-Guruswami-Li（STOC 2024）将字母表规模改进至线性。我们通过类似地建立某些随机矩形矩阵满秩概率的强界，实现了类似的字母表规模缩减。为了在我们的插入删除场景中达成这一目标，我们的证明将列表解码中的随机矩阵技术与最长公共子序列的结构性质相结合。