In this paper, we prove that any `appropriate' folded Reed-Solomon and univariate multiplicity codes achieve relaxed generalized Singleton bound for list size $L\ge1.$ More concretely, we show the following: (1) Any $(s,\gamma)$-folded RS code over the alphabet $\mathbb{F}_q^s$ of block length $n$ and rate $R$ with pair-wise distinct evaluation points $\{\gamma^i\alpha_j\}_{(i,j)\in\left(\{0\}\sqcup[s-1],[n]\right)}\subset\mathbb{F}_q$ are $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ (average-radius) list-decodable for list size $L\in[s]$. (2) Any $s$-order univariate multiplicity code over the alphabet $\mathbb{F}_p^s$ ($p$ is a prime) of block length $n$ and rate $R$ with pair-wise distinct evaluation points $\{\alpha_i\}_{i\in[n]}\subset\mathbb{F}_p$ are $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ (average-radius) list-decodable for list size $L\in[s]$. Choose $s=\Theta(1/\epsilon^2)$ and $L=O(1/\epsilon)$, our results imply that both explicit folded RS codes and explicit univariate multiplicity codes achieve list decoding capacity $1-R-\epsilon$ with evidently optimal list size $O(1/\epsilon)$, which exponentially improves the previous state-of-the-art $(1/\epsilon)^{O(1/\epsilon)}$ established by Kopparty, Ron-Zewi, Saraf, and Wootters (FOCS 2018 or SICOMP, 2023) and Tamo (IEEE TIT, 2024). In particular, our results on folded Reed--Solomon codes fully resolve a long-standing open problem originally proposed by Guruswami and Rudra (STOC 2006 or IEEE TIT, 2008). Furthermore, our results imply the first explicit constructions of $(1-R-\epsilon,O(1/\epsilon))$ (average-radius) list-decodable codes of rate $R$ with polynomial-sized alphabets in the literature.

翻译：本文证明，任何“适当”的折叠里德-所罗门码与一元重数码在列表大小$L\ge1$时均能达到松弛广义Singleton界。具体而言，我们证明以下结论：(1) 对于字母表$\mathbb{F}_q^s$上分组长度为$n$、码率为$R$的$(s,\gamma)$折叠RS码，若其互异估值点集$\{\gamma^i\alpha_j\}_{(i,j)\in\left(\{0\}\sqcup[s-1],[n]\right)}\subset\mathbb{F}_q$满足适当条件，则该码对任意列表大小$L\in[s]$是$\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$（平均半径）列表可译的。(2) 对于字母表$\mathbb{F}_p^s$（$p$为素数）上分组长度为$n$、码率为$R$的$s$阶一元重数码，若其互异估值点集$\{\alpha_i\}_{i\in[n]}\subset\mathbb{F}_p$满足适当条件，则该码对任意列表大小$L\in[s]$是$\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$（平均半径）列表可译的。选取$s=\Theta(1/\epsilon^2)$与$L=O(1/\epsilon)$，我们的结果表明显式折叠RS码与显式一元重数码均能以显著最优的列表大小$O(1/\epsilon)$达到列表解码容量$1-R-\epsilon$，这相对于Kopparty、Ron-Zewi、Saraf与Wootters（FOCS 2018或SICOMP 2023）以及Tamo（IEEE TIT 2024）建立的先前最佳结果$(1/\epsilon)^{O(1/\epsilon)}$实现了指数级改进。特别地，我们关于折叠里德-所罗门码的结果完整解决了由Guruswami与Rudra（STOC 2006或IEEE TIT 2008）提出的长期开放问题。此外，我们的结果意味着在文献中首次显式构造出具有多项式规模字母表、码率为$R$的$(1-R-\epsilon,O(1/\epsilon))$（平均半径）列表可译码。