In this paper, we prove that explicit FRS codes and multiplicity codes achieve relaxed generalized Singleton bounds for list size $L\ge1.$ Specifically, we show the following: (1) FRS code of length $n$ and rate $R$ over the alphabet $\mathbb{F}_q^s$ with distinct evaluation points is $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ list-decodable (LD) for list size $L\in[s]$. (2) Multiplicity code of length $n$ and rate $R$ over the alphabet $\mathbb{F}_p^s$ with distinct evaluation points is $\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ LD for list size $L\in[s]$. Choosing $s=\Theta(1/\epsilon^2)$ and $L=O(1/\epsilon)$, our results imply that both FRS codes and multiplicity codes achieve LD capacity $1-R-\epsilon$ with optimal list size $O(1/\epsilon)$. This exponentially improves the previous state of the art $(1/\epsilon)^{O(1/\epsilon)}$ established by Kopparty et. al. (FOCS 2018) and Tamo (IEEE TIT, 2024). In particular, our results on FRS codes fully resolve a open problem proposed by Guruswami and Rudra (STOC 2006). Furthermore, our results imply the first explicit constructions of $(1-R-\epsilon,O(1/\epsilon))$ LD codes of rate $R$ with poly-sized alphabets. Our method can also be extended to analyze the list-recoverability (LR) of FRS codes. We provide a tighter radius upper bound that FRS codes cannot be $(\frac{L+1-\ell}{L+1}(1-\frac{mR}{m-1})+o(1),\ell, L)$ LR where $m=\lceil\log_{\ell}{(L+1)}\rceil$. We conjecture this bound is almost tight when $L+1=\ell^a$ for any $a\in\mathbb{N}^{\ge 2}$. To give some evidences, we show FRS codes are $\left(\frac{1}{2}-\frac{sR}{s-2},2,3\right)$ LR, which proves the tightness in the smallest non-trivial case. Our bound refutes the possibility that FRS codes could achieve LR capacity $(1-R-\epsilon, \ell, O(\frac{\ell}{\epsilon}))$. This implies an intrinsic separation between LD and LR of FRS codes.

翻译：本文证明了显式FRS码与重数码在列表大小$L\ge1$时达到松弛广义Singleton界。具体而言，我们证明：(1) 定义在字母表$\mathbb{F}_q^s$上、具有相异求值点、长度为$n$、码率为$R$的FRS码对于列表大小$L\in[s]$是$\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$列表可译(LD)的。(2) 定义在字母表$\mathbb{F}_p^s$上、具有相异求值点、长度为$n$、码率为$R$的重数码对于列表大小$L\in[s]$是$\left(\frac{L}{L+1}\left(1-\frac{sR}{s-L+1}\right),L\right)$ LD的。选取$s=\Theta(1/\epsilon^2)$和$L=O(1/\epsilon)$，我们的结果表明FRS码与重数码均能以最优列表大小$O(1/\epsilon)$达到LD容量$1-R-\epsilon$。这相对于Kopparty等人(FOCS 2018)和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FRS码的结果完全解决了Guruswami与Rudra(STOC 2006)提出的一个开放问题。此外，我们的结果意味着首次显式构造出具有多项式规模字母表、码率为$R$的$(1-R-\epsilon,O(1/\epsilon))$ LD码。我们的方法也可扩展用于分析FRS码的列表可恢复性(LR)。我们给出了更紧的半径上界：FRS码不可能是$(\frac{L+1-\ell}{L+1}(1-\frac{mR}{m-1})+o(1),\ell, L)$ LR的，其中$m=\lceil\log_{\ell}{(L+1)}\rceil$。我们推测当$L+1=\ell^a$（$a\in\mathbb{N}^{\ge 2}$）时该界几乎是紧的。作为证据，我们证明FRS码是$\left(\frac{1}{2}-\frac{sR}{s-2},2,3\right)$ LR的，这证明了最小非平凡情形下的紧性。我们的界否定了FRS码可能达到LR容量$(1-R-\epsilon, \ell, O(\frac{\ell}{\epsilon}))$的可能性。这揭示了FRS码的LD与LR之间存在本质分离。