AI 中文总结
研究实现列表恢复容量的码的界与局限性，建立广义Singleton界给出速率限制，通过对基于AEL框架的构造进行元分析，证明其无法突破加性和线性码的列表恢复障碍。
AI 中文摘要
在编码理论中，列表可恢复性是一个基本概念，它有力地刻画了码字在码中的‘分散程度’。更正式地，给定一个码\(C \subseteq \Sigma^n\)和大小至多为\(\ell\)的输入列表\(S_1, \dots, S_n \subseteq \Sigma\)，列表可恢复性要求至多有\(L\)个码字\(c \in C\)，使得对于至少\((1 - \rho)n\)个\(i \in [n]\)的选择，\(c_i \in S_i\)。列表恢复在许多领域有应用。首先建立了一个紧密的‘广义Singleton界’，表明对于常数\(\ell, L, \rho\)和足够大的字母表\(\Sigma\)，\((\rho, \ell, L)\)列表可恢复码的速率有界。其次，证明了现有构造显式、最优列表可恢复码的方法存在根本缺陷，通过对基于Alon - Edmonds - Luby（AEL）框架的构造进行元分析，表明没有基于AEL的码能突破最近确定的加性和线性码的列表恢复障碍。
英文摘要
In coding theory, list recoverability is a fundamental concept which robustly captures how ``spread-out'' codewords are in a code. More formally, given a code $C \subseteq Σ^n$ and input lists $S_1, \dots, S_n \subseteq Σ$ of size at most $\ell$, list recoverability requires that there are at most $L$ codewords $c \in C$ such that $c_i \in S_i$ for at least $(1-ρ)n$ choices of $i \in [n]$. List recovery is an important question which has found applications in many areas, including complexity theory, property testing, compressed sensing, streaming algorithms, and cryptography. As our first main result, we establish a tight ``generalized singleton bound''. Formally, we show that for constant $\ell, L,ρ$ and sufficiently large alphabets $Σ$, if we define $R^*=\frac{L+1-\ell}{L}-\frac{L+1}{L}ρ$, it is possible for a $(ρ,\ell,L)$ list-recoverable code to have rate $R^*-ε$ but impossible to have rate $R^*+ε$. One direction of our result already directly generalizes and improves a weaker impossibility result due to Goldberg, Shangguan, and Tamo. For our second main result, we prove that there is a fundamental shortcoming in existing methods that aim to construct explicit, optimal list-recoverable codes. Indeed, recent work has constructed explicit codes achieving list-decoding capacity (along with other related properties) using a framework introduced in the work of Alon--Edmonds--Luby (AEL). We give a meta-analysis of such constructions by presenting an ``AEL framework'' which captures all such recent constructions in the literature. Within this framework, we show that no AEL-based code can break a recently-identified list-recovery barrier for additive and linear codes.
Comments1 figure, 29 pages, abstract abridged
