Given a family $X$ of subsets of $[n]$ and an ensemble of local functions $\{f_s:s\to\Sigma\; | \; s\in X\}$, an agreement test is a randomized property tester that is supposed to test whether there is some global function $G:[n]\to\Sigma$ such that $f_s=G|_s$ for many sets $s$. A "classical" small-soundness agreement theorem is a list-decoding $(LD)$ statement, saying that \[\tag{$LD$} Agree(\{f_s\}) > \varepsilon \quad \Longrightarrow \quad \exists G^1,\dots, G^\ell,\quad P_s[f_s\overset{0.99}{\approx}G^i|_s]\geq poly(\varepsilon),\;i=1,\dots,\ell. \] Such a statement is motivated by PCP questions and has been shown in the case where $X=\binom{[n]}k$, or where $X$ is a collection of low dimensional subspaces of a vector space. In this work we study small the case of on high dimensional expanders $X$. It has been an open challenge to analyze their small soundness behavior. Surprisingly, the small soundness behavior turns out to be governed by the topological covers of $X$.We show that: 1. If $X$ has no connected covers, then $(LD)$ holds, provided that $X$ satisfies an additional expansion property. 2. If $X$ has a connected cover, then $(LD)$ necessarily fails. 3. If $X$ has a connected cover (and assuming the additional expansion property), we replace the $(LD)$ by a weaker statement we call lift-decoding: \[ \tag{$LFD$} Agree(\{f_s\})> \varepsilon \Longrightarrow \quad \exists\text{ cover }\rho:Y\twoheadrightarrow X,\text{ and }G:Y(0)\to\Sigma,\text{ such that }\] \[P_{{\tilde s\twoheadrightarrow s}}[f_s \overset{0.99}{\approx} G|_{\tilde s}] \geq poly(\varepsilon),\] where ${\tilde s\twoheadrightarrow s}$ means that $\rho(\tilde s)=s$. The additional expansion property is cosystolic expansion of a complex derived from $X$ holds for the spherical building and for quotients of the Bruhat-Tits building.

翻译：给定集合$[n]$的子集族$X$及局部函数族$\{f_s:s\to\Sigma\; | \; s\in X\}$，一致性检验是一种随机性质检测器，旨在检验是否存在全局函数$G:[n]\to\Sigma$使得对于多个集合$s$有$f_s=G|_s$。“经典”的小可靠性一致性定理是列表解码（$LD$）表述，即：\[\tag{$LD$} Agree(\{f_s\}) > \varepsilon \quad \Longrightarrow \quad \exists G^1,\dots, G^\ell,\quad P_s[f_s\overset{0.99}{\approx}G^i|_s]\geq poly(\varepsilon),\;i=1,\dots,\ell.\]此类表述源于PCP问题，已在$X=\binom{[n]}k$或$X$为向量空间低维子空间集合的情形中得到证明。本文研究高维扩展图$X$的小可靠性情形。分析其小可靠性行为一直是一个开放挑战。令人惊讶的是，小可靠性行为由$X$的拓扑覆盖决定。我们证明：1. 若$X$无连通覆盖，则当$X$满足额外扩展性质时，$(LD)$成立；2. 若$X$有连通覆盖，则$(LD)$必然不成立；3. 若$X$有连通覆盖（假定额外扩展性质成立），我们将$(LD)$替换为较弱的称为提升译码的表述：\[ \tag{$LFD$} Agree(\{f_s\})> \varepsilon \Longrightarrow \quad \exists\text{ 覆盖 }\rho:Y\twoheadrightarrow X,\text{ 及 }G:Y(0)\to\Sigma,\text{ 使得 }\]\[P_{{\tilde s\twoheadrightarrow s}}[f_s \overset{0.99}{\approx} G|_{\tilde s}] \geq poly(\varepsilon),\]其中${\tilde s\twoheadrightarrow s}$表示$\rho(\tilde s)=s$。额外扩展性质指源于$X$的复形具有余循环扩展性，其对球面建筑及Bruhat-Tits建筑的商空间成立。