In the noisy $k$-XOR problem, one is given $y \in \mathbb{F}_2^M$ and must distinguish between $y$ uniform and $y = A x + e$, where $A$ is the adjacency matrix of a $k$-left-regular bipartite graph with $N$ variables and $M$ constraints, $x\in \mathbb{F}_2^N$ is random, and $e$ is noise with rate $η$. Lower bounds in restricted computational models such as Sum-of-Squares and low-degree polynomials are closely tied to the expansion of $A$, leading to conjectures that expansion implies hardness. We show that such conjectures are false by constructing an explicit family of graphs with near-optimal expansion for which noisy $k$-XOR is solvable in polynomial time. Our construction combines two powerful directions of work in pseudorandomness and coding theory that have not been previously put together. Specifically, our graphs are based on the lossless expanders of Guruswami, Umans and Vadhan (JACM 2009). Our key insight is that by an appropriate interpretation of the vertices of their graphs, the noisy XOR problem turns into the problem of decoding Reed-Muller codes from random errors. Then we build on a powerful body of work from the 2010s correcting from large amounts of random errors. Putting these together yields our construction. Concretely, we obtain explicit families for which noisy $k$-XOR is polynomial-time solvable at constant noise rate $η= 1/3$ for graphs with $M = 2^{O(\log^2 N)}$, $k = (\log N)^{O(1)}$, and $(N^{1-α}, 1-o(1))$-expansion. Under standard conjectures on Reed--Muller codes over the binary erasure channel, this extends to families with $M = N^{O(1)}$, $k=(\log N)^{O(1)}$, expansion $(N^{1-α}, 1-o(1))$ and polynomial-time algorithms at noise rate $η= N^{-c}$.

翻译：在含噪$k$-XOR问题中，给定$y \in \mathbb{F}_2^M$，需区分$y$为均匀分布与$y = A x + e$两种情况，其中$A$是$k$左正则二分图的邻接矩阵，包含$N$个变量节点和$M$个约束节点，$x\in \mathbb{F}_2^N$为随机向量，$e$是噪声率为$η$的噪声项。在平方和与低次多项式等受限计算模型中，下界与$A$的扩展性密切相关，由此引发了扩展性意味着难解性的猜想。我们通过构造显式图族证明此类猜想不成立——该图族具有近最优扩展性，且其上的含噪$k$-XOR问题可在多项式时间内求解。我们的构造融合了伪随机性与编码理论两个方向此前未曾结合的重要研究成果。具体而言，我们的图基于Guruswami、Umans与Vadhan（JACM 2009）提出的无损扩展图。关键洞见在于：通过适当诠释图中顶点的含义，含噪XOR问题可转化为从随机错误中解码里德-穆勒码的问题。进而我们借鉴2010年代关于纠正大量随机错误的系统性研究成果，将两者结合完成构造。具体实现中，我们获得显式图族：当$M = 2^{O(\log^2 N)}$、$k = (\log N)^{O(1)}$且扩展性为$(N^{1-α}, 1-o(1))$时，含噪$k$-XOR问题可在恒定噪声率$η= 1/3$下多项式求解。基于二进制擦除信道上里德-穆勒码的标准猜想，该结果可推广至$M = N^{O(1)}$、$k=(\log N)^{O(1)}$、扩展性为$(N^{1-α}, 1-o(1))$的图族，且噪声率$η= N^{-c}$时仍存在多项式时间算法。