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 ∈ 𝔽₂^M，需区分y均匀随机分布与y = A x + e两种情况，其中A是k-左正则二部图的邻接矩阵，包含N个变量和M个约束，x∈𝔽₂^N为随机向量，e为噪声率η的噪声项。在平方和与低阶多项式等受限计算模型下，下界与A的扩展性紧密相关，因此有猜想认为扩展性意味着计算困难性。我们通过构造一个具有近最优扩展性的显式图族，证明此类猜想不成立——该图族上的含噪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中两个先前未结合的重要研究方向。具体而言，我们的图基于Guruswami、Umans和Vadhan（JACM 2009）的无损扩展子图。关键洞察在于：通过适当解释图中顶点，含噪XOR问题转化为从随机错误中解码里德-穆勒码的问题。随后，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。将二者结合即得所需构造。具体而言，我们获得显式图族，使得在噪声率η=1/3的常数噪声率下，含噪k-XOR可在多项式时间内求解，其中M = 2^{O(log² N)}，k = (log N)^{O(1)}，且图具有(N^{1-α}, 1-o(1))-扩展性。基于二进制擦除信道上里德-穆勒码的标准猜想，该结论可推广至M = N^{O(1)}、k=(log N)^{O(1)}、扩展性为(N^{1-α}, 1-o(1))的图族，并在噪声率η = N^{-c}下存在多项式时间算法。