Expanders Meet Reed--Muller: Easy Instances of Noisy k-XOR

2026-04-05

AI Summary

This work investigates the efficient solvability of the noisy k-XOR problem on expander graph structures under high noise rates. By reframing the problem as a random error decoding task for Reed–Muller codes, the authors introduce the first approach that combines lossless expander graphs with low-degree polynomial decoding techniques, yielding an explicit family of graphs with nearly optimal expansion properties. Their method achieves polynomial-time solvability even at constant noise rates (e.g., η = 1/3), thereby refuting the prevailing conjecture that strong expansion necessarily entails computational intractability. Specifically, the algorithm is efficient for k = (log N)^{O(1)}, M = 2^{O(log² N)}, and (N^{1−α}, 1−o(1)) expansion, and extends under standard complexity assumptions to settings with M = N^{O(1)} and subconstant noise η = N^{−c}.

Abstract

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 $η= N^{-c}$.

Problem

• noisy k-XOR
• expander graphs
• computational hardness
• Reed-Muller codes
• polynomial-time solvability

Innovation

• lossless expanders
• Reed-Muller codes
• noisy k-XOR
• polynomial-time decoding
• pseudorandomness