Superlogarithmic-Rank Matrix Rigidity for the Walsh-Hadamard Transform

Josh Alman

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arXiv · 2608.06592

Superlogarithmic-Rank Matrix Rigidity for the Walsh-Hadamard Transform

Josh Alman

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Abstract

For sufficiently large $N$ which is a power of 2, we prove that changing at most one percent of the entries of the $N\times N$ Walsh-Hadamard Transform cannot reduce its rank over $\mathbb{F}_3$ to $\lfloor \log^2 N/80\rfloor$ or below. To the best of our knowledge, this is the first constant-fraction rigidity lower bound for an explicit matrix family at a superlogarithmic target rank over any choice of field, inching toward the parameters in Razborov's program for communication complexity lower bounds.

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Josh Alman. 2026-08-06. Superlogarithmic-Rank Matrix Rigidity for the Walsh-Hadamard Transform. https://arxiv.org/abs/2608.06592

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