A Lower Bound of 21 for $3\times3$ Matrix Multiplication over $\mathbb{F}_2$

2026-09-06

AI Summary

本文通过结合Wang的自动框架与D'Ambrosio的策略，证明了在$\mathbb{F}_2$上$3\times3$矩阵乘法的双线性复杂度至少为21，改进了原有下界。

Abstract

We prove that $3\times3$ matrix multiplication over $\mathbb{F}_2$ has bilinear complexity at least $21$, improving the lower bound $20$. Lower bounds for restrictions of the first input constrain how many first factors of a decomposition can lie in each subspace. The strengthened restriction bounds force all first factors of matrix rank at least two into one coset of a three-dimensional rank-one subspace. An exhaustive computation finds no admissible first-factor profile with $20$ terms. We also prove that $2\times3$ by $3\times3$ matrix multiplication over $\mathbb{F}_3$ has rank exactly $15$: three profiles survive the corresponding computation, and short restriction arguments exclude them. We combine Wang's automated framework for tensor-rank lower bounds with D'Ambrosio's capacity-and-profile strategy. Our computational contributions are rank-one-span searches that strengthen the subspace lower-bound table and a direct, symmetry-reduced profile enumerator that enforces all subspace capacities simultaneously.

Problem

• bilinear complexity
• matrix multiplication
• lower bound
• subspace restrictions

Innovation

• bilinear complexity
• matrix multiplication
• tensor rank lower bounds
• subspace capacity