We analyze rank decompositions of the $3\times 3$ matrix multiplication tensor over $\mathbb{Z}/2\mathbb{Z}$. We restrict our attention to decompositions of rank $\le 21$, as only those decompositions will yield an asymptotically faster algorithm for matrix multiplication than Strassen's algorithm. To reduce search space, we also require decompositions to have certain symmetries. Using Boolean SAT solvers, we show that under certain symmetries, such decompositions do not exist.

翻译：我们分析了在$\mathbb{Z}/2\mathbb{Z}$上的$3\times 3$矩阵乘法张量的秩分解。我们将注意力限制在秩不超过21的分解上，因为只有这些分解才能产生比Strassen算法渐近更快的矩阵乘法算法。为缩小搜索空间，我们还要求分解具有特定的对称性。通过使用布尔SAT求解器，我们证明在某些对称性条件下，这样的分解不存在。