Ruling Out Low-rank Matrix Multiplication Tensor Decompositions with Symmetries via SAT

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.