  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.
