Decision version of matrix multiplication problem Is there any known decision version of matrix multiplication problem such that the time complexity of the best known algorithm for this decision problem is $O(n^k)$ for $n \times n$-dimensional matrices, where $k$ is bigger than 2 and preferable very close to $ \omega $ or the best known candidate of $\omega$ ($\omega$ is the optimal matrix-product constant)? 

P.S. I am aware of matrix product verification problem (e.g. this post).
Let $H$ be some complex hash function (almost any function will do), mapping long bit strings down to a single bit. Then to decide whether $H( A \times B ) = 0$, you will basically need to multiply $A \times B$ and compute $H$ on the resulting product. Unless $H$ has very special properties, there won't be any short cut to this.