Testing the Binary Rank with Polynomial Query Complexity

We provide an adaptive two-sided error testing algorithm for the binary rank of a $0,1$ matrix $M$ with query complexity $O(d^3\log (d+1)\mathrm{/}{\epsilon }^2)$, where $d$ is the tested binary rank bound and $\epsilon$ is the distance parameter. This answers an open question posed by Parnas, Ron and Shraibman~\cite{parnas2021property}, who asked whether the binary rank can be tested with query complexity polynomial in $d$ and $1\mathrm{/}\epsilon$. Furthermore, our testing algorithm can be used to find an approximate binary decomposition of $M$ with an additional $d(n+m)$ queries. That is, under the promise that the binary rank of $M$ is at most $d$, we show how to find, with probability at least $5\mathrm{/}6$, two $0,1$ matrices $A^{\mathrm{\prime }},B^{\mathrm{\prime }}$ such that $M^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cdot B^{\mathrm{\prime }}$ is a $0,1$ matrix which differs from $M$ on at most an $O(\epsilon )$ fraction of its entries.