A Note on Property Testing of the Binary Rank

Let $M$ be a $n\times m$ $(0,1)$-matrix. We define the $s$-binary rank, $br_s(M)$, of $M$ to be the minimal integer $d$ such that there are $d$ monochromatic rectangles that cover all the $1$-entries in the matrix, and each $1$-entry is covered by at most $s$ rectangles. When $s=1$, this is the binary rank,~$br(M)$, known from the literature. Let $R(M)$ and $C(M)$ be the set of rows and columns of~$M$, respectively. We use the result of Sgall (Comb. 1999) to prove that if $M$ has $s$-binary rank at most~$d$, then $|R(M)|\cdot |C(M)|\le {d\choose \le s}2^{d}$ where ${d\choose \le s}=\sum_{i=0}^s{d\choose i}$. This bound is tight; that is, there exists a matrix $M'$ of $s$-binary rank $d$ such that $|R(M')|\cdot |C(M')|= {d\choose \le s}2^{d}$. Using this result, we give a new one-sided adaptive and non-adaptive testers for $(0,1)$-matrices of $s$-binary rank at most $d$ (and exactly $d$) that makes $\tilde O\left({d\choose \le s}2^d/\epsilon\right)$ and $\tilde O\left({d\choose \le s}2^d/\epsilon^2\right)$ queries, respectively. For a fixed $s$, this improves the query complexity of the tester of Parnas et al. (Theory Comput. Syst. 2021) by a factor of $\tilde \Theta (2^d)$.