The binary rank of the $0,1$ matrix is the minimum size of the division into squares of monochromatic combinations. The matrix $M$ is for each $i \in [m]The $i$th diagonal block of $,$M$ is a cyclic block with $k_i$ 1s followed by $n_i-k_i$ zeros in the first row and all other entries are zeros. matrix. In this work, we examine the binary rank of these matrices and their complements. In particular, we compare the binary rank of these matrices with the real rank to form a lower bound for the former.

We present a general method to prove an upper bound on the binary rank of block matrices with diagonal blocks of certain structures and diagonal blocks of other structures. Using this method, the binary rank of the complement of the $(k_1, \ldots, k_m ; n_1, \ldots, n_m)$ cyclic block diagonal matrix yields $n_i>k_i>0$ for each $i. Prove that it is an integer that satisfies \of [m]$ never exceeds the maximum of $\gcd(n_i,k_i)-1$ over all $i \in . [m]$. In addition, we present some sufficient conditions for the binary rank of these matrices to strictly exceed their actual rank. Combining upper and lower bounds, we determine the exact binary rank of different families of matrices, further generalizing Gregory’s result significantly.

Motivated by Pullman’s question, we examine the binary rank of the $k$-regular $0,1$ matrix and its complement. As an application of the result to circulant block-diagonal matrices, we show that for every $k \geq 2$ there exists a $k$-regular $0,1$ matrix whose binary rank is strictly greater than the rank of its complement. indicate. Moreover, for all integer $r$, we determine exactly the smallest possible binary rank of the complement of the $2$-regular $0,1$ matrix with binary rank $r$.