On the Binary Rank of Matrices with Constant Real Rank

Abstract: We continue the study initiated by Parnas and Shraibman~\cite{PARNAS2026264} who gave upper bounds on the binary rank of $0,1$ matrices which have a small rank over the reals. We give alternative completely mathematical proofs of results proved in~\cite{PARNAS2026264} with the assistance of a computer program, and also solve one of the open problems presented there regarding the maximal binary rank of a matrix with real rank $5$. Moreover, our techniques provide a general method for giving non-trivial upper bounds on the maximal binary rank of a matrix with constant real rank. Our results also imply bounds on the equivalent problem of finding the minimum number of bicliques needed to partition the edges of a bipartite graph whose reduced adjacency matrix has real rank at most $d$.