  A $0,1$ matrix is said to be regular if all of its rows and columns have the
same number of ones. We prove that for infinitely many integers $k$, there
exists a square regular $0,1$ matrix with binary rank $k$, such that the
Boolean rank of its complement is $k^{\widetilde{\Omega}(\log k)}$.
Equivalently, the ones in the matrix can be partitioned into $k$ combinatorial
rectangles, whereas the number of rectangles needed for any cover of its zeros
is $k^{\widetilde{\Omega}(\log k)}$. This settles, in a strong form, a question
of Pullman (Linear Algebra Appl., 1988) and a conjecture of Hefner, Henson,
Lundgren, and Maybee (Congr. Numer., 1990). The result can be viewed as a
regular analogue of a recent result of Balodis, Ben-David, G\"{o}\"{o}s, Jain,
and Kothari (FOCS, 2021), motivated by the clique vs. independent set problem
in communication complexity and by the (disproved) Alon-Saks-Seymour conjecture
in graph theory. As an application of the produced regular matrices, we obtain
regular counterexamples to the Alon-Saks-Seymour conjecture and prove that for
infinitely many integers $k$, there exists a regular graph with biclique
partition number $k$ and chromatic number $k^{\widetilde{\Omega}(\log k)}$.
