On depth-3 circuits and covering number: an explicit counter-example

We give a simple construction of $n\times n$ Boolean matrices with $\mathrm{\Omega }(n^{4\mathrm{/}3})$ zero entries that are free of $2\times 2$ all-zero submatrices and have covering number $O({\log }^4(n))$. This construction provides an explicit counterexample to a conjecture of Pudlák, Rödl and Savický and Research Problems 1.33, 4.9, 11.17 of Jukna [Boolean function complexity]. These conjectures were previously refuted by Katz using a probabilistic construction.