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

TL;DR

This paper presents an explicit construction of Boolean matrices with many zeros, no 2x2 all-zero submatrices, and small covering number, providing a counterexample to a longstanding conjecture in Boolean function complexity.

Contribution

It offers a simple, explicit counterexample to a conjecture about Boolean matrices, improving upon previous probabilistic constructions.

Findings

Constructed matrices with ^{4/3} zeros and small covering number

Counterexample refutes conjecture by Pudle1k, Rf6dl, and Savickfd

Provides explicit example previously only known probabilistically

Abstract

We give a simple construction of $n\times n$ Boolean matrices with $\Omega(n^{4/3})$ 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 Pudl\'{a}k, R\"{o}dl and Savick\'{y} 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.