arXiv · 2210.08300
On depth-3 circuits and covering number: an explicit counter-example
Abstract
We give a simple construction of $n\times n$ Boolean matrices with $Ω(n^{4/3})$ 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 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.
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Lianna Hambardzumyan, Hamed Hatami, Ndiamé Ndiaye. 2022-10-15. On depth-3 circuits and covering number: an explicit counter-example. https://arxiv.org/abs/2210.08300
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