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

10/15/2022

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by   Lianna Hambardzumyan, et al.

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We give a simple construction of n× n Boolean matrices with Ω(n^4/3) zero entries that are free of 2 × 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.