New Lower Bounds For Essential Covers Of The Cube

Abstract

An essential cover of the vertices of the $n$-cube $\{0,1\}^n$ by hyperplanes is a minimal covering where no hyperplane is redundant and every variable appears in the equation of at least one hyperplane. Linial and Radhakrishnan gave a construction of an essential cover with $\lceil \frac{n}{2} \rceil + 1$ hyperplanes and showed that $\Omega(\sqrt{n})$ hyperplanes are required. Recently, Yehuda and Yehudayoff improved the lower bound by showing that any essential cover of the $n$-cube contains at least $\Omega(n^{0.52})$ hyperplanes. In this paper, building on the method of Yehuda and Yehudayoff, we prove that $\Omega \left( \frac{n^{5/9}}{(\log n)^{4/9}} \right)$ hyperplanes are needed.