Covering Hypercube $mB^n$

Abstract

The Alon--Füredi theorem determines the minimum number of hyperplanes needed to cover every nonzero vertex of the Boolean cube while avoiding the origin. We study hyperplane coverings with multiplicity for the generalized hypercube $mB^n=\{0,1,\ldots,m\}^n$. Our main result is a polynomial multiplicity theorem extending the theorem of Sauermann and Wigderson to $mB^n$, with sharp degree bounds for polynomials having prescribed high-order zeros. We apply this result to obtain lower and upper bounds for hyperplane coverings with multiplicity and determine the exact minimum number of hyperplanes for coverings with multiplicities $k=1,2$. In the special case of the Boolean cube, our constructions improve the upper bound of Clifton and Huang for certain ranges of the parameters.