Exact hyperplane covers for subsets of the hypercube

TL;DR

This paper explores the minimal number of hyperplanes needed to cover specific subsets of the hypercube without covering certain points, extending previous results and providing exact solutions for small cases.

Contribution

It introduces the study of exact hyperplane covers for various hypercube subsets, offering solutions for small cases and asymptotic bounds for the general case.

Findings

Exact solutions for covering or our points

Asymptotic bounds for general hypercube subsets

Extension of previous hyperplane covering results

Abstract

Alon and F\"{u}redi (1993) showed that the number of hyperplanes required to cover $\{0,1\}^n\setminus \{0\}$ without covering $0$ is $n$. We initiate the study of such exact hyperplane covers of the hypercube for other subsets of the hypercube. In particular, we provide exact solutions for covering $\{0,1\}^n$ while missing up to four points and give asymptotic bounds in the general case. Several interesting questions are left open.