Exact hyperplane covers for subsets of the hypercube

Abstract

Alon and Füredi (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.

Small fixes and updated code