Exact hyperplane covers for subsets of the hypercube

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.