A subspace of F₂ⁿ is called cyclically covering if every vector in F₂ⁿ has a cyclic shift which is inside the subspace. Let h₂(n) denote the largest possible codimension of a cyclically covering subspace of F₂ⁿ. We show that h₂(p)=2 for every prime p such that 2 is a primitive root modulo p, which, assuming Artin's conjecture, answers a question of Peter Cameron from 1991. We also prove various bounds on h₂(ab) depending on h₂(a) and h₂(b) and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud.

Alon and Füredi (1993) showed that the number of hyperplanes required to cover {0,1}ⁿ∖{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.