Cyclically covering subspaces in F2n

A subspace of F-2(n) is called cyclically covering if every vector in F-2(n) has a cyclic shift which is inside the subspace. Let h(2)(n) denote the largest possible codimension of a cyclically covering subspace of F-2(n). We show that h(2)(p) = 2for every prime psuch 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 h(2)(ab) depending on h(2)(a) and h(2)(b) and extend some of our results to a more general set-up proposed by Cameron, Ellis and Raynaud. (C) 2021 Elsevier Inc. All rights reserved.