Cyclically covering subspaces in F₂ⁿ

Abstract

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.