Cyclically covering subspaces in ${\mathbb{F}}_2^n$

Abstract

A subspace of ${\mathbb{F}}_2^n$ is called cyclically covering if every vector in ${\mathbb{F}}_2^n$ has a cyclic shift which is inside the subspace. Let $h_2(n)$ denote the largest possible codimension of a cyclically covering subspace of ${\mathbb{F}}_2^n$. We show that $h_2(p)=2$ 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 $h_2(ab)$ 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.

36 pages