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

TL;DR

This paper investigates the properties of cyclically covering subspaces in binary vector spaces, establishing exact values for primes where 2 is primitive and providing bounds for composite cases, advancing understanding in combinatorial and algebraic structures.

Contribution

It determines the exact value of h_2(p) for primes with 2 as a primitive root, and extends bounds and generalizations for cyclically covering subspaces.

Findings

h_2(p)=2 for primes with 2 as a primitive root

Bounds on h_2(ab) based on h_2(a) and h_2(b)

Extensions to generalized set-ups

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.