On cyclically covering subspaces of ${\mathbb{F}}_q^n$

For a prime power $q$ and a positive integer $n$, a subspace $U\subseteq {\mathbb{F}}_q^n$ is called cyclically covering if the union of all its cyclic shifts covers the whole space ${\mathbb{F}}_q^n$. Let $h_q(n)$ denote the maximum possible codimension of such a subspace. This paper focuses on the case $h_q(n)=0$. We provide necessary and sufficient conditions under which $h_q(n)=0$ holds. As an application, we show that $h_q({\mathrm{\ell }}^t)=0$ whenever $q$ is a primitive root modulo ${\mathrm{\ell }}^t$. Moreover, we prove that if $n$ is odd and $h_q(n)=0$, then also $h_q(2n)=0$. As an example, we show that $h_3(11)=h_3(16)=1$. Furthermore, we investigate the relationship between the coverings of ${\mathbb{F}}_{q^m}^n$ and ${\mathbb{F}}_q^{mn}$, and obtain several sufficient conditions for $h_{q^m}(n)=0$. Specifically, we derive that if $n=3$ or $n=2^d$ (where $d$ is a nonnegative integer), then $h_4(n)=0$.