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

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 $h_q(\ell^t) = 0$ whenever $q$ is a primitive root modulo $\ell^t$. 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 $h_4(n) = 0$.