On Trivial Cyclically Covering Subspaces of ${\mathbb{F}}_q^n$ in Non-Coprime Characteristic

A subspace $U$ of ${\mathbb{F}}_q^n$ is called $\mathit{\text{cyclically}}\mathit{\text{covering}}$ if the whole space ${\mathbb{F}}_q^n$ is the union of the cyclic shifts of $U$. The case ${\mathbb{F}}_q^n$ itself is the only covering subspace, is of particular interest. Recently, Huang solved this problem completely under the condition $\gcd (n,q)=1$ using primitive idempotents and trace functions, and explicitly posed the non-coprime case as an open question. This paper provides a complete answer to Huang’s question. We prove that if $n=p^{k}m$ where $p=\mathrm{char}({\mathbb{F}}_q)$ and $\gcd (m,p)=1$, then $h_q(p^{k}m)=0$ if and only if $h_q(m)=0$. This result fully reduces the non-coprime case to the coprime case settled by Huang. Our proof employs the structure theory of cyclic group algebras in modular characteristic.