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

TL;DR

This paper completely characterizes when trivial cyclically covering subspaces exist in vector spaces over finite fields in non-coprime characteristic, reducing the problem to the previously solved coprime case.

Contribution

It proves that the existence of trivial cyclically covering subspaces in non-coprime characteristic cases depends solely on the coprime component, extending Huang's results.

Findings

If n=p^k m with gcd(m,p)=1, then h_q(p^k m)=0 iff h_q(m)=0.

The non-coprime case reduces to the coprime case solved by Huang.

The proof uses the structure theory of cyclic group algebras in modular characteristic.

Abstract

A subspace $U$ of $\mathbb{F}_q^n$ is called \textit{cyclically 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 $p = \operatorname{char}(\mathbb{F}_q)$ 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.