Discrete Fourier Transform Approach to Cyclically Covering Subspaces of ${\mathbb{F}}_q^n$

Let $q$ be a prime power and $n$ a positive integer. 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. When $\gcd (q,n)=1$, we derive necessary and sufficient conditions for $h_q(n)=0$ via Discrete Fourier Transforms, and prove this equality is equivalent to the existence of full-weight codewords in cyclic codes of ${\mathbb{F}}_q^n$. We also characterize codimension-$k$ cyclically covering subspaces. Based on these results, we give a unified characterization of $h_q(n)$ in the case where $q$ and $n$ are primes with $n>q$ and $q$ being a primitive root modulo $n$. Specifically, $h_2(n)\ge 2$ and $h_q(n)=0$ for $q\mathrm{\ne }2$. We prove that $h_3(n)\ge 1$ for every prime $n>3$ with odd ${\mathrm{ord}}_n(3)$. Moreover, for any prime $q>3$, the Generalized Riemann Hypothesis implies the existence of infinitely many primes $n>q$ such that $q$ is not a primitive root modulo $n$ and $h_q(n)=0$. We provide algebraic interpretations for the inequalities $h_q(mn)\ge \max \{h_q(m),h_q(n)\}$ and $h_q(mn)\ge h_q(m)+h_q(n)$. Using Galois descent, we prove $h_{q^m}(n)\le h_q(n)$. Furthermore, we generalize a class of constructions that achieve the upper bound $\lfloor {\log }_q(n)\rfloor$. Finally, under the Generalized Riemann Hypothesis, we obtain average lower bounds of $h_q(n)$ for $q=2,3$.