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

Abstract

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. {Under suitable coprimality conditions on \(m,n\) and on the multiplicative orders of \(q\), we prove that the vanishing of \(h_q(m)\) and \(h_q(n)\) is preserved under their product.} 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, \(h_2(n) \geq 2\) and \(h_q(n) = 0\) for \(q \neq 2\). We prove that \(h_3(n) \ge 1\) for every prime \(n > 3\) with odd \(\operatorname{ord}_n(3)\). 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 $q=2,3$.