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

Abstract

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 \(h_4(n) = 0\).