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

TL;DR

This paper characterizes when cyclically covering subspaces of finite fields exist with zero codimension, providing conditions related to primitive roots and field extensions, and explores their properties for various parameters.

Contribution

It offers necessary and sufficient conditions for the existence of cyclically covering subspaces with zero codimension, including new results on prime power and extension fields.

Findings

For prime powers, h_q(n)=0 if q is a primitive root modulo n.

If n is odd and h_q(n)=0, then h_q(2n)=0.

Examples show h_3(11)=h_3(16)=1, indicating specific cases where the property fails.

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 \(h_4(n) = 0\).