On trivial cyclically covering subspaces of Fqn

Abstract

A subspace of $F_q^n$ is called a cyclically covering subspace if for every vector of $F_q^n$, operating a certain number of cyclic shifts on it, the resulting vector lies in the subspace. In this paper, we study the problem of under what conditions $F_q^n$ is itself the only covering subspace of $F_q^n$, symbolically, $h_q\left(n\right)=0$, which is an open problem posed in Cameron et al. (2019) [3] and Aaronson et al. (2021) [1]. We apply the primitive idempotents of the cyclic group algebra to attack this problem; when q is relatively prime to n, we obtain a necessary and sufficient condition under which $h_q\left(n\right)=0$, which completely answers the problem in this case. Our main result reveals that the problem can be fully reduced to that of determining the values of the trace function over finite fields. As consequences, we explicitly determine several infinitely families of $F_q^n$ which satisfy $h_q\left(n\right)=0$.