Cyclically covering subspaces of Fqn

Abstract

Let q be a prime power, n be a positive integer, and $F_q^n$ be the n-dimensional row vector space over finite field $F_q$. We say a subspace U of $F_q^n$ is cyclically covering if the union of the cyclic shifts of U is equal to $F_q^n$. Recently, the largest possible codimension of a cyclically covering subspace of $F_q^n$, denoted by $h_q\left(n\right)$, has attracted the attention of many scholars. In this paper, we introduce cyclically covering subspaces of finite field $F_{q^n}$. By virtue of the theory of direct sum decomposition of finite fields, we describe a method for constructing cyclically covering subspaces of $F_{q^n}$, and determine the value of $h_2\left(n\right)$ for some special n. In particular, we prove $h_2\left(21\right)=4$. Finally, several lower bounds of $h_q\left(n\right)$ are given when $(q,n)=1$, which generalizes results of the existing results in [2].