Cyclic Codes and Cyclically Covering Subspaces over Finite Fields

Let $q$ be a power of a prime $p$, and let $n$ be a positive integer. A subspace $U\subseteq \mathbb F_q^n$ is called cyclically covering if the union of all its cyclic shifts covers $\mathbb F_q^n$, and $h_q(n)$ denotes the maximum possible codimension of such a subspace. This paper studies cyclically covering subspaces via cyclic codes. We first prove that $h_q(n)=0$ if and only if every nonzero cyclic code in $\mathbb F_q^n$ contains a full-weight codeword. We also relate $h_q(n)$ to the maximum weights of cyclic codes. In particular, when $h_q(n)>0$, we obtain sharp bounds for the maximum weight of cyclic codes without full-weight codewords and provide explicit examples attaining these bounds. Moreover, we study the number of cyclic codes containing no full-weight codeword. We determine this number completely over $\mathbb F_2$, and give lower bounds over $\mathbb F_3$. From this, we prove that if $q\ge 3$ is an odd prime and $m\ge 4$ is an integer, then $h_q\left(\frac{q^m+1}{2}\right)>0$.