arXiv:2611.13044v1 Announce Type: new Abstract: Let $h_q(n)$ be the largest codimension of a subspace of $\mathbb{F}_q^n$ whose cyclic shifts cover the whole space. For $r>0$, we characterize the inequality $h_q(n)\ge r$ in terms of cyclic $\mathbb{F}_q$-linear codes over vector alphabets with covering radius one, no nonzero word of block weight one, and a scalar dual generated by the shifts of one word. For $k,m\ge2$, let $I_q(j)$ count the monic irreducible polynomials of degree $j$ over $\mathbb{F}_q$. If $k$ is prime or $m\ge2\sum_{j=1}^{k-1}I_q(j)+2$, we prove that a cyclic $\mathbb{F}_q$ linear perfect code of length $\frac{q^{km}-1}{q^k-1}$ over $\mathbb{F}_q^k$ exists if and only if $\gcd(m,q-1)=1$. When this condition holds, we also obtain \(h_q\!\left(\frac{q^{km}-1}{q^k-1}\right)=k(m-1).\) Restriction of scalars gives the same criterion for every $m\ge2$ when $q$ is odd, $q+1$ is a power of two, and $k=2p$ with $p$ prime. We also show that $h_q(q^k+1)=k$ exactly when $q$ is even, for every $k\ge1$.