Comment by Gro-Tsen on Linear independence in $\mathbb{Z}_q^n$

For $n=1$, there are $\varphi(q)$ elements of $\mathbb{Z}/q\mathbb{Z}$ that are (each, separately) “linearly independent”, so to find a single vector you need $q-\varphi(q)+1$ vectors. This does not bode well for your question.