Given a $n$ dimensional vector space over the finite field $F_q$, called $V(F_q)$, and a set of $M$ vectors $\vec c_m=(c_0,c_1,...c_{n-1})^T$ that fulfill $(\sum c_k )=0$. Further the complete set of $\{\vec c_m\}$ is symmetric under all permutations of elements of a given $\vec c_m$.
I'm searching $\vec s\in V(F_q)$, the smallest (in the sense of the taxicab norm) element of $V(F_q)$ which has:
$$ \vec s \cdot \vec c_m \neq 0 \bmod q, \forall m$$
If each $\vec c_m$ makes up an $n-1$ dimensional subspace, orthogonal to $\vec c_m$, my feeling says that $\vec s$ lies in the complement of union of all these subspaces. If and how is this related to Grassmanians? I ask because the application of Gaussian Binomial Coefficients tells us that
the Gaussian binomial coefficient ${\displaystyle {n \choose k}_{q}}$ counts the number of $k$-dimensional vector subspaces of an $n$-dimensional vector space over $F_q$ (a Grassmannian).
Is the union of all these subspaces a Grassmanian?