The following is a special case of my earlier question which is still not solved.
Suppose both $\mathbf{a_i}$ and $\mathbf{v}$ are $1\times N$ vectors over a finite field $\mathbb{F_q}$, where $i\in \{1,2,\ldots\,K\}$. $\mathbf{a_i}$ are very sparse, i.e. there are only two or three non-zero entries in each $\mathbf{a_i}$. Now, I have $K$ inequations $\mathbf{a_i}\cdot\mathbf{v}\neq 0$.
What is the necessary and sufficient condition that there exists at one such $\mathbf{v}$ that make the $K$ inequations hold simultaneously? Thanks.