Let $V_n=\{-1,1\}^n$ be the hypercube and let $C_n$ be a collection $\{x_1,...,x_n\}$ of $n$ distinct elements of $V_n.$

Question: what is the smallest number $N(n)$ of non-zero vectors with integer coefficients are needed to check that $C_n$ is linearly independent over the integers? That is, what is the smallest set of such vectors $\{v_1,...v_{N(n)}\}$ that if $C_n$ is linearly dependent over the integers then there exists some element $v_{i}$ with coordinates $(k_1,...,k_n)$ such that $k_1x_1+...+k_nx_n$ is zero.