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 non-zero.