I have an algebraic problem related to vectors in the field GF(2).Let $v_1, v_2, \ldots, v_m$ be  (0,1)-vectors of dimension $n$, and $m=n^{O(1)}$. Find a polynomial time algorithm that finds a (0,1)-vector $u$ of the same dimension such that $u$ is not the sum of any $(\log n)^{O(1)}$ vectors among $v_1,v_2, \ldots, v_m$. The addition of vectors is over the field GF(2), which has two elements 0 and 1 ($0+1=0+1=1$, and $0+0=1+1=0$).It is easy to see the existence of such a vector u by a simple counting argument.  Can we find $u$ in a polynomial time?  It is trivial to find $u$ in exponential time.   I will send a $200 check award for the first correct solution.
Algorithmic Vector Problem
ds.algorithms;linear algebra