Finding a Vector Orthogonal to Roughly Half a Collection of Vectors

13 pagesDimitri Grigoriev has shown that for any family of N vectors in the d-dimensional linear space E = (F_2)^d, there exists a vector in E which is orthogonal to at least N/3 and at most 2N/3 vectors of the family. We show that the range [N/3, 2N/3] can be replaced by the much smaller range [N/2 − √N /2, N/2 + √N /2] and we give an efficient, deterministic parallel algorithm which finds a vector achieving this bound. The optimality of the bound is also investigated