Abstract
Dimitri Grigoriev has shown that for any family of $N$ vectors in the $d$-dimensional linear space $E=(\ff{2})^d$, 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 $[N/2-\sqrt{N}/2,N/2+\sqrt{N}/2]$ and we give an efficient, deterministic parallel algorithm which finds a vector achieving this bound. The optimality of the bound is also investigated.