How to prove one-to-one correspondence in vector space?

Question

I am given a vector subspace $S$ over the space $\{0, 1\}^n$, and a vector space $S^{\perp}$ which contains the set of vectors in $\{0, 1\}^n$ which are perpendicular to all the vectors in $S$. I have to prove that is the a vector is a part of $\{0, 1\}^n$, either it belongs to $S^{\perp}$, or it is perpendicular to exactly half of the element in S. I believe one way to solve it is to prove one-to-one correspondence between elements $\textbf{s}$ or $S$ which satisfy $\textbf{s}\cdot\textbf{z}=0$ and others which satisfy $\textbf{s}\cdot\textbf{z}=1$, for some $\textbf{z} \notin S^{\perp}$, but a member of $\{0, 1\}^n$. But I am not sure on how to carry on this problem. I need some hint. Any help will be appreciated.