A threshold property of quadratic Boolean functions

Let $f$ be a Boolean function in $n$ variables and for any affine subspace $L$ of dimension $\lceil n/2\rceil$ either $f$ is affine on all shifts of $L$ or $f$ is not affine on any shift of $L$. It is proved that the algebraic degree of $f$ can be more than 2 only if there is no affine subspace of dimension $\lceil n/2\rceil$ that $f$ is affine on. Bibliogr. 8.