  We continue our study of the Johnson-Lindenstrauss lemma and its connection
to circulant matrices started in \cite{HV}. We reduce the bound on $k$ from
$k=O(\epsilon^{-2}\log^3n)$ proven there to $k=O(\epsilon^{-2}\log^2n)$. Our
technique differs essentially from the one used in \cite{HV}. We employ the
discrete Fourier transform and singular value decomposition to deal with the
dependency caused by the circulant structure.
