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Résumé : 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 $k=O(ε^{-2}\log^3n)$ proven there to $k=O(ε^{-2}\log^2n)$. 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.