An upper bound on the number of perfect quadratic forms

In a recent preprint on arXiv Roland Bacher showed that the number \(p_d\) of non-similar perfect \(d\)-dimensional quadratic forms satisfies \(e^{\Omega(d)} < p_d < e^{O(d^3\log(d))}\). We improve the upper bound to \(e^{O(d^2\log(d))}\) by a volumetric argument based on Voronoi's first reduction theory.