In a recent publication Roland Bacher showed that the number p d of non-similar perfect d-dimensional quadratic forms satisfies e Ω ( 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.
Perfect quadratic form, Perfect lattice
doi.org/10.1016/j.aim.2020.107031
Advances in Mathematics
Cryptology

van Woerden, W.P.J. (2020). An upper bound on the number of perfect quadratic forms. Advances in Mathematics, 365. doi:10.1016/j.aim.2020.107031