Perfect lattice

In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by. A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by.

proved that a lattice is extreme if and only if it is both perfect and eutactic.

The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by 1, 1, 1, 2, 3, 7, 33, 10916. summarize the properties of perfect lattices of dimension up to 7. verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.