Low-dimensional lattices. III. Perfect forms

Abstract

The perfect lattices or quadratic forms in dimensions n ≼ 7 found by Barnes, Stacey and others are studied, and their automorphism groups, orbits of minimal vectors and eutactic coefficients are determined. It is shown that just 30 of the 33 known seven-dimensional perfect lattices are extreme. A simple and self-contained proof is given of the classification of perfect lattices in dimensions n ≼ 4 and hence of the determination of the densest lattice packings in these dimensions.