The covering radius of the Leech lattice

Abstract

We investigate the points in 24-dimensional space at maximum distance from the Leech lattice, i. e. the ‘deepest holes’ in that lattice. The maximum distance of any such point from the Leech lattice is shown to be 1/√2 times the minimum distance between the lattice points. Furthermore there are 23 types of ‘deepest hole’, one for each of the 23 even unimodular 24-dimensional lattices found by Niemeier.