Lattice Coverings ofn-Space By Spheres

Abstract

Alattice∧ in euclideann-space,Enis a group of vectors under vector addition generated bynindependent vectors,X1, X2… ,Xn, called abasisfor the lattice. The absolute value of then×ndeterminant the rows of which are the co-ordinates of a basis is called thedeterminant of the latticeand is denoted byd(∧). For any lattice ∧ there is a unique minimal positive numberrsuch that, if spheres of radiusrare placed with centres at all points of ∧, the entire space is covered. Thedensityof this covering may be defined as(Jnrn)/(d(∧))whereJnis the volume of the unit sphere inn-dimensional euclidean space. This density will be denoted by θn(∧). Thedensity of the most efficient lattice coveringofn-space by spheres,θnis the absolute minimum of θn(∧) considered as a function from the space of all lattices to the real numbers.