Voronoi Cells of Lattices and Quantization Errors

Abstract

In n-dimensional space, what is the average squared distance of a random point from the closest point of the lattice A _n (or D _n , E _n , A ^* _n or D ^* _n )? If a point is picked at random inside a regular simplex, octahedron, 600-cell or other polytope, what is its average squared distance from the centroid? The answers are given here, together with a description of the Voronoi cells of the above lattices. The results have applications to quantization and to the design of codes for a bandlimited channel. For example, a quantizer based on the eight-dimensional lattice E ₈ has a mean squared error per symbol of 0.0717... when applied to uniformly distributed data, compared with 0.08333... for the best one-dimensional quantizer.