A lattice quantizer maps a point of Euclidean space to a nearest point of a point lattice spanning that space. A fixed rule may resolve ties on Voronoi cell boundaries (Conway and Sloane 1982). The inverse images of lattice points are their Voronoi cells, apart from the assignment of boundary points.
For a point lattice 
of covolume 
, the mean squared error per coordinate
for a uniform point in a Voronoi
cell is 
,
where 
is the normalized second moment. Minimizing
therefore minimizes this error among
point lattices of fixed covolume.
The D4 lattice minimizes the normalized
second moment among four-dimensional point lattices
(Liu 2026).
