In digital communications, the cubic lattice plays the role of a simple uniform quantizer – for source coding, or an “uncoded” constellation – for channel coding. Better source-channel coding schemes can be associated with more complex lattices. This relation requires a definition of the notion of lattice goodness.

We shall develop two figures of merit of a lattice in the context of digital communication: (i) the normalized second moment – a measure of its goodness as a vector quantizer under a squared-error distortion measure, in Section 3.2, and (ii) the volume to noise ratio – a measure of its goodness as a (coded) constellation for the AWGN channel, in Section 3.3. Before that, in Section 3.1, we introduce two more fundamental quantities associated with lattice goodness for sphere packing and covering. As we may expect, all these quantities are invariant to scaling and rotation of the lattice.

Sphere packing and covering

Which shape minimizes the surface area among all shapes of a given volume in ℝn? Which shape minimizes the diameter or the second moment? The iso-perimetric inequality implies that for all these questions – and many more – the unique solution is a ball. Two of the most fundamental questions about balls (or spheres) are how efficiently they can be packed in the Euclidean space, and how efficiently they can cover it.