As the dimension increases, lattices can form richer arrangements of points in space. Richness, though, comes at the cost of a higher complexity. Are high-dimensional lattices better?

The advantage of going to higher dimensions is questionable if we are only interested in sphere packing and covering. The one-dimensional lattice is already perfect for both problems, whereas higher-dimensional lattices are not. To be more specific, the best lattice packing efficiency ρpack gets worse (roughly monotonically) as the dimension increases, and decreases from 1 to the Minkowski bound of onehalf. The best lattice-covering efficiency ρcov exhibits anomalous behavior. First it increases (i.e., deteriorates) but then, by the Rogers bound, it becomes asymptotically perfect again and approaches 1 as the dimension goes to infinity. See more on that later in this chapter.

While mathematicians paid attention to these two hard problems, Shannon and his followers were more interested in the “softer” questions of quantization and modulation. Shannon's theory demonstrates the advantage of high-dimensional source and channel codes. The underlying principle is the law of large numbers. Does this principle apply also to lattice codes?

Lattices indeed improve when it comes to the quantization and the modulation problems.