  This work investigates linear precoding over non-singular linear channels
with additive white Gaussian noise, with lattice-type inputs. The aim is to
maximize the minimum distance of the received lattice points, where the
precoder is subject to an energy constraint. It is shown that the optimal
precoder only produces a finite number of different lattices, namely perfect
lattices, at the receiver. The well-known densest lattice packings are
instances of perfect lattices, however it is analytically shown that the
densest lattices are not always the solution. This is a counter-intuitive
result at first sight, since previous work in the area showed a tight
connection between densest lattices and minimum distance. Since there are only
finitely many different perfect lattices, they can theoretically be enumerated
off-line. A new upper bound on the optimal minimum distance is derived, which
significantly improves upon a previously reported bound. Based on this bound,
we propose an enumeration algorithm that produces a finite codebook of optimal
precoders.
