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This work investigates linear precoding over non-singular linear channelswith additive white Gaussian noise, with lattice-type inputs. The aim is tomaximize the minimum distance of the received lattice points, where theprecoder is subject to an energy constraint. It is shown that the optimalprecoder only produces a finite number of different lattices, namely perfectlattices, at the receiver. The well-known densest lattice packings areinstances of perfect lattices, however it is analytically shown that thedensest lattices are not always the solution. This is a counter-intuitiveresult at first sight, since previous work in the area showed a tightconnection between densest lattices and minimum distance. Since there are onlyfinitely many different perfect lattices, they can theoretically be enumeratedoff-line. A new upper bound on the optimal minimum distance is derived, whichsignificantly improves upon a previously reported bound. Based on this bound,we propose an enumeration algorithm that produces a finite codebook of optimalprecoders.

A Lattice-Theoretic Characterization of Optimal Minimum-Distance Linear Precoders