Optimal Lattices for MIMO Precoding

In English Consider the communication model (y) over bar = H F (x) over bar + (n) over bar, where H; F are real-valued matrices, (x) over bar is a data vector drawn from some real-valued lattice (e.g. M-PAM), (n) over bar is additive white Gaussian noise and (y) over bar is the received vector. It is assumed that the transmitter and the receiver have perfect knowledge of the channel matrix H (perfect CSI) and that the transmitted signal F (x) over bar is subject to an average energy constraint. The columns of the matrix H F can be viewed as basis vectors that span a lattice, and we are interested in the minimum distance of this lattice. More precisely, for a given H, which F under an average energy constraint will maximize the minimum distance of the lattice H F ? This particular question remains open within the theory of lattices. This work provides the solution for 2 x 2 matrices H; F. The answer is an F such that H F is a hexagonal lattice.