Optimal lattices for MIMO precoding

Consider the communication model y̅ = HF x̅ + n̅, where H; F are real-valued matrices, x̅ is a data vector drawn from some real-valued lattice (e.g. M-PAM), n̅ is additive white Gaussian noise and y̅ 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 x̅ is subject to an average energy constraint. The columns of the matrix HF 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 HF? This particular question remains open within the theory of lattices. This work provides the solution for 2×2 matrices H; F. The answer is an F such that HF is a hexagonal lattice.