Optimal Two-Dimensional Lattices for Precoding of Linear Channels

Consider the communication system model y =
HFx+n, where H and F are the channel and precoder matrices,
x is a vector of data symbols drawn from some lattice-type
constellation, such as M-QAM, n is an additive white Gaussian
noise vector and y is the received vector. It is assumed that
both the transmitter and the receiver have perfect knowledge
of the channel matrix H and that the transmitted signal Fx is
subject to an average energy constraint. The columns of the
matrix HF can be viewed as the basis vectors that span a
lattice, and we are interested in the precoder F that maximizes
the minimum distance of this lattice. This particular problem
remains open within the theory of lattices and the communication
theory. This paper provides the complete solution for any nonsingular
M ×2 channel matrix H. For real-valued matrices and
vectors, the solution is that HF spans the hexagonal lattice. For
complex-valued matrices and vectors, the solution is that HF,
when viewed in four-dimensional real-valued space, spans the
Schlafli lattice D4.