Constellations matched to the Rayleigh fading channel

We introduce a new technique for designing signal sets matched to the Rayleigh fading channel. In particular, we look for n-dimensional (n greater than or equal to 2) lattices whose structure provides nth-order diversity. Our approach is based on a geometric formulation of the design problem which in turn can be solved by using a number-geometric approach. Specifically, a suitable upper bound on the pairwise error probability makes the design problem tantamount to the determination of what is called a critical lattice of the body S = {x=(x(1),...,x(n))is an element of R(n),\Pi(i=1)(n) x(i)\less than or equal to 1}. The lattices among which we search for an optimal solution are the standard embeddings in R(n) of the number ring of some totally real number field of degree n over Q. Simulation results confirm that this approach yields lattices with considerable coding gains.