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⩾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=(x₁, ···, x_n)∈Rⁿ, |Π_i=1ⁿx_i|⩽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