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/spl ges/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/sub 1/, /spl middot//spl middot//spl middot/, x/sub n/)/spl isin/R/sup n/, |/spl Pi//sub i=1//sup n/x/sub i/|/spl les/1}. The lattices among which we search for an optimal solution are the standard embeddings in R/sup 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.