Systematic construction of full diversity algebraic constellations

A simple and systematic approach for constructing full diversity m-dimensional constellations, carved from lattices over a number ring R, is proposed for an arbitrary dimension m. When R=Z[w_n], the nth cyclotomic number ring, all the possible dimensions that allow for achieving the optimal minimum product distances using the proposed approach are determined. It turns out that one can construct optimal unitary transformations using our construction if and only if m factors into a power of 2 and powers of the primes dividing n. For m not satisfying these conditions, a method based on Diophantine approximation theory is proposed to "optimize" the minimum product distance. A lower bound on the product distance is given in this case, thus ensuring full diversity with "good" minimum product distances. Furthermore, the proposed approach subsumes the optimal unitary transformations proposed by Giraud et al. over R=Z[w₄] and R=Z[w₃], while giving optimal unitary transformations for infinitely many new values of n and m.