The equivalence of space-time codes and codes defined over finite fields and Galois rings

Space-time codes for a wide variety of channels have the property that the diversity of a pair of codeword matrices is measured by the vanishing or non-vanishing of polynomials in the entries of the matrices. We show that for every such channel: I) There is an appropriately-defined notion of approximation of space-time codes such that each code is arbitrarily well approximated by one whose alphabet lies in the field of algebraic numbers; II) Each space-time code whose alphabet lies in the field of algebraic numbers is an appropriately-defined lift from a corresponding space-time code defined over a finite field or a “scaled” lift from a galois ring of arbitrary characteristic. This implies that all space-time codes can be designed over finite fields or over galois rings of arbitrary characteristic and then lifted to complex matrices with entries in a number field.