Abstract
Space-time codes are sets Of Complex All x T matrices used to describe the amplitude-phase modulation of a radio signal transmitted over T time slots from each of M transmit antennas. Under certain assumptions on the transmission channel, useful examples when M = T have been built by taking as codes M-dimensional vector spaces V of M x M matrices over number fields with every non-zero element of V nonsingular. All such spaces arise as representations of M-dimensional non-associative division algebras over number fields. We introduce new 3-dimensional non-associative division algebras over any perfect field k associated to elliptic curves E over k, which allows us to classify all such algebras over k. We give a finer classification over number fields in terms of the Tate-Shafarevich group of E.