A complete family of quasi-orthogonal space-time codes

For two-transmitter systems, there are (full-rate) orthogonal space-time block codes (O-STBC). For four-transmitter systems, there are (full-rate) quasi-orthogonal space-time block codes (QO-STBC). The orthogonality of such codes makes a decoder attractively simple with only little compromise of optimality of coding performance. A complete family of O-STBC is well understood. But for QO-STBC, only ad hoc examples have been reported in the literature. In this paper, we provide a systematic construction of a complete family of 4 x 4 QO-STBC. We show that there are only three independent QO-STBC and all other QO-STBC can be constructed by trivial variations of any three independent codes. Indeed, all 4 x 4 QO-STBC in the literature can be constructed in such a way. Furthermore, we show a connection between three independent QO-STBC and the Hurwitz-Radon families of matrices. A complete set of the HR families of size four is also discovered.