STBC-schemes with nonvanishing determinant for certain number of transmit antennas

A space-time block-code scheme (STBC-scheme) is a family of STBCs {C(SNR)}, indexed by the signal-to-noise ratio (SNR) such that the rate of each STBC scales with SNR. An STBC-scheme is said to have a nonvanishing determinant if the coding gain of every STBC in the scheme is lower-bounded by a fixed nonzero value. The nonvanishing determinant property is important from the perspective of the diversity multiplexing- gain (DM-G) tradeoff: a concept that characterizes the maximum diversity gain achievable,by any STBC-scheme transmitting at a particular rate. This correspondence presents a systematic technique for constructing STBC-schemes with nonvanishing determinant, based on cyclic division algebras. Prior constructions of STBC-schemes from cyclic division algebra have either used transcendental elements, in which case the scheme may have vanishing determinant, or is available with nonvanishing determinant only for two, three, four, and six transmit antennas. In this correspondence, we construct STBC-schemes with nonvanishing determinant for the number of transmit antennas of the form 2(k) 3 (.) 2(k) 2 (.) 3(k), and q(k) (q - 1)/2, where q is any prime of the form 4s + 3. For cyclic division algebra based STBC-schemes, in a recent work by Elia et al, the nonvanishing determinant property has been shown to be sufficient for achieving DM-G tradeoff. In particular, it has been shown that the class of STBC-schemes constructed in this correspondence achieve the optimal DM-G tradeoff. Moreover, the results presented in this correspondence have been used for constructing optimal STBC-schemes for arbitrary number of transmit antennas, by Elia et al.