On the nonexistence of rate-one generalized complex orthogonal designs
IEEE Transactions on Information Theory
Orthogonal space-time block coding proposed recently by Alamouti  and Tarokh, Jafarkhani, and Calderbank  is a promising scheme for information transmission over Rayleigh-fading channels using multiple transmit antennas due to its favorable characteristics of having full transmit diversity and a decoupled maximum-likelihood (ML) decoding algorithm. Tarokh, Jafarkhani, and Calderbank extended the theory of classical orthogonal designs to the theory of generalized, real, or complex, linear
... or complex, linear processing orthogonal designs and then applied the theory of generalized orthogonal designs to construct space-time block codes (STBCs) with the maximum possible diversity order while having a simple decoding algorithm for any given number of transmit and receive antennas. It has been known that the STBCs constructed in this way can achieve the maximum possible rate of one for every number of transmit antennas using any arbitrary real constellation and for two transmit antennas using any arbitrary complex constellation. Contrary to this, in this correspondence we prove that there does not exist rate-one STBC from generalized complex linear processing orthogonal designs for more than two transmit antennas using any arbitrary complex constellation. Index Terms-Alamouti scheme, complex orthogonal designs, full rate, generalized complex orthogonal designs, Hurwitz-Radon theory, orthogonal designs, space-time block codes (STBCs), transmit diversity.