Optimal norm form integer space-time codes for two antenna MIMO systems

In this paper, we introduce the definitions of a full diversity integer generating matrix and the corresponding norm form space-time code for MIMO systems. Subject to a power constraint, we characterize all full diversity integer generating matrices with the first three largest gains in the Gaussian integer ring and the Eisenstein integer ring for two transmitter antennas. Using this generating matrix family to separately design space-time codes layer by layer for two transmitter antenna and two receiver antenna MIMO systems, we obtain the optimal norm form integer space-time codes both in the Gaussian integer ring and the Eisenstein integer ring in the sense of maximizing the minimum determinant of codeword matrices. As a consequence, we prove that the golden code constructed by Dayal and Varanasi, Belfiore, Rekaya and Viterbo is optimal in the Gaussian integer ring. Also, we find the optimal code in the Eisentein integer ring, the coding gain of which is greater than that of the golden code.