Generalization of the Σ0-rank criteria for space-time codes

Recently, Liu et al., developed the Σ₀-rank criteria for space-time codes. It provides a sufficient condition on codeword and generator matrices defined over finite rings Z_2k(j) to ensure full spatial diversity with 2^2k-QAM modulation. Here, we generalize the Σ₀-rank criteria and derive a sufficient condition on the generator matrices defined over finite rings Z_2l(j) to ensure full spatial diversity with 2^2k-QAM modulation for any positive integer l≤k. We also show that generator matrices defined over GF(2) satisfying the BPSK stacking construction constraint of Mammons and El Gamal achieve full spatial diversity when used with 2^2k-QAM modulation.