Some designs and normalized diversity product upper bounds for lattice-based diagonal and full-rate space-time block codes

In this paper, we first present two tight upper bounds for the normalized diversity products (or product distances) of 2 × 2 diagonal space-time block codes from quadratic extensions on Q(i) and Q(ζ_ε, where i = → -1 and Ζ_ε. Two such codes are shown to reach the tight upper bounds and therefore have the maximal normalized diversity products. We present two new diagonal space-time block codes from higher order algebraic extensions on Q(i) and Q(ζ_ε for three and four transmit antennas. We also present a nontight upper bound for normalized diversity products of 2 × 2 diagonal space-time block codes with QAM information symbols, i.e., in Z[i], from general 2 × 2 complex-valued generating matrices. We then present an n × n-diagonal space-time code design method directly from 2n real integers based on extended complex lattices (of generating matrix size n × 2 ) that are shown to have better normalized diversity products than the optimal diagonal cyclotomic codes do. We finally use the optimal 2 × 2 diagonal space-time codes from the optimal quadratic extensions to construct two 2 7times; 2 full-rate space-time block codes and find that both of them have better normalized diversity products than the Golden code does.