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 <i>2&nbsp;&times;&nbsp;2</i> diagonal space-time block codes from quadratic extensions on <i>BBQ (</i>i<i>) </i> and <i> BBQ ( mmbzeta₆)</i>, where i <i>=radic{-1}</i> and <i>mmbzeta₆=exp( </i>i<i> 2pi/6)</i>. 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 <i>BBQ (</i>i<i>) </i> and <i> BBQ ( mmbzeta₆)</i> for three and four transmit antennas. We also present a nontight upper bound for normalized diversity products of <i>2&nbsp;&times;&nbsp;2</i> diagonal space-time block codes with QAM information symbols, i.e., in <i>BBZ [</i>i<i> ]</i>, from general <i>2&nbsp;&times;&nbsp;2 </i> complex-valued generating matrices. We then present an <i><i>n</i>&nbsp;&times;&nbsp;<i>n</i></i>-diagonal space-time code design method directly from <i> 2<i>n</i></i> real integers based on extended complex lattices (of generating matrix size <i><i>n</i>&nbsp;&times;&nbsp;2<i>n</i></i>) that are shown to have better normalized diversity products than the optimal diagonal cyclotomic codes do. We finally use the optimal <i>2&nbsp;&times;&nbsp;2</i> diagonal space-time codes from the optimal quadratic extensions to construct two <i>2&nbsp;&times;&nbsp;- - 2 </i> full-rate space-time block codes and find that both of them have better normalized diversity products than the Golden code does.