Optimal STBCs from codes over Galois rings

A space-time block code (STBC) C/sub ST/ is a finite collection of n/sub t/ /spl times/ l complex matrices. If S is a complex signal set, then C/sub ST/ is said to be completely over S if all the entries of each of the codeword matrices are restricted to S. The transmit diversity gain of such a code is equal to the minimum of the ranks of the difference matrices (X - X'), for any X /spl ne/ X' /spl isin/ C/sub ST/, and the rate is R = (log /sub |s|/|C/sub ST/|)/l complex symbols per channel use, where |C/sub ST/| denotes the cardinality of C/sub ST/. For a STBC completely over S achieving transmit diversity gain equal to d, the rate is upper-bounded as R /spl les/ n/sub t/ - d + 1. An STBC which achieves equality in this tradeoff is said to be optimal. A rank-distance (RD) code C/sub FF/ is a linear code over a finite field F/sub q/, where each codeword is a n/sub t/ /spl times/ l matrix over F/sub q/. RD codes have found applications as STBC by using suitable rank-preserving maps from F/sub p/ to S. In this paper, we generalize these rank-preserving maps, leading to generalized constructions of STBC from codes over Galois ring GR(p/sup a/, k). To be precise, for any given value of d, we construct n/sub t/ /spl times/ l matrices over GR(p/sup a/, k) and use a rank-preserving map that yields optimal STBC with transmit diversity gain equal to d. Galois ring includes the finite field F/sub p//sup k/ when a = 1 and the integer ring Z/sub p//sup a/ when k = 1. Our construction includes as a special case, the earlier construction by Lusina et al. (2003) which is applicable only for RD codes over F/sub p/ (p = 4s + 1) and transmit diversity gain d = n/sub t/.