Optimal STBCs from codes over Galois rings

A space-time block code (STBC) C_ST is a finite collection of n_t × l complex matrices. If S is a complex signal set, then C_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 ≠ X´ ∈ C_ST, and the rate is R = (log _|s||C_ST|)/l complex symbols per channel use, where |C_ST| denotes the cardinality of C_ST. For a STBC completely over S achieving transmit diversity gain equal to d, the rate is upper-bounded as R ≤ n_t - d + 1. An STBC which achieves equality in this tradeoff is said to be optimal. A rank-distance (RD) code C_FF is a linear code over a finite field F_q, where each codeword is a n_t × l matrix over F_q. RD codes have found applications as STBC by using suitable rank-preserving maps from F_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^a, k). To be precise, for any given value of d, we construct n_t × l matrices over GR(p^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_p^k when a = 1 and the integer ring Z_p^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_p (p = 4s + 1) and transmit diversity gain d = n_t.