Optimal STBCs from codes over galois rings

Abstract

A Space-Time Block Code (STBC) CST is a finite collection of nt × l complex matrices. If S is a complex signal set, then CST 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 − X0 ), for any X 6 = X0 ∈ CST and the rate is R = log|S| |CST| l complex symbols per channel use, where |CST| denotes the cardinality of CST . For a STBC completely over S achieving transmit diversity gain equal to d, the rate is upper-bounded as R ≤ nt − d + 1. An STBC which achieves equality in this tradeoff is said to be optimal. A Rank-Distance (RD) code CF F is a linear code over a finite field Fq, where each codeword is a nt×l matrix over Fq. RD codes have found applications as STBCs by using suitable rank-preserving maps from Fp to S. In this paper, we generalize these rank-preserving maps, leading to generalized constructions of STBCs from codes over Galois ring GR (p a , k). To be precise, for any given value of d, we construct nt ×l matrices over GR (p a , k) and use a rank-preserving map that yields optimal STBCs with transmit diversity gain equal to d. Galois ring includes the finite field Fp k when a = 1 and the integer ring Zpa when k = 1. Our construction includes as a special case, the earlier construction by Lusina et. al. which is applicable only for RD codes over Fp (p = 4s + 1) and transmit diversity gain d.