# Optimal STBCs from Codes Over Galois Rings

Kiran, T and Rajan, Sundar B (2005) Optimal STBCs from Codes Over Galois Rings. In: IEEE International Conference on Personal Wireless Communications, 2005. ICPWC 2005, 23-25 January, New Delhi, India, 120 -124.

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A Space-Time Block Code (STBC) $C_S_T$ is a finite collection of $\mathbf{n_t}\times\mathbf{l}$ complex matrices. If S is a complex signal set, then $C_S_T$ 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 $X \not= X'\in C_S_T$, and the rate is $$R=\frac{log_{\mid{s}\mid}\mid{C_S_T}\mid}{l}$$ complex symbols per channel use, where $\mid{C_S_T}\mid$ denotes the cardinality of $C_S_T$. For a STBC completely over S achieving transmit diversity gain equd to d, the rate is upper-bounded as $R\leq\mathbf{n_t} - \mathbf{d}+\mathbf{1}$. An STBC which achieves equality in this tradeoff is said to be optimal. A Rank-Distance (RD) code $C_F_F$ is a linear code over a finite field $F_q$ where each code-word is a $\mathbf{n_t}\times\mathbf{l}$ matrix over $F_q$. RD codes have found applications as STBCs 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 STBCs fiom codes over Galois ring $GR(p^a,k)$. To be precise, for any given value of d, we construct $\mathbf{n_t}\times\mathbf{l}$ 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 $F_{p^a}$ 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. which is applicable only for RD codes over $F_p$ $(p=\mathbf{4s}+\mathbf{1})$ and transmit diversity gain $d=n_t$.