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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 $d=n_t$.

Item Type: Conference Paper
Publisher: IEEE
Additional Information: �©1990 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.
Department/Centre: Division of Electrical Sciences > Electrical Communication Engineering
Date Deposited: 25 Nov 2005
Last Modified: 19 Sep 2010 04:21
URI: http://eprints.iisc.ac.in/id/eprint/4088

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