Abstract-We present some general techniques for constructing full-rank, minimal-delay, rate at least one space-time block codes (STBCs) over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras (field extensions) as well as using noncommutative division algebras of the rational field embedded in matrix rings. The first half of the paper deals with constructions using field extensions of . Working with cyclotomic field extensions, we construct several families of STBCs over a wide range of signal sets that are of full rank, minimal delay, and rate at least one appropriate for any number of transmit antennas. We study the coding gain and capacity of these codes. Using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas. We also present a method of constructing STBCs using noncyclotomic field extensions. In the later half of the paper, we discuss two ways of embedding noncommutative division algebras into matrices: left regular representation, and representation over maximal cyclic subfields. The 4 4 real orthogonal design is obtained by the left regular representation of quaternions. Alamouti's code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it. Also, we discuss a general principle for constructing cyclic division algebras using the th root of a transcendental element and study the capacity of the STBCs obtained from this construction. Another family of cyclic division algebras discovered by Brauer is discussed and several examples of STBCs derived from each of these constructions are presented.

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This paper 1 deals with low maximum likelihood (ML) decoding complexity, full-rate and full-diversity space-time block codes (STBCs), which also offer large coding gain, for the 2 transmit antenna, 2 receive antenna (2 × 2) and the 4 transmit antenna, 2 receive antenna (4 × 2) MIMO systems. Presently, the best known STBC for the 2 × 2 system is the Golden code and that for the 4 × 2 system is the DjABBA code. Following the approach by Biglieri, Hong and Viterbo, a new STBC is presented in this paper for the 2 × 2 system. This code matches the Golden code in performance and ML-decoding complexity for square QAM constellations while it has lower ML-decoding complexity with the same performance for non-rectangular QAM constellations. This code is also shown to be information-lossless and diversitymultiplexing gain (DMG) tradeoff optimal. This design procedure is then extended to the 4 × 2 system and a code, which outperforms the DjABBA code for QAM constellations with lower ML-decoding complexity, is presented. So far, the Golden code has been reported to have an ML-decoding complexity of the order of M 4 for square QAM of size M . In this paper, a scheme that reduces its ML-decodingsingle-output (SISO) system. The Alamouti code [1] for two transmit antennas, due to its orthogonality property, allows a low complexity ML-decoder. This scheme led to the development of the generalized orthogonal designs [2]. These designs are famous for the simplified ML-decoding that they provide.They allow all the symbols to be decoupled from one another and hence, are said to be single-symbol decodable. Another bright aspect about these codes is that they have full transmit diversity for arbitrary complex constellation. However, the limiting factor of these designs is the low code rate (refer Section II for a definition of code rate) that they support.At the other extreme are the well known codes from division algebra, first introduced in [3]. The well known perfect codes [4] have also been evolved from division algebra with large coding gains. These codes have full transmit diversity and have the advantage of a very high symbol rate, equal to that of the VBLAST scheme, which, incidentally doesn't have full transmit diversity. But unfortunately, the codes from division algebra including perfect codes have a very high ML-decoding complexity (refer Section II for a definition of ML-decoding complexity), making their use prohibitive in practice.The class of single-symbol decodable codes also includes the codes constructed using co-ordinate interleaving, called co-ordinate interleaved orthogonal designs (CIODs) [5], and the Clifford-UnitaryWeight single-symbol decodable designs (CUW-SSD) [6]. These designs allow a symbol rate higher than that of the orthogonal designs, although not as much as that provided by the codes from division algebra.The disadvantage with these codes when compared with the Orthogonal designs is that they have full transmit diversity for only specific complex constellations.The Golden code [7], developed from divi...

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Abstract-In this paper, we deal with low-complexity near-optimal detection/equalization in large-dimension multiple-input multiple-output inter-symbol interference (MIMO-ISI) channels using message passing on graphical models. A key contribution in the paper is the demonstration that near-optimal performance in MIMO-ISI channels with large dimensions can be achieved at low complexities through simple yet effective simplifications/approximations, although the graphical models that represent MIMO-ISI channels are fully/densely connected (loopy graphs). These include 1) use of Markov random field (MRF)-based graphical model with pairwise interaction, in conjunction with message damping, and 2) use of factor graph (FG)-based graphical model with Gaussian approximation of interference (GAI). The per-symbol complexities are ( 2 2 ) and ( ) for the MRF and the FG with GAI approaches, respectively, where and denote the number of channel uses per frame, and number of transmit antennas, respectively. These low-complexities are quite attractive for large dimensions, i.e., for large . From a performance perspective, these algorithms are even more interesting in large-dimensions since they achieve increasingly closer to optimum detection performance for increasing . Also, we show that these message passing algorithms can be used in an iterative manner with local neighborhood search algorithms to improve the reliability/performance of -QAM symbol detection.Index Terms-Factor graphs, graphical models, large dimensions, low-complexity detection, Markov random fields, multiple-input multiple-output inter-symbol interference (MIMO-ISI) channels, pairwise interaction, severe delay spreads.