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For the quasi-static, Rayleigh-fading multiple-input multiple-output (MIMO) channel with [n.sub.t] transmit and [n.sub.r], receive antennas, Zheng and Tse showed that there exists a fundamental tradeoff between diversity and spatial-muitiplexing gains, referred to as the diversity-multiplexing gain (D-MG) tradeoff. Subsequently, El Gamal, Caire, and Damen considered signaling across the same channel using an L-round automatic retransmission request (ARQ) protocol that assumes the presence of a noiseless feedback channel capable of conveying one bit of information per use of the feedback channel. They showed that given a fixed number L of ARQ rounds and no power control, there is a tradeoff between diversity and multiplexing gains, termed the diversity-multiplexing-delay (DMD) tradeoff. This tradeoff indicates that the diversity gain under the ARQ scheme for a particular information rate is considerably larger than that obtainable in the absence of feedback. In this paper, a set of sufficient conditions under which a space-time (ST) code will achieve the DMD tradeoff is presented. This is followed by two classes of explicit constructions of ST codes which meet these conditions. Constructions belonging to the first class achieve minimum delay and apply to a broad class of fading channels whenever [n.sub.r], [greater than or equal to] [n.sub.t] and either Ll [n.sub.t] or [n.sub.t] . The second class of constructions do not achieve minimum delay, but do achieve the DMD tradeoff of the fading channei for all statistical descriptions of the channel and for all values of the parameters Index Terms--Automatic retransmission request (ARQ), cyclic division algebra, diversity-multiplexing-delay (DMD) tradeoff, diversity-multiplexing gain(D-MG) tradeoff, explicit construction, multiple-input multiple-output (MIMO) feedback, space-time (ST) codes.

In a recent paper, perfect (n x n) space-time codes were introduced as the class of linear dispersion space-time (ST) codes having full rate, nonvanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2[n.sup.2] dimensions, and uniform average transmitted energy per antenna. Consequence of these conditions include optimality of perfect codes with respect to the Zheng-Tse diversity-multiplexing gain tradeoff (DMT), as well as excellent low signal-to-noise ratio (SNR) performance. Yet perfect space-time codes have been constructed only for two, three, four, and six transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of ST codes, and extend the notion of a perfect code to the rectangular case. Index Terms--Diversity-multiplexing tradeoff (DMT), division algebras, multiple-input multiple-output (MIMO), perfect space-time (ST) codes. Digital Object Identifier 10.1109/TIT.2007.907502

It is known that the Alamouti code is the only complex orthogonal design (COD) which achieves capacity and that too for the case of two transmit and one receive antenna only. Damen et al. proposed a design for two transmit antennas, which achieves capacity for any number of receive antennas, calling the resulting space--time block code (STBC) when used with a signal set an information-lossless STBC. In this paper, using crossed-product central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an a priori specified signal set. Alamouti code and quasi-orthogonal designs are the simplest special cases of our constructions. We obtain a condition under which these STBCs from crossed-product algebras are information-lossless. We give some classes of crossed-product algebras, from which the STBCs obtained are information-lossless and also of full rank. We present some simulation results for two, three, and four transmit antennas to show that our STBCs perform better than some of the best known STBCs and also that these STBCs are approximately 1 dB away from the capacity of the channel with quadrature amplitude modulation (QAM) symbols as input. Index Terms--Division algebras, information-lossless codes, multiple-input multiple-output (MIMO), space-time block codes.

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 Q embedded in matrix rings. The first half of the paper deals with constructions using field extensions of Q. 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 x 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 nth 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. Index Terms--Division algebras, mutual information, space-time block codes (STBCs), transmit diversity, wireless communications.