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FullDiversity, HighRate SpaceTime Block Codes from Division Algebras
 IEEE TRANS. INFORM. THEORY
, 2003
"... We present some general techniques for constructing fullrank, minimaldelay, rate at least one spacetime 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 algeb ..."
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Cited by 177 (55 self)
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We present some general techniques for constructing fullrank, minimaldelay, rate at least one spacetime 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.
Perfect SpaceTime Codes for Any Number of Antennas
"... In a recent paper, perfect (n × n) spacetime codes were introduced as the class of linear dispersion spacetime codes having full rate, nonvanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted en ..."
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Cited by 37 (3 self)
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In a recent paper, perfect (n × n) spacetime codes were introduced as the class of linear dispersion spacetime codes having full rate, nonvanishing determinant, a signal constellation isomorphic to either the rectangular or hexagonal lattices in 2n 2 dimensions and uniform average transmitted energy per antenna. Consequence of these conditions include optimality of perfect codes with respect to the ZhengTse DiversityMultiplexing Gain tradeoff (DMT), as well as excellent lowSNR performance. Yet perfect spacetime codes have been constructed only for 2, 3, 4 and 6 transmit antennas. In this paper, we construct perfect codes for all channel dimensions, present some additional attributes of this class of spacetime codes and extend the notion of a perfect code to the rectangular case.
Perfect spacetime codes with minimum and nonminimum delay for any number of antennas
 IEEE Trans. Inform. Theory
, 2005
"... Abstract — Perfect spacetime codes were first introduced by Oggier et. al. to be the spacetime codes that have full rate, full diversitygain, nonvanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. These d ..."
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Cited by 26 (8 self)
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Abstract — Perfect spacetime codes were first introduced by Oggier et. al. to be the spacetime codes that have full rate, full diversitygain, nonvanishing determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping of the constellation. These defining conditions jointly correspond to optimality with respect to the ZhengTse DMG tradeoff, independent of channel statistics, as well as to near optimality in maximizing mutual information. All the above traits endow the code with error performance that is currently unmatched. Yet perfect spacetime codes have been constructed only for 2, 3,4 and 6 transmit antennas. We construct minimum and nonminimum delay perfect codes for all channel dimensions. A. Definition of Perfect Codes I.
InformationLossless SpaceTime Block Codes from CrossedProduct Algebras
 IEEE TRANS. INFORM. THEORY
, 2006
"... 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, callin ..."
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Cited by 17 (11 self)
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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 informationlossless STBC. In this paper, using crossedproduct central simple algebras, we construct STBCs for arbitrary number of transmit antennas over an a priori specified signal set. Alamouti code and quasiorthogonal designs are the simplest special cases of our constructions. We obtain a condition under which these STBCs from crossedproduct algebras are informationlossless. We give some classes of crossedproduct algebras, from which the STBCs obtained are informationlossless 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.
STBCs using Capacity Achieving Designs from CrossedProduct Division Algebras
 in ICC 2004
, 2004
"... We construct fullrank, raten SpaceTime Block Codes (STBC), over any a priori specified signal set for n transmit antennas using crossedproduct division algebras and give a sufficient condition for these STBCs to be information lossless. A class of division algebras for which this sufficient con ..."
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Cited by 7 (2 self)
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We construct fullrank, raten SpaceTime Block Codes (STBC), over any a priori specified signal set for n transmit antennas using crossedproduct division algebras and give a sufficient condition for these STBCs to be information lossless. A class of division algebras for which this sufficient condition is satisfied is identified. Simulation results are presented to show that STBCs constructed in this paper perform better than the best known codes, including those constructed from cyclic division algebras and also to show that they are very close to the capacity of the channel with QAM input.
AsymptoticInformationLossless Designs and the Diversity–Multiplexing Tradeoff
, 2009
"... It is known that neither the Alamouti nor the VBLAST scheme achieves the Zheng–Tse diversity–multiplexing tradeoff (DMT) of the multipleinput multipleoutput (MIMO) channel. With respect to the DMT curve, the Alamouti scheme achieves the point corresponding to maximum diversity gain only, whereas ..."
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It is known that neither the Alamouti nor the VBLAST scheme achieves the Zheng–Tse diversity–multiplexing tradeoff (DMT) of the multipleinput multipleoutput (MIMO) channel. With respect to the DMT curve, the Alamouti scheme achieves the point corresponding to maximum diversity gain only, whereas VBLAST meets only the point corresponding to maximum multiplexing gain. It is also known that DBLAST achieves the optimal DMT for n transmit and n receive antennas, but only under the assumption that the leading and trailing zeros are ignored. When these zeros are taken into account, DBLAST achieves the point corresponding to zero multiplexing gain, but not the point corresponding to zero diversity gain. The first scheme to achieve the DMT is the coding scheme of Yao and Wornell for the case of two transmit and two receive antennas. In this paper, we introduce the notion of an asymptoticinformationlossless (AILL) design and obtain a necessary and sufficient condition under which a design is AILL. Analogous to the result that fullrank designs achieve the point corresponding to the zero multiplexing gain of the optimal DMT curve, we show AILL to be a necessary and sufficient condition for a design to achieve the point on the DMT curve corresponding to zero diversity gain. We also derive a lower bound on the tradeoff achieved by designs from field extensions and show that the tradeoff is very close to the optimal tradeoff in the case of a single receive antenna. A lower bound to the tradeoff achieved by designs from division algebras is presented which indicates that these designs achieve both extreme points (corresponding to zero diversity and zero multiplexing gain) of the optimal DMT curve. Finally, we present simulations results for n transmit and n receive antennas, for n = 2; 3; 4, which suggest that designs from division algebras are likely to have the property of being DMT achieving.
AsymptoticInformationLossless Designs and DiversityMultiplexing tradeoff
"... Abstract — It is well known that in the ZhengTse optimal diversitymultiplexing tradeoff curve, the Alamouti scheme meets the point corresponding to the maximum diversity gain only, whereas VBLAST meets only the point corresponding to the maximum multiplexing gain. In this paper, we define Asympto ..."
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Abstract — It is well known that in the ZhengTse optimal diversitymultiplexing tradeoff curve, the Alamouti scheme meets the point corresponding to the maximum diversity gain only, whereas VBLAST meets only the point corresponding to the maximum multiplexing gain. In this paper, we define AsymptoticInformationLossless (AILL) designs and obtain a necessary and sufficient condition under which a design is AILL. Analogous to the condition that fullrank designs achieve the point corresponding to the zero multiplexing gain of the optimal tradeoff, we show that it is a necessary and sufficient condition for a design to be AILL to achieve the point corresponding to the zero diversity gain of the optimal tradeoff curve. Also, we obtain a lower bound on the tradeoff achieved by the designs from field extensions and division algebras. The lower bound for the designs from division algebras indicates that they achieve both