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27
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.
DMT optimality of LRaided linear decoders for a general class of channels, lattice designs, and system models
 IEEE Trans. Infom. Theory
, 2010
"... Abstract—The work identifies the first general, explicit, and nonrandom MIMO encoderdecoder structures that guarantee optimality with respect to the diversitymultiplexing tradeoff (DMT), without employing a computationally expensive maximumlikelihood (ML) receiver. Specifically, the work establi ..."
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Cited by 33 (4 self)
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Abstract—The work identifies the first general, explicit, and nonrandom MIMO encoderdecoder structures that guarantee optimality with respect to the diversitymultiplexing tradeoff (DMT), without employing a computationally expensive maximumlikelihood (ML) receiver. Specifically, the work establishes the DMT optimality of a class of regularized lattice decoders, and more importantly the DMT optimality of their latticereduction (LR)aided linear counterparts. The results hold for all channel statistics, for all channel dimensions, and most interestingly, irrespective of the particular latticecode applied. As a special case, it is established that the LLLbased LRaided linear implementation of the MMSEGDFE lattice decoder facilitates DMT optimal decoding of any lattice code at a worstcase complexity that grows at most linearly in the data rate. This represents a fundamental reduction in the decoding complexity when compared to ML decoding whose complexity is generally exponential in rate. The results ’ generality lends them applicable to a plethora of pertinent communication scenarios such as quasistatic MIMO, MIMOOFDM, ISI, cooperativerelaying, and MIMOARQ channels, in all of which the DMT optimality of the LRaided linear decoder is guaranteed. The adopted approach yields insight, and motivates further study, into joint transceiver designs with an improved SNR gap to ML decoding. Index Terms—Diversitymultiplexing tradeoff, lattice decoding, linear decoding, lattice reduction, regularization, multipleinput multipleoutput (MIMO), spacetime codersdecoders. I.
DIVISION ALGEBRAS AND WIRELESS COMMUNICATION
"... The aim of this note is to bring to the attention of a wide mathematical audience the recent application of division algebras to wireless communication. The application occurs in the context of communication involving multiple transmit and receive antennas, a context known in engineering as MIMO, sh ..."
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The aim of this note is to bring to the attention of a wide mathematical audience the recent application of division algebras to wireless communication. The application occurs in the context of communication involving multiple transmit and receive antennas, a context known in engineering as MIMO, short for multiple input, multiple output. While the use of multiple receive antennas goes back to the time of Marconi, the basic theoretical framework for communication using multiple transmit antennas was only published about ten years ago. The progress in the field has been quite rapid, however, and MIMO communication is widely credited with being one of the key emerging areas in telecommunication. Our focus here will be on one aspect of this subject: the formatting of transmit information for optimum reliability. Recall that a division algebra is an (associative) algebra with a multiplicative identity in which every nonzero element is invertible. The center of a division algebra is the set of elements in the algebra that commute with every other element in the algebra; the center is itself just a commutative field, and the division algebra
Inverse Determinant Sums and Connections between Fading Channel Information Theory and Algebra
 IEEE Transactions on Information Theory
, 2013
"... Abstract—This work considers inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic spacetime block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse ..."
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Cited by 6 (2 self)
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Abstract—This work considers inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic spacetime block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversitymultiplexing gain tradeoff is investigated. It is proven that the growth of the inverse determinant sum of a division algebrabased spacetime code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recentmethods from ergodic theory, a complete classification of the inverse determinant sums of the most wellknown algebraic spacetime codes is provided. The approach reveals an interesting and tight relation between diversitymultiplexing gain tradeoff and point counting in Lie groups. Index Terms—Algebra, diversitymultiplexing gain tradeoff (DMT), division algebra, Lie groups, multipleinput multipleoutput (MIMO), number theory, spacetime block codes
New Space–Time Code Constructions for TwoUser Multiple Access Channels
"... Abstract—This paper addresses the problem of constructing multiuser multipleinput multipleoutput (MUMIMO) codes for two users. The users are assumed to be equipped with transmit antennas, and there are antennas available at the receiving end. A general scheme is proposed and shown to achieve the ..."
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Cited by 3 (2 self)
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Abstract—This paper addresses the problem of constructing multiuser multipleinput multipleoutput (MUMIMO) codes for two users. The users are assumed to be equipped with transmit antennas, and there are antennas available at the receiving end. A general scheme is proposed and shown to achieve the optimal diversitymultiplexing gain tradeoff (DMT). Moreover, an explicit construction for the special case of =2and =2is given, based on the optimization of the code shape and density. All the proposed constructions are based on cyclic division algebras and their orders and take advantage of the multiblock structure. Computer simulations show that both the proposed schemes yield codes with excellent performance improving upon the best previously known codes. Finally, it is shown that the previously proposed design criteria for DMT optimal MUMIMO codes are sufficient but in general too strict and impossible to fulfill. Relaxed alternative design criteria are then proposed and shown to be still sufficient for achieving the multipleaccess channel diversitymultiplexing tradeoff. Index Terms—Cyclic division algebras (CDAs), diversitymultiplexing gain tradeoff (DMT), multiple access channel (MAC), multipleinput multipleoutput (MIMO) channel, space–time block codes (STBCs). I.
Higher dimensional perfect spacetime coded modulation”, ITW 09
"... dimensional perfect spacetime coded modulation ..."
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J.V.: Arithmetic Fuchsian groups and space– time block codes
 Comput. Appl. Math
, 2011
"... Abstract. In the context of space time block codes (STBCs) the theory of arithmetic Fuchsian groups is presented. Additionally, in this work we present a new class of STBCs based on arithmetic Fuchsian groups. This new class of codes satisfies the property fulldiversity, linear dispersion and full ..."
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Abstract. In the context of space time block codes (STBCs) the theory of arithmetic Fuchsian groups is presented. Additionally, in this work we present a new class of STBCs based on arithmetic Fuchsian groups. This new class of codes satisfies the property fulldiversity, linear dispersion and fullrate.
Golden Space–Time BlockCoded Modulation
"... Abstract—In this paper, blockcoded modulation is used to design a 2 2 2 multipleinput multipleoutput (MIMO) space–time code for slow fading channels. The Golden Code is chosen as the inner code; the scheme is based on a set partitioning of the Golden Code using twosided ideals whose norm is a po ..."
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Abstract—In this paper, blockcoded modulation is used to design a 2 2 2 multipleinput multipleoutput (MIMO) space–time code for slow fading channels. The Golden Code is chosen as the inner code; the scheme is based on a set partitioning of the Golden Code using twosided ideals whose norm is a power of two. In this case, a lower bound for the minimum determinant is given by the minimum Hamming distance. The description of the ring structure of the quotients suggests further optimization in order to improve the overall distribution of determinants. Simulation results show that the proposed schemes achieve a significant gain over the uncoded Golden Code.
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.