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34
On Optimal Multilayer Cyclotomic Space–Time Code Designs
, 2005
"... High rate and large diversity product (or coding advantage, or coding gain, or determinant distance, or minimum product distance) are two of the most important criteria often used for good space–time code designs. In recent (linear) latticebased space–time code designs, more attention is paid to t ..."
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Cited by 24 (7 self)
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High rate and large diversity product (or coding advantage, or coding gain, or determinant distance, or minimum product distance) are two of the most important criteria often used for good space–time code designs. In recent (linear) latticebased space–time code designs, more attention is paid to the high rate criterion but less to the large diversity product criterion. In this paper, we consider these two criteria together for multilayer cyclotomic space–time code designs. In a previous paper, we recently proposed a systematic cyclotomic diagonal space–time code design over a general cyclotomic number ring that has infinitely many designs for a fixed number of transmit antennas, where diagonal codes correspond to singlelayer codes in this paper. In this paper, we first propose a general multilayer cyclotomic space–time codes. We present a general optimality theorem for these infinitely many cyclotomic diagonal (or singlelayer) space–time codes over general cyclotomic number rings for a general number of transmit antennas. We then present optimal multilayer (fullrate) cyclotomic space–time code designs for two and three transmit antennas. We also present an optimal twolayer cyclotomic space–time code design for three and four transmit antennas. The optimality here is in the sense that, for a fixed mean transmission signal power, its diversity product is maximized, or equivalently, for a fixed diversity product, its mean transmission signal power is minimized. It should be emphasized that all the optimal multilayer cyclotomic space–time codes presented in this paper have the nonvanishing determinant property.
Systemetic and optimal cyclotomic lattices and diagonal spacetime block code designs
 IEEE Trans. Inform. Theory
, 2004
"... In this correspondence, a new and systematic design of cyclotomic lattices with full diversity is proposed by using some algebraic number theory. This design provides innitely many full diversity cyclotomic lattices for a given lattice size. Based on the packing theory and the concrete form of the ..."
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Cited by 10 (5 self)
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In this correspondence, a new and systematic design of cyclotomic lattices with full diversity is proposed by using some algebraic number theory. This design provides innitely many full diversity cyclotomic lattices for a given lattice size. Based on the packing theory and the concrete form of the design, optimal cyclotomic lattices are presented by minimizing the mean transmission signal power for a given minimum (diversity) product (or equivalently maximizing the minimum product for a given mean transmission signal power). The newly proposed cyclotomic lattices can be applied to both spacetime code designs for multiantenna systems and linear precode design for signal space diversity in single antenna systems over fast Rayleigh fading channels. Although there are some cyclotomic lattices/spacetime codes existed in the literature, most of them are not optimal.
An algebraic family of complex lattices for fading channels with application to spacetime codes
 IEEE Trans. Inform. Theory
, 2005
"... Abstract—A new approach is presented for the design of full modulation diversity (FMD) complex lattices for the Rayleighfading channel. The FMD lattice design problem essentially consists of maximizing a parameter called the normalized minimum product distance P of the finite signal set carved out o ..."
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Cited by 7 (1 self)
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Abstract—A new approach is presented for the design of full modulation diversity (FMD) complex lattices for the Rayleighfading channel. The FMD lattice design problem essentially consists of maximizing a parameter called the normalized minimum product distance P of the finite signal set carved out of the lattice. We approach the problem of maximizing P by minimizing the average energy of the signal constellation obtained from a new family of FMD lattices. The unnormalized minimum product distance for every lattice in the proposed family is lowerbounded by a nonzero constant. Minimizing the average energy of the signal set translates to minimizing the Frobenius norm of the generator matrices within the proposed family. The two strategies proposed for the Frobenius norm reduction are based on the concepts of successive minima (SM) and basis reduction of an equivalent real lattice. The lattice constructions in this paper provide significantly larger normalized minimum product distances compared to the existing lattices in certain dimensions. The proposed construction is general and works for any dimension as long as a list of number fields of the same degree is available. Index Terms—Algebraic number theory, energy minimization, fading channels, lattices, Lenstra–Lenstra–Lovász (LLL) algorithm, modulation diversity, number fields, product distance, signal space diversity, space–time coding, successive minima (SM). I.
MORDELLWEIL GROUPS AND THE RANK OF ELLIPTIC CURVES OVER LARGE FIELDS
, 2004
"... Abstract. Let K be a number field, K an algebraic closure of K and E/K an elliptic curve defined over K. In this paper, we prove that if E/K has a Krational point P such that 2P ̸ = O and 3P ̸ = O, then for each σ ∈ Gal(K/K), the MordellWeil group E(K σ) of E over the fixed subfield of K under σ h ..."
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Cited by 7 (1 self)
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Abstract. Let K be a number field, K an algebraic closure of K and E/K an elliptic curve defined over K. In this paper, we prove that if E/K has a Krational point P such that 2P ̸ = O and 3P ̸ = O, then for each σ ∈ Gal(K/K), the MordellWeil group E(K σ) of E over the fixed subfield of K under σ has infinite rank. 1.
Highrate Fullrank SpaceTime Block Codes from Cayley Algebra
 Intl Conf on Signal Proc and Comm (SPCOM 2004
, 2004
"... For a quasistatic, multipleinput multipleoutput (MIMO) fading channel, fulldiversity, highrate spacetime block codes (STBCs) for any number of transmit antennas have been constructed in [1] by using the regular matrix representation of an associative division algebra. While the 2 × 2 as well a ..."
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Cited by 5 (1 self)
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For a quasistatic, multipleinput multipleoutput (MIMO) fading channel, fulldiversity, highrate spacetime block codes (STBCs) for any number of transmit antennas have been constructed in [1] by using the regular matrix representation of an associative division algebra. While the 2 × 2 as well as 4×4 realorthogonal design (ROD) [2] and Alamouti code [3] were obtained as a special case of this construction, the 8 × 8 ROD could not be obtained. In this paper, starting with a nonassociative division algebra (Cayley algebra or more popularly known as octonion algebra) over an arbitrary characteristic zero field F, a method of embedding this algebra into the ring of matrices over F is described, and highrate fullrank STBCs for 8m (m, an arbitrary integer) antennas are obtained. We also give a closed form expression for the coding gain of these STBCs. This embedding when specialized to F = R and m = 1, gives the 8 × 8 ROD. 1.
Fast and slow solutions in general relativity: the initialization procedure
 J. Math. Phys
, 1998
"... We apply recent results in the theory of PDE, specifically in problems with two different time scales, on Einstein’s equations near their Newtonian limit. The results imply a justification to Postnewtonian approximations when initialization procedures to different orders are made on the initial data ..."
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Cited by 4 (0 self)
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We apply recent results in the theory of PDE, specifically in problems with two different time scales, on Einstein’s equations near their Newtonian limit. The results imply a justification to Postnewtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned the first Postnewtonian equations and their solutions in terms of Green functions are presented in order to indicate how to proceed in calculations with this approach. 1
Spacetime codes meeting the diversitymultiplexing gain tradeoff with low signalling complexity
 the 39th Annual CISS 2005 Conference on Information Sciences and
, 2005
"... In the recent landmark paper of Zheng and Tse it is shown that there exists a fundamental tradeoff between diversity gain and multiplexing gain, referred to as the DiversityMultiplexing gain(DMG) tradeoff. It is shown in [5] that ST codes with nonvanishing determinant (NVD) constructed from cycli ..."
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Cited by 4 (3 self)
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In the recent landmark paper of Zheng and Tse it is shown that there exists a fundamental tradeoff between diversity gain and multiplexing gain, referred to as the DiversityMultiplexing gain(DMG) tradeoff. It is shown in [5] that ST codes with nonvanishing determinant (NVD) constructed from cyclicdivisionalgebras (CDA), are optimal under the DMG tradeoff for any number nt, nr of transmit and receive antennas and with minimum delay T = nt. CDAbased ST codes with NVD have been previously constructed for restricted values of nt. This paper presents an explicit construction of spacetime (ST) codes for arbitrary number of transmit antennas that achieve the DMG tradeoff. A unified construction of DMG optimal CDAbased ST codes with NVD is given here, for any number nt of transmit antennas. Index Terms Spacetime codes, diversitymultiplexing gain (DMG) tradeoff, and algebraic number theory. I.
On the nontriviality of G(D) and the existence of maximal subgroups of GL1(D
 Journal of Algebra
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Maximal Diversity Algebraic Space–Time Codes With Low PeaktoMean Power Ratio
, 2005
"... The design requirements for space–time coding typically involves achieving the goals of good performance, high rates, and low decoding complexity. In this paper, we introduce a further constraint on space–time code design in that the code should also lead to low values of the peaktomean envelope p ..."
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Cited by 3 (0 self)
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The design requirements for space–time coding typically involves achieving the goals of good performance, high rates, and low decoding complexity. In this paper, we introduce a further constraint on space–time code design in that the code should also lead to low values of the peaktomean envelope power ratio (PMEPR) for each antenna. Towards that end, we propose a new class of space–time codes called the “low PMEPR space–time ” (LPST) codes. The LPST codes are obtained using the properties of certain cyclotomic number fields. The LPST codes achieve a performance identical to that of the threaded algebraic space–time (TAST) codes but at a much smaller PMEPR. With antennas and a rate of one symbol per channel use, the LPST codes lead to a decrease in PMEPR by at least a factor of relative to a Hadamard spread version of the TAST code. For rates beyond one symbol per channel use and up to a guaranteed amount, the LPST codes have provably smaller PMEPR than the corresponding TAST codes. Additionally, with the concept of punctured LPST codes proposed in this paper, significant performance improvement is obtained over the full diversity TAST schemes of comparable complexity. Numerical examples are provided to illustrate the advantage of the proposed codes in terms of PMEPR reduction and performance improvement for very high rate wireless communications.
Quantum Behaviors on an Excreting Black Hole
, 810
"... Often, geometries with horizons offer insights into the intricate relationships between general relativity and quantum physics. However, some subtle aspects of gravitating quantum systems might be difficult to ascertain using static backgrounds, since quantum mechanics incorporates dynamic measurabi ..."
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Cited by 2 (1 self)
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Often, geometries with horizons offer insights into the intricate relationships between general relativity and quantum physics. However, some subtle aspects of gravitating quantum systems might be difficult to ascertain using static backgrounds, since quantum mechanics incorporates dynamic measurability constraints (such as the uncertainty principle, etc.). For this reason, the behaviors of quantum systems on a dynamic black hole background are explored in this paper. The velocities and trajectories of representative outgoing, ingoing, and stationary classical particles are calculated and contrasted, and the dynamics of simple quantum fields (both massless and massive) on the spacetime are examined. Invariant densities associated with the quantum fields are exhibited on the Penrose diagram that represents the excreting black hole. Furthermore, a generic approach for the consistent mutual gravitation of quanta in a manner that reproduces the given geometry is developed. The dynamics of the mutually gravitating quantum fields are expressed in terms of the affine parameter that describes local motions of a given quantum type on the spacetime. Algebraic equations that relate the energymomentum densities of the quantum fields to Einstein’s tensor can then be developed. An example mutually gravitating system of macroscopically coherent quanta along with a core gravitating field is demonstrated. Since the approach is generic and algebraic, it can be used to represent a variety of systems with specified boundary conditions.