on optimal multilayer cyclotomic space time code designs 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 lattice based 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 single layer 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 single layer space time codes over general cyclotomic number rings for a general number of transmit antennas we then present optimal multilayer full rate cyclotomic space time code designs for two and three transmit antennas we also present an optimal two layer 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

the golden code a 2 spl times 2 full rate space time code with nonvanishing determinants in this paper the golden code for a 2 spl times 2 multiple input multiple output mimo system is presented this is a full rate 2 spl times 2 linear dispersion algebraic space time code with unprecedented performance based on the golden number 1 spl radic 5 2

fast decodable asymmetric space time codes from division algebras multiple input double output mido codes are important in the near future wireless communications where the portable end user device is physically small and will typically contain at most two receive antennas especially tempting is the 4 2 channel due to its immediate applicability in the digital video broadcasting dvb such channels optimally employ rate two space time st codes consisting of 4 4 matrices unfortunately such codes are in general very complex to decode hence setting forth a call for constructions with reduced complexity recently some reduced complexity constructions have been proposed but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes in this paper it will be shown that a family of division algebra based mido codes will always result in at least 37 5 worst case complexity reduction while maintaining full diversity and for the first time the nonvanishing determinant nvd property the reduction follows from the fact that similarly to the alamouti code the codes will be subsets of matrix rings of the hamiltonian quaternions hence allowing simplified decoding at the moment such reductions are among the best known for rate two mido codes 5 6 several explicit constructions are presented and shown to have excellent performance through computer simulations simulations

constructions of multiblock space time coding schemes that achieve the diversity multiplexing tradeoff constructions of multiblock space time coding schemes that are optimal with respect to diversity multiplexing d m tradeoff when coding is applied over any number of fading blocks are presented in this correspondence the constructions are based on a left regular representation of elements in some cyclic division algebra in particular the main construction applies to the case when the quasi static fading interval equals the number of transmit antennas hence the resulting scheme is termed a minimal delay multiblock space time coding scheme constructions corresponding to the cases of nonminimal delay are also provided as the number of coded blocks approaches infinity coding schemes derived from the proposed constructions can be used to provide a reliable multiple input multiple output mimo communication with vanishing error probability

on the numerical modelling of quasi static crack growth in linear elastic fracture mechanics in this paper we present a strategy for the simulation of a propagating crack under mixed mode linear elastic conditions using a discontinuous finite element method a key issue is to accurately compute the incremental change in the kink angle of the propagating crack during subsequent steps we have chosen to work with a domain formulation of the material force vector as a criteria for the propagation direction we describe the theoretical background together with the numerical implementation in detail and show some results for different loading conditions copyright 2005 john wiley sons ltd

generalized silver codes for an nt transmit nr receive antenna system nt nr system a full rate space time block code stbc transmits at least nmin min nt nr complex symbols per channel use the well known golden code is an example of a full rate full diversity stbc for two transmit antennas its ml decoding complexity is of the order of m2 5 for square m qam the silver code for two transmit antennas has all the desirable properties of the golden code except its coding gain but offers lower ml decoding complexity of the order of m2 importantly the slight loss in coding gain is negligible compared to the advantage it offers in terms of lowering the ml decoding complexity for higher number of transmit antennas the best known codes are the perfect codes which are full rate full diversity information lossless codes for nr nt but have a high ml decoding complexity of the order of mntnmin for nr nt the punctured perfect codes are considered in this paper a scheme to obtain full rate stbcs for 2a transmit antennas and any nr with reduced ml decoding complexity of the order of mnt nmin 3 4 0 5 is presented the codes constructed are also information lossless for nr nt like the perfect codes and allow higher mutual information than the comparable punctured perfect codes for nr nt these codes are referred to as the generalized silver codes since they enjoy the same desirable properties as the comparable perfect codes except possibly the coding gain with lower ml decoding complexity analogous to the silver code and the golden code for two transmit antennas simulation results of the symbol error rates for four and eight transmit antennas show that the generalized silver codes match the punctured perfect codes in error performance while offering lower ml decoding complexity

a construction of a space time code based on number theory we construct a full data rate space time st block code over m 2 transmit antennas and t 2 symbol periods and we prove that it achieves a transmit diversity of 2 over all constellations carved from z i sup 4 further we optimize the coding gain of the proposed code and then compare it to the alamouti code it is shown that the new code outperforms the alamouti see ieee j select areas commun vol 16 p 1451 58 1998 code at low and high signal to noise ratio snr when the number of receive antennas n 1 the performance improvement is further enhanced when n or the size of the constellation increases we relate the problem of st diversity gain to algebraic number theory and the coding gain optimization to the theory of simultaneous diophantine approximation in the geometry of numbers we find that the coding gain optimization is equivalent to finding irrational numbers the furthest from any simultaneous rational approximations

the coding gain of real matrix lattices bounds and existence results the paper considers the question of the normalized minimum determinant or asymptotic coding gain of real matrix lattices the coding theoretic motivation for such study arises for example from the questions considering multiple input multiple output mimo ultra wideband uwb transmission at the beginning totally general coding gain bounds for real mimo lattice codes is given by translating the problem into geometric language then code lattices that are produced from division algebras are considered by applying methods from the theory of central simple algebras coding gain bounds for code lattices coming from orders of division algebras are given finally it is proven that these bounds can be reached by using maximal orders in the case of 2 2 matrix lattices this existence result proves that the general geometric bound derived earlier can be reached

on the densest mimo lattices from cyclic division algebras it is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant using results from class field theory a lower bound to the minimum discriminant of a maximal order with a given center and index the number of tx rx antennas is derived also numerous examples of division algebras achieving the bound are given for example a matrix lattice with quadrature amplitude modulation qam coefficients that has 2 5 times as many codewords as the celebrated golden code of the same minimum determinant is constructed also a general algorithm due to ivanyos and ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed

full diversity high rate space time block codes from division algebras 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 spl qopf embedded in matrix rings the first half of the paper deals with constructions using field extensions of spl qopf 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 spl times 4 real orthogonal design is obtained by the left regular representation of quaternions alamouti s 1998 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 1933 is discussed and several examples of stbcs derived from each of these constructions are presented

an enhanced dmt optimality criterion for stbc schemes for asymmetric mimo systems for any nt transmit nr receive antenna nt nr multiple input multiple output mimo system in a quasi static rayleigh fading environment it was shown by elia that linear space time block code schemes lstbc schemes that have the nonvanishing determinant nvd property are diversity multiplexing gain tradeoff dmt optimal for arbitrary values of nr if they have a code rate of nt complex dimensions per channel use however for asymmetric mimo systems where with the exception of a few lstbc schemes it is unknown whether general lstbc schemes with nvd and a code rate of nr complex dimensions per channel use are dmt optimal in this paper an enhanced sufficient criterion for any stbc scheme to be dmt optimal is obtained and using this criterion it is established that any lstbc scheme with nvd and a code rate of min nt nr complex dimensions per channel use is dmt optimal this result settles the dmt optimality of several well known low ml decoding complexity lstbc schemes for certain asymmetric mimo systems

euclidean and space time block codes relationship optimality performance analysis revisited an equivalent model for a multiple input multiple output communication system with space time block codes stbcs is proposed based on a revealed connection between stbcs and euclidean codes examples of distance spectra signal constellations and signal coordinate diagrams of euclidean codes equivalent to simplest orthogonal stbcs are given a new asymptotic upper bound for the symbol error rate ser of stbcs based on the distance spectra of the equivalent euclidean codes is derived and new general design criteria for signal constellations of the optimal code are proposed some bounds relating distance properties dimensionality and cardinality of stbcs with constituent signals of equal energy are given and new signal constellations with cardinalities of 8 and 16 for alamouti s code are designed a general methodology for performance analysis of stbcs is revisited as an example of the application of this methodology an exact evaluation of the ser of an orthogonal stbc is given namely a new expression for the ser of alamouti s code with binary phase shift keying signals is derived

application of material forces to hyperelastostatic fracture mechanics ii computational setting the concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics to this end we consider the continuum mechanics of material forces as outlined in part i of this work p steinmann int j solid struct 37 7371 7391 which act contrary to the common physical forces on the material manifold or rather in the material space in the sequel it is proposed to discretize the corresponding quasi static balance of pseudo momentum by a standard galerkin finite element procedure as a result we obtain global discrete node point quantities the material node point forces which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical j integral

perfect space time codes for any number of antennas in a recent paper perfect n times 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 2n 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

almost universal codes achieving ergodic mimo capacity within a constant gap this paper addresses the question of achieving capacity with lattice codes in multi antenna block fading channels when the number of fading blocks tends to infinity a design criterion based on the normalized minimum determinant is proposed for division algebra multi block space time codes over fading channels this plays a similar role to the hermite invariant for gaussian channels under maximum likelihood decoding it is shown that this criterion is sufficient to guarantee transmission rates within a constant gap from capacity both for deterministic channels and ergodic fading channels moreover if the number of receive antennas is greater than or equal to the number of transmit antennas the same constant gap is achieved under naive lattice decoding as well in the case of independent identically distributed rayleigh fading the error probability vanishes exponentially fast in contrast to the standard approach in the literature which employs random lattice ensembles the existence results in this paper are derived from the number theory first the gap to capacity is shown to depend on the discriminant of the chosen division algebra then class field theory is applied to build families of algebras with small discriminants the key element in the construction is the choice of a sequence of division algebras whose centers are number fields with small root discriminants

optimal diversity multiplexing tradeoff and code constructions of some constrained asymmetric mimo systems in multiple input multiple output mimo communications the notion of asymmetric channel refers to the situation when the number of transmit antennas is strictly larger than the number of receive antennas such channels can often be found in mimo downlink transmissions while existing cyclic division algebra cda based codes can still be employed to achieve the optimal diversity multiplexing tradeoff dmt at high signal to noise ratio snr regime such codes cannot be directly decoded using for example the pure sphere decoding method although other means of decoding methods such as minimum mean square error generalized decision feedback equalizer mmse gdfe with lattice search and regularized lattice decoding are available an alternative approach is to constrain the number of active transmit antennas in each channel use to be no larger than the number of receive antennas the resulting system is coined constrained asymmetric mimo system two general types of asymmetrical channels are considered in this paper namely 1 when there are two receive antennas and the number of transmit antennas is arbitrary and 2 when the number of transmit antennas is one larger than the number of receive antennas explicit optimal transmission schemes as well as the corresponding code constructions for such constrained asymmetric mimo channels are presented and are shown to achieve the same dmt performance as their unconstrained counterparts

space time codes from structured lattices we present constructions of space time st codes based on lattice coset coding first we focus on st code constructions for the short block length case i e when the block length is equal to or slightly larger than the number of transmit antennas we present constructions based on dense lattice packings and nested lattice voronoi shaping our codes achieve the optimal diversity multiplexing tradeoff dmt of quasi static multiple input multiple output mimo fading channels for any fading statistics and perform very well also at practical moderate values of signal to noise ratios snr then we extend the construction to the case of large block lengths by using trellis coset coding we provide constructions of trellis coded modulation tcm schemes that are endowed with good packing and shaping properties both short block and trellis constructions allow for a reduced complexity decoding algorithm based on minimum mean squared error generalized decision feedback equalizer mmse gdfe lattice decoding and a combination of this with a viterbi tcm decoder for the tcm case beyond the interesting algebraic structure we exhibit codes whose performance is among the state of the art considering codes with similar encoding decoding complexity

fast decodable mido codes with large coding gain in this paper a new method is proposed to obtain full diversity rate 2 rate of two complex symbols per channel use space time block codes stbcs that are full rate for multiple input double output mido systems using this method rate 2 stbcs for 4 2 6 2 8 2 and 12 2 systems are constructed and these stbcs are fast ml decodable have large coding gains and stbc schemes consisting of these stbcs have a non vanishing determinant nvd so that they are dmt optimal for their respective mido systems it is also shown that the srinath rajan code for the 4 2 system which has the lowest ml decoding complexity among known rate 2 stbcs for the 4 2 mido system with a large coding gain for 4 16 qam has the same algebraic structure as the stbc constructed in this paper for the 4 2 system this also settles in positive a previous conjecture that the stbc scheme that is based on the srinath rajan code has the nvd property and hence is dmt optimal for the 4 2 system

construction methods for asymmetric and multiblock space time codes in this paper the need for the construction of asymmetric and multiblock space time codes is discussed above the trivial puncturing method i e switching off the extra layers in the symmetric multiple input multiple output mimo setting two more sophisticated asymmetric construction methods are proposed the first method called the block diagonal method bdm can be converted to produce multiblock space time codes that achieve the diversity multiplexing tradeoff dmt it is also shown that maximizing the density of the newly proposed block diagonal asymmetric space time ast codes is equivalent to minimizing the discriminant of a certain order a result that also holds as such for the multiblock codes an implicit lower bound for the density is provided and made explicit for an important special case that contains e g the systems equipped with 4tx 2rx antennas further an explicit scheme achieving the bound is given another method proposed here is the smart puncturing method spm that generalizes the subfield construction method proposed in earlier work by hollanti and ranto and applies to any number of transmitting and lesser receiving antennas the use of the general methods is demonstrated by building explicit sphere decodable codes using different cyclic division algebras cdas computer simulations verify that the newly proposed methods can compete with the trivial puncturing method and in some cases clearly outperform it the conquering construction exploiting maximal orders improves upon the punctured perfect code and the djabba code as well as the icosian code also extensive dmt analysis is provided

information lossless space time block codes from crossed product algebras 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 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

stbc schemes with nonvanishing determinant for certain number of transmit antennas a space time block code scheme stbc scheme is a family of stbcs c snr indexed by the signal to noise ratio snr such that the rate of each stbc scales with snr an stbc scheme is said to have a nonvanishing determinant if the coding gain of every stbc in the scheme is lower bounded by a fixed nonzero value the nonvanishing determinant property is important from the perspective of the diversity multiplexing gain dm g tradeoff a concept that characterizes the maximum diversity gain achievable by any stbc scheme transmitting at a particular rate this correspondence presents a systematic technique for constructing stbc schemes with nonvanishing determinant based on cyclic division algebras prior constructions of stbc schemes from cyclic division algebra have either used transcendental elements in which case the scheme may have vanishing determinant or is available with nonvanishing determinant only for two three four and six transmit antennas in this correspondence we construct stbc schemes with nonvanishing determinant for the number of transmit antennas of the form 2 sup k 3 spl middot 2 sup k 2 spl middot 3 sup k and q sup k q 1 2 where q is any prime of the form 4s 3 for cyclic division algebra based stbc schemes in a recent work by elia et al the nonvanishing determinant property has been shown to be sufficient for achieving dm g tradeoff in particular it has been shown that the class of stbc schemes constructed in this correspondence achieve the optimal dm g tradeoff moreover the results presented in this correspondence have been used for constructing optimal stbc schemes for arbitrary number of transmit antennas by elia et al

perfect space time block codes in this paper we introduce the notion of perfect space time block codes stbcs these codes have full rate full diversity nonvanishing constant minimum determinant for increasing spectral efficiency uniform average transmitted energy per antenna and good shaping we present algebraic constructions of perfect stbcs for 2 3 4 and 6 antennas

maximal orders in the design of dense space time lattice codes in this paper we construct explicit rate one full diversity geometrically dense matrix lattices with large nonvanishing determinants nvds for four transmit antenna multiple input single output miso space time st applications the constructions are based on the theory of rings of algebraic integers and related subrings of the hamiltonian quaternions and can be extended to a larger number of tx antennas the usage of ideals guarantees an nvd larger than one and an easy way to present the exact proofs for the minimum determinants the idea of finding denser sublattices within a given division algebra is then generalized to a multiple input multiple output mimo case with an arbitrary number of tx antennas by using the theory of cyclic division algebras cdas and maximal orders it is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding related to the decoding complexity the notion of sensitivity is introduced and experimental evidence indicating a connection between sensitivity decoding complexity and performance is provided simulations in a quasi static rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated quasi orthogonal abba lattice as well as the diagonal algebraic space time dast lattice we also show that our quaternionic lattice is better than the dast lattice in terms of the diversity multiplexing gain tradeoff dmt

explicit space time codes achieving the diversity multiplexing gain tradeoff a recent result of zheng and tse states that over a quasi static channel there exists a fundamental tradeoff referred to as the diversity multiplexing gain d mg tradeoff between the spatial multiplexing gain and the diversity gain that can be simultaneously achieved by a space time st code this tradeoff is precisely known in the case of independent and identically distributed i i d rayleigh fading for tgesnt nr 1 where t is the number of time slots over which coding takes place and nt nr are the number of transmit and receive antennas respectively for t nt case we present two general techniques for building d mg optimal rectangular st codes from their square counterparts a byproduct of our results establishes that the d mg tradeoff for all tgesnt is the same as that previously known to hold for tgesnt n r 1

global gradient estimates for non quadratic vector valued parabolic quasi minimizers abstract we establish local and global higher integrability estimates for the gradient of vector valued parabolic quasi minimizers of integral functionals with polynomial p growth p 2 n n 2 on a parabolic domain t in r n 1 n 2 and under a weak regularity assumption on the parabolic boundary of t

a noncommutative analogue of the odlyzko bounds and bounds on performance for space time lattice codes this paper considers space time coding over several independently rayleigh faded blocks in particular we will concentrate on giving upper bounds for the coding gain of lattice space time codes as the number of blocks grow this problem was previously considered in the single antenna case by bayer fluckiger et al in 2006 crucial to their work was odlyzko s bound on the discriminant of an algebraic number field as this provides an upper bound for the normalized coding gain of number field codes in the mimo context natural codes are constructed from division algebras defined over number fields and the coding gain is measured by the discriminant of the corresponding noncommutative algebra in this paper we will develop analogues of the odlyzko bounds in this context and show how these bounds limit the normalized coding gain of a very general family of division algebra based space time codes these bounds can also be used as benchmarks in practical code design and as tools to analyze asymptotic bounds of performance as the number of independently faded blocks increases

diagonal algebraic space time block codes we construct a new family of linear space time st block codes by the combination of rotated constellations and the hadamard transform and we prove them to achieve the full transmit diversity over a quasi static or fast fading channels the proposed codes transmit at a normalized rate of 1 symbol s when the number of transmit antennas n 1 2 or n is a multiple of four we spread a rotated version of the information symbol vector by the hadamard transform and send it over n transmit antennas and n time periods for other values of n we construct the codes by sending the components of a rotated version of the information symbol vector over the diagonal of an n spl times n st code matrix the codes maintain their rate diversity and coding gains for all real and complex constellations carved from the complex integers ring z i and they outperform the codes from orthogonal design when using complex constellations for n 2 the maximum likelihood ml decoding of the proposed codes can be implemented by the sphere decoder at a moderate complexity it is shown that using the proposed codes in a multiantenna system yields good performances with high spectral efficiency and moderate decoding complexity