universal space time coding a universal framework is developed for constructing full rate and full diversity coherent space time codes for systems with arbitrary numbers of transmit and receive antennas the proposed framework combines space time layering concepts with algebraic component codes optimized for single input single output siso channels each component code is assigned to a thread in the space time matrix allowing it thus full access to the channel spatial diversity in the absence of the other threads diophantine approximation theory is then used in order to make the different threads transparent to each other within this framework a special class of signals which uses algebraic number theoretic constellations as component codes is thoroughly investigated the lattice structure of the proposed number theoretic codes along with their minimal delay allow for polynomial complexity maximum likelihood ml decoding using algorithms from lattice theory combining the design framework with the cayley transform allows to construct full diversity differential and noncoherent space time codes the proposed framework subsumes many of the existing codes in the literature extends naturally to time selective and frequency selective channels and allows for more flexibility in the tradeoff between power efficiency bandwidth efficiency and receiver complexity simulation results that demonstrate the significant gains offered by the proposed codes are presented in certain representative scenarios

lattice coding and decoding achieve the optimal diversity multiplexing tradeoff of mimo channels this paper considers communication over coherent multiple input multiple output mimo flat fading channels where the channel is only known at the receiver for this setting we introduce the class of lattice space time last codes we show that these codes achieve the optimal diversity multiplexing tradeoff defined by zheng and tse under generalized minimum euclidean distance lattice decoding our scheme is based on a generalization of erez and zamir mod lambda scheme to the mimo case in our construction the scalar scaling of erez zamir and costa gaussian dirty paper schemes is replaced by the minimum mean square error generalized decision feedback equalizer mmse gdfe this result settles the open problem posed by zheng and tse on the construction of explicit coding and decoding schemes that achieve the optimal diversity multiplexing tradeoff moreover our results shed more light on the structure of optimal coding decoding techniques in delay limited mimo channels and hence open the door for novel approaches for space time code constructions in particular 1 we show that mmse gdfe plays a fundamental role in approaching the limits of delay limited mimo channels in the high signal to noise ratio snr regime unlike the additive white gaussian noise awgn channel case and 2 our random coding arguments represent a major departure from traditional space time code designs based on the rank and or mutual information design criteria

approximation algorithms for three dimensional assignment problems with triangle inequalities consider the following classical formulation of the axial three dimensional assignment problem 3da see e g balas and saltzman 1989 given is a complete tripartite graph kn n n i j k i j i k j k where i j k are disjoint sets of size n and a cost c ijk for each triangle i j k i j k the problem 3da is to find a subset a of n triangles a i j k such that every element of i j k occurs in exactly one triangle of a and the total cost c a i j k ac ijk is minimized some recent references to this problem are balas and saltzman 1989 frieze 1974 frieze and yadegar 1981 hansen and kaufman 1973

optimal and lattice codes in a mimo system with complete feedback space time coding for a multiple input multiple output mimo system with complete feedback and maximum likelihood ml decoding is considered first a criterion for designing an optimal codebook with complete feedback in the case of high signal to noise ratio snr is given it turns out that designing an optimal codebook with complete feedback in the case of high snr is equivalent to efficiently packing a finite number of points into a dynamic ellipsoid in a finite dimensional real euclidean space second based on this design criterion an optimal tradeoff between diversity and multiplexing is then derived finally practical design using lattice codes is addressed and surprisingly it is shown that the lattice codes can achieve the optimal tradeoff it is also demonstrated that it is not necessary for an optimal lattice code to use all the subchannels generated by the system this is in striking contrast with a general optimal code which generally utilizes all the subchannels

optimal throughput diversity delay tradeoff in mimo arq block fading channels in this paper we consider an automatic repeat request arq retransmission protocol signaling over a block fading multiple input multiple output mimo channel unlike previous work we allow for multiple fading blocks within each transmission arq round and we constrain the transmitter to fixed rate codes constructed over complex signal constellations in particular we examine the general case of average input power constrained con stellations with a fixed signaling alphabet of finite cardinality this scenario is a suitable model for practical wireless communications systems employing orthogonal frequency division multiplexing ofdm techniques over a mimo arq channel two cases of fading dynamics are considered namely short term static fading where channel fading gains change randomly for each arq round and long term static fading where channel fading gains remain constant over all arq rounds pertaining to a given message as our main result we prove that for the block fading mimo arq channel with a fixed signaling alphabet satisfying a short term power constraint the optimal signal to noise ratio snr exponent is given by a modified singleton bound relating all the system parameters to demonstrate the practical significance of the theoretical analysis we present numerical results showing that practical singleton bound achieving maximum distance separable codes achieve the optimal snr exponent

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

asymptotic behavior of trigonometric series with o regularly varying quasimonotone coefficients ii abstract in this paper the asymptotic behavior of a trigonometric series with o regularly varying quasimonotone coefficients is investigated our results generalize the work done by s aljancic r bojanic and m tomic 1956 publ inst math beograd n s 10 101 120 r bojanic and j karamata 1963 tech rep 432 math research center madison wi g h hardy 1928 j london math soc 3 12 13 1931 proc london math soc 2 32 441 448 g h hardy and w w rogosinski 1945 quart j math oxford 16 49 58 j r nurcombe 1993 j math anal appl 178 63 69 c h yong 1969 publ inst math beograd n s 9 29 35 and a zygmund 1968 trigonometric series vol 1 2nd ed cambridge univ press cambridge uk

hybrid arq in multiple antenna slow fading channels performance limits and optimal linear dispersion code design this paper focuses on studying the fundamental performance limits and linear dispersion code design for the mimo arq slow fading channel the optimal average rate of well known harq protocols is analyzed the optimal design of space time coding for the mimo arq channel is discussed information theoretic measures are used to optimize the rate assignment and derive the optimum design criterion which is then used to evaluate the optimality of existing space time codes a different design criterion which is obtained from the error probability analysis of space time coded mimo harq is presented examples are studied to reveal the gain of arq feedback in space time coded mimo systems

an algorithm for the three index assignment problem we describe a branch and bound algorithm for solving the axial three index assignment problem the main features of the algorithm include a lagrangian relaxation that incorporates a class of facet inequalities and is solved by a modified subgradient procedure to find good lower bounds a primal heuristic based on the principle of minimizing maximum regret plus a variable depth interchange phase for finding good upper bounds and a novel branching strategy that exploits problem structure to fix several variables at each node and reduce the size of the total enumeration tree computational experience is reported on problems with up to 78 equations and 17 576 variables the primal heuristics were tested on problems with up to 210 equations and 343 000 variables

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

lattices which are good for almost everything we define an ensemble of lattices and show that for asymptotically high dimension most of its members are simultaneously good as sphere packings sphere coverings additive white gaussian noise awgn channel codes and mean squared error mse quantization codes these lattices are generated by applying construction a to a random linear code over a prime field of growing size i e by lifting the code to spl ropf sup n

progressive linear precoder optimization for mimo packet retransmissions exploiting channel covariance information this work investigates the design of linear precoders for arq packet retransmissions in multi input multi output mimo systems we consider transmitter precoder design based on partial mimo channel information in the form of their covariance feedback our objective is to maximize the ergodic mutual information provided by multiple re transmissions of a packet subject to transmission power constraint we propose a set of near optimal successive linear arq precoders for flat fading mimo channels this progressive linear arq precoder combines the appropriate power loading and the reverse order pairing of singular values in the current retransmission with previous transmissions this reverse order pairing is a special feature unique to our sequential arq preceding approach with demonstrated performance gains

trigonometric series with general monotone coefficients we study trigonometric series with general monotone coefficients convergence results in the different metrics are obtained also we prove a hardy type result on the behavior of the series near the origin

the mimo arq channel diversity multiplexing delay tradeoff in this paper the fundamental performance tradeoff of the delay limited multiple input multiple output mimo automatic retransmission request arq channel is explored in particular we extend the diversity multiplexing tradeoff investigated by zheng and tse in standard delay limited mimo channels with coherent detection to the arq scenario we establish the three dimensional tradeoff between reliability i e diversity throughput i e multiplexing gain and delay i e maximum number of retransmissions this tradeoff quantifies the arq diversity gain obtained by leveraging the retransmission delay to enhance the reliability for a given multiplexing gain interestingly arq diversity appears even in long term static channels where all the retransmissions take place in the same channel state furthermore by relaxing the input power constraint allowing variable power levels in different retransmissions we show that power control can be used to dramatically increase the diversity advantage our analysis reveals some important insights on the benefits of arq in slow fading mimo channels in particular we show that 1 allowing for a sufficiently large retransmission delay results in an almost flat diversity multiplexing tradeoff and hence renders operating at high multiplexing gain more advantageous 2 mimo arq channels quickly approach the ergodic limit when power control is employed finally we complement our information theoretic analysis with an incremental redundancy lattice space time ir last coding scheme which is shown through a random coding argument to achieve the optimal tradeoff s an integral component of the optimal ir last coding scheme is a list decoder based on the minimum mean square error mmse lattice decoding principle for joint error detection and correction throughout the paper our theoretical claims are validated by numerical results

convergence of fourier series with complex quasimonotone coefficients and coefficients of bounded variation of order m abstract l1 convergence theorems for fourier series with complex quasimonotone coefficients are generalized the class of null sequences of bounded variation of order m is extended and the point wise theorem for fourier series with such coefficients is proved

on maximum likelihood detection and the search for the closest lattice point maximum likelihood ml decoding algorithms for gaussian multiple input multiple output mimo linear channels are considered linearity over the field of real numbers facilitates the design of ml decoders using number theoretic tools for searching the closest lattice point these decoders are collectively referred to as sphere decoders in the literature in this paper a fresh look at this class of decoding algorithms is taken in particular two novel algorithms are developed the first algorithm is inspired by the pohst enumeration strategy and is shown to offer a significant reduction in complexity compared to the viterbo boutros sphere decoder the connection between the proposed algorithm and the stack sequential decoding algorithm is then established this connection is utilized to construct the second algorithm which can also be viewed as an application of the schnorr euchner strategy to ml decoding aided with a detailed study of preprocessing algorithms a variant of the second algorithm is developed and shown to offer significant reductions in the computational complexity compared to all previously proposed sphere decoders with a near ml detection performance this claim is supported by intuitive arguments and simulation results in many relevant scenarios

on communication over unknown sparse frequency selective block fading channels this paper considers the problem of reliable communication over discrete time channels whose impulse responses have length l and exactly s l non zero coefficients and whose support and coefficients remain fixed over blocks of n l channel uses but change independently from block to block here it is assumed that the channel s support and coefficient realizations are both unknown although their statistics are known assuming gaussian non zero coefficients and noise and focusing on the high snr regime it is first shown that the ergodic noncoherent channel capacity has pre log factor 1 s n for any l it is then shown that to communicate with arbitrarily small error probability at rates in accordance with the capacity pre log factor it suffices to use pilot aided orthogonal frequency division multiplexing ofdm with s pilots per fading block in conjunction with an appropriate noncoherent decoder since the achievability result is proven using a noncoherent decoder whose complexity grows exponentially in the number of fading blocks k a simpler decoder based on s 1 pilots is also proposed its e achievable rate is shown to have pre log factor equal to 1 s 1 n with the previously considered channel while its achievable rate is shown to have pre log factor 1 s 1 n when the support of the block fading channel remains fixed over time

on the average rate performance of hybrid arq in quasi static fading channels the problem of efficient communication over a scalar quasi static fading channel is considered the single layer transmission slt and multi layer transmission mlt schemes do not require any knowledge of the channel state information csi at the transmitter but their performance is also limited it is shown that using hybrid arq harq can significantly improve the average rate performance provided that the rate assignment between different arq rounds is carefully chosen the average rate performance of several harq schemes is optimized and compared in addition optimal power allocation among retransmissions is derived and shown to further increase the average rate this power allocation gain is remarkable at low signal to noise ratio snr but becomes negligible at high snr comparison of two different types of limited feedback sequential feedback arq and one shot feedback quantized csi is made from several perspectives although the optimization problem is formed with respect to the average rate simulation results give a comprehensive comparison under different metrics including average rate outage probability and the combination of both substantial performance improvement is observed with even one arq retransmission in all simulations more importantly this gain appears to be robust with respect to the fading distributions

a lagrangian relaxation algorithm for multidimensional assignment problems arising from multitarget tracking the central problem in multitarget tracking is the data association problem of partitioning the observations into tracks in some optimal way so that an accurate estimate of the true tracks can be recovered this work considers what is perhaps the simplest multitarget tracking problem in a setting where the issues are easily delineated i e straight lines in two dimensional space time with an error component introduced into the observations a multidimensional assignment problem is formulated using gating techniques to introduce sparsity into the problem and filtering techniques to generate tracks which are then used to score each assignment of a collection of observations to a filtered track problem complexity is further reduced by decomposing the problem into disjoint components which can then be solved independently a recursive lagrangian relaxation algorithm is developed to obtain high quality suboptimal solutions in real time the algorithms are however applicable to a large class of sparse mul

space time codes achieving the dmd tradeoff of the mimo arq channel for the quasi static rayleigh fading multiple input multiple output mimo channel with n t transmit and n r receive antennas zheng and tse showed that there exists a fundamental tradeoff between diversity and spatial multiplexing 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 nr ges nt and either l nt or nt l the second class of constructions do not achieve minimum delay but do achieve the dmd tradeoff of the fading channel for all statistical descriptions of the channel and for all values of the parameters nr nt l

l 1 approximation of fourier series of complex valued functions v b stanojevic suggested in her recent paper that it would be of interest to prove a corresponding l 1 convergence theorem for fourier series with complex o regularly varying quasimonotonc coefficients the present paper will discuss this question and establish l 1 convergence and furthermore l 1 approximation theorems for complex valued integrable functions

remarks on convergence of trigonometric series with special varying coefficients abstract we give an essential generalization of telyakovskii s results on the convergence of trigonometric series with rarely changing coefficients in the norm of l 1

on the required number of antennas in a point to point large but finite mimo system outage limited scenario this paper investigates the performance of the point to point multiple input multiple output mimo systems in the presence of a large but finite numbers of antennas at the transmitters and or receivers considering the cases with and without hybrid automatic repeat request harq feedback we determine the minimum numbers of the transmit receive antennas which are required to satisfy different outage probability constraints our results are obtained for different fading conditions and the effect of the power amplifiers efficiency feedback error probability on the performance of the mimo harq systems is analyzed then we use some recent results on the achievable rates of finite block length codes to analyze the effect of the codewords lengths on the system performance moreover we derive closed form expressions for the asymptotic performance of the mimo harq systems when the number of antennas increases our analytical and numerical results show that different outage requirements can be satisfied with relatively few transmit receive antennas

on the optimum diversity multiplexing tradeoff of the two user gaussian interference channel with rayleigh fading in this paper the optimum tradeoff between diversity and multiplexing gains in a two user quasi static rayleigh fading interference channel ic is studied the diversity and multiplexing gains are two basic performance measures in wireless networks which characterize the transmission reliability and the data rate respectively it would be of interest to investigate the optimal tradeoff between these two measures first we develop a coding scheme for the two user quasi static rayleigh fading gaussian ic with interference level log inr log snr 1 then for this coding scheme the achievable diversity multiplexing tradeoff dmt is characterized our achievable dmt coincides with its outer bound in the low and high rate regions to be defined later the proposed coding scheme is a one level gaussian code independent of the channel state information csi in the middle rate region to be defined later the proposed coding scheme depending on the partial csi can be a one level or a two level gaussian code we show that the relevant partial csi can be represented by only one bit determined by the absolute value of the channel gain in the middle rate region we assume that the single bit partial csi for all the four channel gains of the gaussian ic are available at both transmitters

uniform convergence of hankel transforms abstract we investigate necessary and or sufficient conditions for the pointwise and uniform convergence of the weighted hankel transforms l f r r 0 r t f t j r t d t 1 2 r 0 where r are such that 0 3 2 we subdivide these transforms into two classes in such a way that the uniform convergence criteria is remarkably different on each class in more detail we have the transforms satisfying 0 such as the classical hankel transform that generalize the cosine transform and those satisfying 0 3 2 generalizing the sine transform

coding and decoding for the dynamic decode and forward relay protocol we study the dynamic decode and forward ddf protocol for a single half duplex relay single antenna channel with quasi static fading the ddf protocol is well known and has been analyzed in terms of the diversity multiplexing tradeoff dmt in the infinite block length limit we characterize the finite block length dmt and give new explicit code constructions the finite block length analysis illuminates a few key aspects that have been neglected in the previous literature 1 we show that one dominating cause of degradation with respect to the infinite block length regime is the event of decoding error at the relay 2 we explicitly take into account the fact that the destination does not generally know a priori the relay decision time at which the relay switches from listening to transmit mode both of the above problems can be tackled by a careful design of the decoding algorithm in particular we introduce a decision rejection criterion at the relay based on forney s decision rule a variant of the neyman pearson rule such that the relay triggers transmission only when its decision is reliable also we show that a receiver based on the generalized likelihood ratio test glrt rule that jointly decodes the relay decision time and the information message achieves the optimal dmt our results show that no cyclic redundancy check crc for error detection or additional protocol overhead to communicate the decision time are needed for ddf finally we investigate the use of minimum mean squared error generalized decision feedback equalizer mmse gdfe lattice decoding at both the relay and the destination and show that it provides near optimal performance at moderate complexity

on the energy delay tradeoff in streaming data finite blocklength analysis this paper investigates basic trade offs between energy and delay in wireless communication systems using finite blocklength theory we first assume that data arrive in constant stream of bits which are put into packets and transmitted over a communications link our results show that depending on exactly how energy is measured in general energy depends on sqrt d 1 or sqrt d 1 log d where d is the delay this means that the energy decreases quite slowly with increasing delay furthermore to approach the absolute minimum of 1 59 db on energy bandwidth has to increase very rapidly much more than what is predicted by infinite blocklength theory we then consider the scenario when data arrive stochastically in packets and can be queued we devise a scheduling algorithm based on finite blocklength theory and develop bounds for the energy delay performance our results again show that the energy decreases quite slowly with increasing delay

the streaming dmt of fading channels we study sequential transmission of a stream of messages over a block fading multi input multi output mimo channel a new message arrives in each coherence block and the decoder is required to output each message after a delay of t coherence blocks we establish the optimal diversity multiplexing tradeoff dmt and show that it can be achieved using a simple interleaving of messages the converse is based on an outage amplification technique which appears to be new we also discuss another coding scheme based on a sequential tree code this coding scheme only requires the knowledge of delay at the decoder and yet realizes the optimal dmt we finally discuss some extensions when multiple messages at uniform intervals arrive within each coherence period

on source transmission over mimo channels with limited feedback the problem of source channel coding over a multiple antenna mimo channel with quantized channel state information at the transmitter csit is considered upper bounds on the distortion exponents achieved with partial csit under a long term power constraint are developed it is shown that the distortion exponent with perfect csit grows unbounded as the ratio between the channel and source bandwidth increases while the exponent achieved with any feedback link of fixed finite resolution is bounded above by a polynomial of the product between the number of transmit and number of receive antennas the resolution of the feedback link should grow with the bandwidth ratio to make the distortion exponent scale as fast as that in the perfect csit case we show that in order to achieve the optimal scaling the csit feedback resolution must grow logarithmically with the bandwidth ratio for mimo channels and faster than linear for the single input single output channel the achievable distortion exponent of some hybrid schemes with heavily quantized feedback is also derived the results demonstrate that dramatic performance improvement over the case of no csit can be achieved by combining simple schemes with a very coarse csit feedback

averaging bounds for lattices and linear codes general random coding theorems for lattices are derived from the minkowski hlawka theorem and their close relation to standard averaging arguments for linear codes over finite fields is pointed out a new version of the minkowski hlawka theorem itself is obtained as the limit for p spl rarr spl infin of a simple lemma for linear codes over gf p used with p level amplitude modulation the relation between the combinatorial packing of solid bodies and the information theoretic soft packing with arbitrarily small but positive overlap is illuminated the soft packing results are new when specialized to the additive white gaussian noise channel they reduce to a version of the de buda poltyrev result that spherically shaped lattice codes and a decoder that is unaware of the shaping can achieve the rate 1 2 log sub 2 p n

on l convergence of trigonometric series abstract in the present paper we consider the trigonometric series with r general monotone and r rest bounded variation coefficients necessary and sufficient conditions of l convergence for such series are obtained in terms of the coefficients moreover we generalize and extend the tikhonov results j math anal appl 347 2008 416 427 to the class gm r or the class rbvs r

achieving 1 2 log 1 snr on the awgn channel with lattice encoding and decoding we address an open question regarding whether a lattice code with lattice decoding as opposed to maximum likelihood ml decoding can achieve the additive white gaussian noise awgn channel capacity we first demonstrate how minimum mean square error mmse scaling along with dithering lattice randomization techniques can transform the power constrained awgn channel into a modulo lattice additive noise channel whose effective noise is reduced by a factor of spl radic 1 snr snr for the resulting channel a uniform input maximizes mutual information which in the limit of large lattice dimension becomes 1 2 log 1 snr i e the full capacity of the original power constrained awgn channel we then show that capacity may also be achieved using nested lattice codes the coarse lattice serving for shaping via the modulo lattice transformation the fine lattice for channel coding we show that such pairs exist for any desired nesting ratio i e for any signal to noise ratio snr furthermore for the modulo lattice additive noise channel lattice decoding is optimal finally we show that the error exponent of the proposed scheme is lower bounded by the poltyrev exponent

on summability of fourier coefficients of functions from lebesgue space abstract in this paper we state new theorems of hardy littlewood type for functions with general monotone fourier coefficients sharpness of stated results is discusses

on trigonometric series with quasimonotone coefficients abstract a result of hardy giving a necessary and sufficient condition for a trigonometric series with decreasing coefficients to have an asymptotic sum is extended to the case of coefficients quasimonotone in either sense it is shown that in a certain sense the results are best possible

on l1 convergence of fourier series abstract in this paper we consider the trigonometric fourier series with the general monotone coefficients necessary and sufficient conditions of l 1 convergence for such a series that is f s n o 1 are obtained in terms of coefficients

tauberian conditions for l 1 convergence of fourier series it is proved that fourier series with asymptotically even coefficients and satisfying lima 1lim supn exn jpi i ajf j p 0 for some i p 2 converge in l norm if and only if i1 f n en f n e n ii o l where en t k e k recent results of stanojevic 1 bojanic and stanojevic 2 and goldberg and stanojevic 3 are special cases of some corollaries to the main theorem

geometric three dimensional assignment problems abstract we investigate two geometric special cases of the three dimensional assignment problem given are three sets b r and g blue red and green each containing n grid points in the euclidean plane we want to find a partition of b r g into n three0colored triangles such that a the total circumference of all triangles or b the total area of all triangles becomes minimum both versions of the problem are proved to be np hard

dmt optimality of lr aided linear decoders for a general class of channels lattice designs and system models this paper identifies the first general explicit and nonrandom mimo encoder decoder structures that guarantee optimality with respect to the diversity multiplexing tradeoff dmt without employing a computationally expensive maximum likelihood ml receiver specifically the work establishes the dmt optimality of a class of regularized lattice decoders and more importantly the dmt optimality of their lattice reduction lr aided linear counterparts the results hold for all channel statistics for all channel dimensions and most interestingly irrespective of the particular lattice code applied as a special case it is established that the lll based lr aided linear implementation of the mmse gdfe lattice decoder facilitates dmt optimal decoding of any lattice code at a worst case 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 the rate the results generality lends them applicable to a plethora of pertinent communication scenarios such as quasi static mimo mimo ofdm isi cooperative relaying and mimo arq channels in all of which the dmt optimality of the lr aided 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