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

distributed source coding using abelian group codes a new achievable rate distortion region a distributed source coding problem with a joint distortion criterion that depends on the sources and the reconstruction is considered in this work while the prevalent trend in information theory has been to prove achievability results using shannon s random coding arguments using structured random codes offer rate gains over unstructured random codes for many problems motivated by this a new achievable rate distortion region an inner bound to the performance limit is presented for this problem for discrete memoryless sources based on good structured random nested codes built over abelian groups for certain sources and distortion functions the new rate region is shown to be strictly bigger than the berger tung rate region which has been the best known achievable rate region for this problem till now this is done using numerical plots achievable rates for single user source coding using abelian group codes are also obtained as a corollary of the main coding theorem it is shown that nested linear codes achieve the shannon rate distortion function in the arbitrary discrete memoryless case

awgn goodness is enough capacity achieving lattice codes based on dithered probabilistic shaping in this paper we show that any sequence of infinite lattice constellations which is good for the unconstrained gaussian channel can be shaped into a capacity achieving sequence of codes for the power constrained gaussian channel under lattice decoding and non uniform signaling unlike previous results in the literature our scheme holds with no extra condition on the lattices e g quantization goodness or vanishing flatness factor thus establishing a direct implication between awgn goodness in the sense of poltyrev and capacity achieving codes our analysis uses properties of the discrete gaussian distribution in order to obtain precise bounds on the probability of error and achievable rates in particular we obtain a simple characterization of the finite blocklength behavior of the scheme showing that it approaches the optimal dispersion coefficient for high signal to noise ratio we further show that for low signal to noise ratio the discrete gaussian over centered lattice constellations cannot achieve capacity and thus a shift or dither is essentially necessary

robust lattice alignment for k user mimo interference channels with imperfect channel knowledge in this paper we consider a robust lattice alignment design for k user quasi static multiple input multiple output mimo interference channels with imperfect channel knowledge with random gaussian inputs the conventional interference alignment ia method has the feasibility problem when the channel is quasi static on the other hand structured lattices can create structured interference as opposed to the random interference caused by random gaussian symbols the structured interference space can be exploited to transmit the desired signals over the gaps however the existing alignment methods on the lattice codes for quasi static channels either require infinite signal to noise ratio snr or symmetric interference channel coefficients furthermore perfect channel state information csi is required for these alignment methods which is difficult to achieve in practice in this paper we propose a robust lattice alignment method for quasi static mimo interference channels with imperfect csi at all snr regimes and a two stage decoding algorithm to decode the desired signal from the structured interference space we derive the achievable data rate based on the proposed robust lattice alignment method where the design of the precoders decorrelators scaling coefficients and interference quantization coefficients is jointly formulated as a mixed integer and continuous optimization problem the effect of imperfect csi is also accommodated in the optimization formulation and hence the derived solution is robust to imperfect csi we also design a low complex iterative optimization algorithm for our robust lattice alignment method by using the existing iterative ia algorithm that was designed for the conventional ia method numerical results verify the advantages of the proposed robust lattice alignment method compared with the time division multiple access tdma two stage maximum likelihood ml decoding generalized han kobayashi hk distributive ia and conventional ia methods in the literature

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

moderate deviations in channel coding we consider block codes whose rate converges to the channel capacity with increasing blocklength at a certain speed and examine the best possible decay of the probability of error for discrete memoryless channels we prove that a moderate deviation principle holds for all convergence rates between the large deviation and the central limit theorem regimes

universal lattice codes for mimo channels we propose a coding scheme that achieves the capacity of the compound mimo channel with algebraic lattices our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill conditioned channel realizations to shape the constellation a discrete gaussian distribution over the lattice points is applied these techniques along with algebraic properties of the proposed lattices are then used to construct a sub optimal de coupled coding scheme that achieves a constant gap to compound capacity by decoding in a lattice that does not dependent on the channel realization the gap is characterized in terms of algebraic invariants of the code and is shown to be significantly smaller than previous schemes in the literature we also exhibit alternative algebraic constructions that achieve the capacity of ergodic siso fading channels

compute and forward harnessing interference through structured codes interference is usually viewed as an obstacle to communication in wireless networks this paper proposes a new strategy compute and forward that exploits interference to obtain significantly higher rates between users in a network the key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise after decoding these linear equations the relays simply send them towards the destinations which given enough equations can recover their desired messages the underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field this scheme is applicable even if the transmitters lack channel state information

universal bound on the performance of lattice codes we present a lower bound on the probability of symbol error for maximum likelihood decoding of lattices and lattice codes on a gaussian channel the bound is tight for error probabilities and signal to noise ratios of practical interest as opposed to most existing bounds that become tight asymptotically for high signal to noise ratios the bound is also universal it provides a limit on the highest possible coding gain that may be achieved at specific symbol error probabilities using any lattice or lattice code in n dimensions in particular it is shown that the effective coding gains of the densest known lattices are much lower than their nominal coding gains the asymptotic as n spl rarr spl infin behavior of the new bound is shown to coincide with the shannon 1948 limit for gaussian channels

lda lattices without dithering achieve capacity on the gaussian channel this paper deals with low density construction a lda lattices which are obtained via construction a from non binary low density parity check codes more precisely a proof is provided that voronoi constellations of lda lattices achieve capacity of the awgn channel under lattice encoding and decoding for every signal to noise ratio greater than 1 this is obtained after showing the same result for more general construction a lattice constellations the theoretical analysis is carried out in a way that allows to describe how the prime number underlying construction a behaves as a function of the lattice dimension moreover no dithering is required in the transmission scheme simplifying some previous solutions of the problem remarkably capacity is achievable with lda lattice codes whose parity check matrices have constant row and column hamming weights some expansion properties of random bipartite graphs constitute an extremely important tool for dealing with sparse matrices and allow to find a lower bound for the minimum euclidean distance of lda lattices in our ensemble

gilbert varshamov bound for euclidean space codes over distance uniform signal sets in this correspondence in an extension of piret s bound for codes over phase shift keying psk signal sets we investigate the application of the gilbert varshamov gv bound to a variety of distance uniform du signal sets in euclidean space it is shown that four dimensional signal sets matched to binary tetrahedral binary octahedral and binary icosahedral groups lead to better bounds compared to the bounds for signal sets matched to dicyclic groups with the same number of signal points and comparable symmetric psk signal sets

solutions of 2 1 dimensional spin systems we use the methods of group theory to reduce the equations of motion of two spin systems in 2 1 dimensions to sets of coupled ordinary differential equations we present solutions of some classes of these sets and discuss their physical significance

on classes of integrable systems and the painleve property the caudrey dodd gibbon equation is found to possess the painleve property investigation of the backlund transformations for this equation obtains the kuperschmidt equation a certain transformation between the kuperschmidt and caudrey dodd gibbon equation is obtained this transformation is employed to define a class of p d e s that identically possesses the painleve property for equations within this class backlund transformations and rational solutions are investigated in particular the sequences of higher order kdv caudrey dobb gibbon and kuperschmidt equations are shown to possess the painleve property

motion of curves and surfaces and nonlinear evolution equations in 2 1 dimensions it is shown that a class of important integrable nonlinear evolution equations in 2 1 dimensions can be associated with the motion of space curves endowed with an extra spatial variable or equivalently moving surfaces geometrical invariants then define topological conserved quantities underlying evolution equations are shown to be associated with a triad of linear equations our examples include ishimori equation and myrzakulov equations which are shown to be geometrically equivalent to davey stewartson and zakharov strachan 2 1 dimensional nonlinear schrodinger equations respectively

spatially coupled ensembles universally achieve capacity under belief propagation we investigate spatially coupled code ensembles for transmission over the binary erasure channel it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble this explains why convolutional ldpc ensembles originally introduced by felstrom and zigangirov perform so well over this channel we show that the equivalent result holds true for transmission over general binary input memoryless output symmetric channels more precisely given a desired error probability and a gap to capacity we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding in fact most codes in this ensemble have this property the quantifier universal refers to the single ensemble code that is good for all channels but we assume that the channel is known at the receiver the key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble we conclude by discussing some interesting open problems

precoded integer forcing universally achieves the mimo capacity to within a constant gap an open loop single user multiple input multiple output mimo communication scheme is considered where a transmitter equipped with multiple antennas encodes the data into independent streams all taken from the same linear code the coded streams are then linearly precoded using the encoding matrix of a perfect linear dispersion space time code at the receiver side integer forcing equalization is applied followed by standard single stream decoding it is shown that this communication architecture achieves the capacity of any gaussian mimo channel up to a gap that depends only on the number of transmit antennas

achievable rates for k user gaussian interference channels the aim of this paper is to study the achievable rates for a k user gaussian interference channel g ifc for any signal to noise ratio using a combination of lattice and algebraic codes lattice codes are first used to transform the g ifc into a discrete input output noiseless channel and subsequently algebraic codes are developed to achieve good rates over this new alphabet in this context a quantity called efficiency is introduced which reflects the effectiveness of the algebraic coding strategy this paper first addresses the problem of finding high efficiency algebraic codes a combination of these codes with construction a lattices is then used to achieve nontrivial rates for the original g ifc

the approximate capacity region of the gaussian y channel via the deterministic approach a full duplex wireless network with three users that want to establish full message exchange via a relay is considered thus this network which is known as the y channel has a total of six messages two outgoing and two incoming at each user the users are not physically connected and thus the relay is essential for their communication the deterministic y channel is considered first its capacity region is characterized and shown not to be given by the cut set bounds the capacity achieving scheme has three different components strategies 1 a bidirectional 2 a cyclic and 3 a unidirectional strategy network coding is used to realize the bidirectional and the cyclic strategies and thus to prove the achievability of the capacity region the result is then extended to the gaussian y channel where the capacity region is characterized within a constant gap independent of the channel parameters

on a coupled system of equations describing pulse propagation in quadratic media in the slowly varying envelope approximation we derive the basic equations that describe the propagation of ultrashort pulses in quadratically nonlinear media in which a wave at a fundamental frequency interacts with its second harmonic in the governing equations we keep linear terms that account for both second and third order dispersion and nonlinear terms describing both nonlinear dispersion and self steepening of the pulse edge we then perform the painleve singularity structure analysis of the most general system of coupled partial differential equations we derived in a specific case when third order dispersion is negligible by using a hirota like method we found zero and one parameter families of bright fundamental frequency and dark second harmonic solitary waves which travel at a locked group velocity

on the evaluation of the polyanskiy poor verdu converse bound for finite block length coding in awgn a tight converse bound to the channel coding rate in the finite block length regime and under additive white gaussian noise conditions was recently proposed by polyanskiy poor verdu ppv the bound is a generalization of a number of other classical results and it was also claimed to be equivalent to shannon s 1959 cone packing bound in this paper we investigate methods for a reliable numerical evaluation of the bound which is troublesome even for not too large values of the block length n by compactly expressing the polyanskiy poor and verdu ppv converse bound in terms of non central chi squared distributions and by evaluating those through an integral expression and a corresponding series expansion which exploit a method proposed by temme as a result a robust evaluation method and new insights on the bound s asymptotics as well as new approximate expressions are obtained

the random coding bound is tight for the average linear code or lattice in 1973 gallager proved that the random coding bound is exponentially tight for the random code ensemble at all rates even below expurgation this result explained that the random coding exponent does not achieve the expurgation exponent due to the properties of the random ensemble irrespective of the utilized bounding technique it has been conjectured that this same behavior holds true for a random ensemble of linear codes this conjecture is proved in this paper in addition it is shown that this property extends to poltyrev s random coding exponent for a random ensemble of lattices

lattice codes for the gaussian relay channel decode and forward and compress and forward lattice codes are known to achieve capacity in the gaussian point to point channel achieving the same rates as i i d random gaussian codebooks lattice codes are also known to outperform random codes for certain channel models that are able to exploit their linearity in this paper we show that lattice codes may be used to achieve the same performance as known i i d gaussian random coding techniques for the gaussian relay channel and show several examples of how this may be combined with the linearity of lattices codes in multisource relay networks in particular we present a nested lattice list decoding technique in which lattice codes are shown to achieve the decode and forward df rate of single source single destination gaussian relay channels with one or more relays we next present two examples of how this df scheme may be combined with the linearity of lattice codes to achieve new rate regions which for some channel conditions outperform analogous known gaussian random coding techniques in multisource relay channels that is we derive a new achievable rate region for the two way relay channel with direct links and compare it to existing schemes and derive a new achievable rate region for the multiple access relay channel we furthermore present a lattice compress and forward cf scheme for the gaussian relay channel which exploits a lattice wyner ziv binning scheme and achieves the same rate as the cover el gamal cf rate evaluated for gaussian random codes these results suggest that structured lattice codes may be used to mimic and sometimes outperform random gaussian codes in general gaussian networks

filter and nested lattice code design for mimo fading channels with side information linear assignment gel fand pinsker coding la gpc is a coding technique for channels with interference known only at the transmitter where the known interference is treated as side information si as a special case of the la gpc dirty paper coding has been shown to be able to achieve the optimal interference free rate for si channels with perfect channel state information at the transmitter csit in the cases where only the channel distribution information at the transmitter cdit is available la gpc also has good sometimes optimal performance in a variety of fast and slow fading si channels in this letter we design filters in the nested lattice based coding to make it achieve the same rate performance as the la gpc in multiple input multiple output mimo channels compared with the random gaussian codebooks used in previous works our resultant coding schemes have algebraic structures and can be implemented in practical systems simulations in slow fading channels are provided and near interference free error performance is obtained the proposed coding schemes can serve as the fundamental building blocks to achieve the promised rate performance of mimo gaussian broadcast channels with cdit or perfect csit

random ensembles of lattices from generalized reductions we propose a general framework to study constructions of euclidean lattices from linear codes over finite fields in particular we prove general conditions for an ensemble constructed using linear codes to contain dense lattices i e with packing density comparable to the minkowski hlawka lower bound specializing to number field lattices we obtain a number of interesting corollaries for instance the best known packing density of ideal lattices and an elementary coding theoretic construction of asymptotically dense hurwitz lattices all results are algorithmically effective in the sense that for any dimension a finite family containing dense lattices is exhibited for suitable constructions based on craig s lattices this family is smaller in terms of alphabet size than previous ensembles in the literature

non random coding error bounds for lattices an upper bound on the error probability of specific lattices based on their distance spectrum is constructed the derivation is accomplished using a simple alternative to the minkowski hlawka mean value theorem of the geometry of numbers in many ways the new bound greatly resembles the shulman feder bound for linear codes based on the new bound error exponent and channel dispersion expressions are derived for specific lattice sequences of increasing dimension over the awgn channel measuring a sequence s gap to capacity using the new asymptotics is demonstrated additional finite dimension results encountered along the way are presented

beyond graph energy norms of graphs and matrices abstract in 1978 gutman introduced the energy of a graph as the sum of the absolute values of graph eigenvalues and ever since then graph energy has been intensively studied since graph energy is the trace norm of the adjacency matrix matrix norms provide a natural background for its study thus this paper surveys research on matrix norms that aims to expand and advance the study of graph energy the focus is exclusively on the ky fan and the schatten norms both generalizing and enriching the trace norm as it turns out the study of extremal properties of these norms leads to numerous analytic problems with deep roots in combinatorics the survey brings to the fore the exceptional role of hadamard matrices conference matrices and conference graphs in matrix norms in addition a vast new matrix class is studied a relaxation of symmetric hadamard matrices the survey presents solutions to just a fraction of a larger body of similar problems bonding analysis to combinatorics thus open problems and questions are raised to outline topics for further investigation

on the sum of two largest eigenvalues of a symmetric matrix abstract gernert conjectured that the sum of two largest eigenvalues of the adjacency matrix of any simple graph is at most the number of vertices of the graph this can be proved in particular for all regular graphs gernert s conjecture was recently disproved by one of the authors v nikiforov linear combinations of graph eigenvalues electron j linear algebra 15 2006 329 336 who also provided a nontrivial upper bound for the sum of two largest eigenvalues in this paper we improve the lower and upper bounds to near optimal ones and extend results from graphs to general non negative matrices

lower bounds on the error probability of block codes based on improvements on de caen s inequality new lower bounds on the error probability of block codes with maximum likelihood decoding are proposed the bounds are obtained by applying a new lower bound on the probability of a union of events derived by improving on de caen s lower bound the new bound includes an arbitrary function to be optimized in order to achieve the tightest results since the optimal choice of this function is known but leads to a trivial and useless identity we find several useful approximations for it each resulting in a new lower bound for the additive white gaussian noise awgn channel and the binary symmetric channel bsc the optimal choice of the optimization function is stated and several approximations are proposed when the bounds are further specialized to linear codes the only knowledge on the code used is its weight enumeration the results are shown to be tighter than the latest bounds in the current literature such as those by seguin 1998 and by keren and litsyn 2001 moreover for the bsc the new bounds widen the range of rates for which the union bound analysis applies thus improving on the bound to the error exponent compared with the de caen based bounds

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

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

fundamental limits of infinite constellations in mimo fading channels the fundamental and natural connection between the infinite constellation ic dimension and the best diversity order it can achieve is investigated in this paper in the first part of this paper we develop an upper bound on the diversity order of ics for any dimension and any number of transmit and receive antennas by choosing the right dimensions we prove in the second part of this paper that ics in general and lattices in particular can achieve the optimal diversity multiplexing tradeoff of finite constellations this paper gives a framework for designing lattices for multiple antenna channels using lattice decoding

the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

achieving awgn channel capacity with lattice gaussian coding we propose a new coding scheme using only one lattice that achieves the 1 2 log 1 snr capacity of the additive white gaussian noise awgn channel with lattice decoding which is provable for signal to noise ratio snr e at present the scheme applies a discrete gaussian distribution over an awgn good lattice but otherwise does not require a shaping lattice or dither thus it significantly simplifies the default lattice coding scheme of erez and zamir which involves a quantization good lattice as well as an awgn good lattice using the flatness factor we show that the error probability of the proposed scheme under minimum mean square error lattice decoding is almost the same as that of erez and zamir for any rate up to the awgn channel capacity we introduce the notion of good constellations which carry almost the same mutual information as that of continuous gaussian inputs we also address the implementation of gaussian shaping for the proposed lattice gaussian coding scheme

nested lattice codes for gaussian relay networks with interference in this paper we consider a class of single source multicast relay networks we assume that all outgoing channels of a node in the network to its neighbors are orthogonal while the incoming signals from its neighbors can interfere with each other we first focus on gaussian relay networks with interference and find an achievable rate using a lattice coding scheme we show that the achievable rate of our scheme is within a constant bit gap from the information theoretic cut set bound where the constant depends only on the network topology but not on the transmit power noise variance and channel gains this is similar to a recent result by avestimehr diggavi and tse who showed an approximate capacity characterization for general gaussian relay networks however our achievability uses a structured code instead of a random one using the idea used in the gaussian case we also consider a linear finite field symmetric network with interference and characterize its capacity using a linear coding scheme

constraint gain in digital storage systems where the input to the noisy channel is required to satisfy a modulation constraint the constrained code and error control code ecc are typically designed and decoded independently the achievable rate for this situation is evaluated as the rate of average intersection of the constraint and the ecc the gap from the capacity of the noisy constrained channel is called the constraint gain which represents the potential improvement in combining the design and decoding of the constrained code and the ecc the constraint gain is computed for various constraints over the binary input additive white gaussian noise awgn channel biawgnc as well as over intersymbol interference isi channels finally it is shown that an infinite cascade of reverse concatenation with independent decoding of constraint and ecc has a capacity equal to the rate of average intersection

corrected proof of de buda s theorem lattice channel codes an error in de buda s see ieee j select areas commun vol 7 no 6 p 893 1989 proof on the asymptotic optimality of lattice channel codes is pointed out and corrected using a modification of de buda s approach comments are given on the correct interpretation and the limitations of this result

finite dimensional infinite constellations in the setting of a gaussian channel without power constraints proposed by poltyrev in 1994 the codewords are points in an n dimensional euclidean space an infinite constellation and the tradeoff between their density and the error probability is considered the normalized log density nld plays the role of the communication rate and capacity as well as error exponent bounds for this setting are known this paper considers the infinite constellation setting in the finite block length dimension regime a simplified expression for poltyrev s achievability bound is found and it is shown to be closely related to the sphere converse bound and to a recently proposed achievability bound based on point processes the bounds are then analyzed asymptotically for growing n for fixed nld the bounds turn out to be extremely tight compared to previous error exponent analysis for fixed error probability e it is shown that the gap of the highest achievable nld to the optimal nld poltyrev s capacity is approximately 1 2n q 1 e where q is the standard complementary gaussian cumulative distribution function thus extending the channel dispersion analysis to infinite constellations connections to the error exponent of the power constrained gaussian channel and to the volume to noise ratio as a figure of merit are discussed finally the new tight bounds are compared to state of the art coding schemes

the degrees of freedom of compute and forward we analyze the asymptotic behavior of compute and forward relay networks in the regime of high signal to noise ratios we consider a section of such a network consisting of k transmitters and k relays the aim of the relays is to reliably decode an invertible function of the messages sent by the transmitters an upper bound on the capacity of this system can be obtained by allowing full cooperation among the transmitters and among the relays transforming the network into a k k multiple input multiple output mimo channel the number of degrees of freedom of compute and forward is hence at most k in this paper we analyze the degrees of freedom achieved by the lattice coding implementation of compute and forward proposed recently by nazer and gastpar we show that this lattice implementation achieves at most 2 1 1 k 2 degrees of freedom thus exhibiting a very different asymptotic behavior than the mimo upper bound this raises the question if this gap of the lattice implementation to the mimo upper bound is inherent to compute and forward in general we answer this question in the negative by proposing a novel compute and forward implementation achieving k degrees of freedom

sphere bound achieving coset codes and multilevel coset codes a simple sphere bound gives the best possible tradeoff between the volume per point of an infinite array l and its error probability on an additive white gaussian noise awgn channel it is shown that the sphere bound can be approached by a large class of coset codes or multilevel coset codes with multistage decoding including certain binary lattices these codes have structure of the kind that has been found to be useful in practice capacity curves and design guidance for practical codes are given exponential error bounds for coset codes are developed generalizing poltyrev s 1994 bounds for lattices these results are based on the channel coding theorems of information theory rather than the minkowski hlawka theorem of lattice theory

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

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

refinement of the sphere packing bound asymmetric channels we provide a refinement of the sphere packing bound for constant composition codes over asymmetric discrete memoryless channels that improves the subexponential factor in front of the exponent the order of our subexponential factor is n 0 5 1 e r for any 0 where r is the left derivative of the sphere packing exponent at rate r and n is the blocklength

the gaussian many to one interference channel with confidential messages the many to one interference channel has received interest by virtue of embodying the essence of an interference network while being more tractable than the general k user interference channel in this paper we introduce information theoretic secrecy to this model and consider the many to one interference channel with confidential messages in which each receiver in particular the one subject to interference is also one from which the interfering users messages need to be kept secret from we derive the achievable secrecy sum rate for this channel using nested lattice codes as well as an upper bound on the secrecy sum rate for all possible channel gain configurations we identify several nontrivial cases where the gap between the upper bound and the achieved secrecy sum rate is only a function of the number of the users k and is uniform over all possible channel gain configurations in each case in addition we identify the secure degrees of freedom for this channel and show it to be equivalent to its degrees of freedom i e the secrecy in high snr comes for free

a tight upper bound for the third order asymptotics for most discrete memoryless channels this paper shows that the logarithm of the e error capacity average error probability for n uses of a discrete memoryless channel dmc is upper bounded by the normal approximation plus a third order term that does not exceed 1 2 logn o 1 if the e dispersion of the channel is positive this matches a lower bound by y polyanskiy 2010 for dmcs with positive reverse dispersion if the e dispersion vanishes the logarithm of the e error capacity is upper bounded by n times the capacity plus a constant term except for a small class of dmcs and e 1 2

lattice coding and decoding for multiple antenna ergodic fading channels for ergodic fading a lattice coding and decoding strategy is proposed and its performance is analyzed for the single input single output siso and multiple input multiple output mimo point to point channel as well as the multiple access channel mac with channel state information available only at the receiver csir at the decoder a novel strategy is proposed consisting of a time varying equalization matrix followed by decision regions that depend only on channel statistics not individual realizations our encoder has a similar structure to that of erez and zamir for the siso channel the gap to capacity is bounded by a constant under a wide range of fading distributions for the mimo channel under rayleigh fading the rate achieved is within a gap to capacity that does not depend on the signal to noise ratio snr and diminishes with the number of receive antennas the analysis is extended to the k user mac where similar results hold achieving a small gap to capacity while limiting the use of csir to the equalizer highlights the scope for efficient decoder implementations since decision regions are fixed i e independent of channel realizations

on the integrability and singularity structure aspects of deformed nonlinear evolution equations of akns type in this paper a singularity structure analysis of some important inhomogeneous nonlinear evolution equations nlees of akns type introduced by burtsev zakharov and mikhailov 1987 is carried out and they are shown to possess the painleve property the other integrability properties such as lax pair backlund transformation and soliton solutions of these systems are also brought out in detail from the painleve analysis we also point out that the non painleve nature of the system of partial differential equations satisfied by the variable spectral parameter may not affect the painleve property and hence the integrability of the associated nlees

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

on lattice sequential decoding for the unconstrained awgn channel in this paper the performance limits and the computational complexity of the lattice sequential decoder are analyzed for the unconstrained additive white gaussian noise channel the performance analysis available in the literature for such a channel has been studied only under the use of the minimum euclidean distance decoder that is commonly referred to as the lattice decoder lattice decoders based on solutions to the np hard closest vector problem are very complex to implement and the search for low complexity receivers for the detection of lattice codes is considered a challenging problem however the low computational complexity advantage that sequential decoding promises makes it an alternative solution to the lattice decoder in this work we characterize the performance and complexity tradeoff via the error exponent and the decoding complexity respectively of such a decoder as a function of the decoding parameter the bias term for the above channel we derive the cut off volume to noise ratio that is required to achieve a good error performance with low decoding complexity

on the sum of k largest singular values of graphs and matrices in the recent years the trace norm of graphs has been extensively studied under the name of graph energy the trace norm is just one of the ky fan k norms given by the sum of the k largest singular values which are studied more generally in the present paper several relations to chromatic number spectral radius spread and to other fundamental parameters are outlined some results are extended to more general matrices

asymptotic capacity of the separated mimo two way relay channel a multiple input multiple output two way relay channel consisting of two communication nodes and a full duplex relay node in which no direct link exists between the two communication nodes is considered we propose an achievable scheme that employs horizontally encoded lattice codes combined with generalized singular value decomposition based precoding for the first phase the second phase of the proposed scheme follows the fundamentals of the previous scheme which uses vertically encoded structural bining with the only difference that the added codeword of the two codewords from the communication nodes instead of those two individual codewords is decoded and retransmitted in the proposed scheme we show that the proposed scheme achieves the cut set bound asymptotically as the signal to noise ratios of the channels tend to infinity

painleve analysis of new higher dimensional soliton equation in this note we prove that the recently proposed new higher dimensional nonlinear partial differential equation admits the painleve property we briefly discuss the integrability properties of the equation

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

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

techniques of bounding the probability of decoding error for block coded modulation structures two techniques for upper bounding the average probability of decoding error in coded modulation structures are presented the first bound which is applicable to the additive white gaussian noise awgn channel is tighter than the well known union bound and the minimum distance bound especially for low signal to noise ratio it is shown that for the leech lattice this upper bound is very close to a sphere lower bound for the second upper bound which is applicable to any memoryless channel not necessarily awgn a method of random coset coding is presented for the awgn channel a tighter upper bound is obtained by employing the method of random coset coding for calculating the average spectrum of distances of the code which is required for the computation of the first upper bound

asymptotic improvement of the gilbert varshamov bound on the size of binary codes given positive integers n and d let a sub 2 n d denote the maximum size of a binary code of length n and minimum distance d the well known gilbert varshamov bound asserts that a sub 2 n d spl ges 2 sup n v n d l where v n d spl sigma sub i 0 sup d sub i sup n is the volume of a hamming sphere of radius d we show that in fact there exists a positive constant c such that a sub 2 n d spl ges c2 sup n v n d 1 log sub 2 v n d 1 whenever d n spl les 0 499 the result follows by recasting the gilbert varshamov bound into a graph theoretic framework and using the fact that the corresponding graph is locally sparse generalizations and extensions of this result are briefly discussed

a lower bound on packing density if subseteq n is a bounded convex body which is symmetric through each of the coordinate hyperplanes then there exist codes which give rise via construction a of leech and sloane to lattice packings of whose density satisfies the logarithmic minkowski hlawka bound lim inf n to infty log 2 sqrt n delta geqq 1 this follows as a corollary of our main result theorem 9 a general way of obtaining lower bounds on the lattice packing densities of various bodies unfortunately whenn is at all large it is computationally prohibitive although theoretically possible to exhibit the arrangements explicitly

on the dynamics of the radially symmetric heisenberg ferromagnetic spin system by considering the geometrical equivalence of the radially symmetric heisenberg ferromagnetic spin system in n arbitrary spatial dimensions and the generalized nonlinear schrodinger equation gnlse with radial symmetry it is shown that they possess the painleve property only for the n 2 circularly planar radially symmetric case for the circularly symmetric case suitable 2 2 matrix eigenvalue equations are constructed involving nonisospectral flows and their gauge equivalence is shown the connection with inhomogeneous systems and in particular the linearly x dependent system is pointed out appropriate backlund transformations bt and explicit soliton solutions for both the spin systems and the gnlses are also derived

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

nested linear lattice codes for structured multiterminal binning network information theory promises high gains over simple point to point communication techniques at the cost of higher complexity however lack of structured coding schemes limited the practical application of these concepts so far one of the basic elements of a network code is the binning scheme wyner 1974 1978 and other researchers proposed various forms of coset codes for efficient binning yet these schemes were applicable only for lossless source or noiseless channel network coding to extend the algebraic binning approach to lossy source or noisy channel network coding previous work proposed the idea of nested codes or more specifically nested parity check codes for the binary case and nested lattices in the continuous case these ideas connect network information theory with the rich areas of linear codes and lattice codes and have strong potential for practical applications we review these developments and explore their tight relation to concepts such as combined shaping and precoding coding for memories with defects and digital watermarking we also propose a few novel applications adhering to a unified approach

on integrability of the inhomogeneous heisenberg ferromagnet model examination of a new test we present an improved version of the integrability test based on application of lie point transformations acting on the bare linear problem the test is then applied to the inhomogeneous heisenberg ferromagnet model equation the integrable cases selected by this test are identical with those obtained by the standard painleve test

on the generalized degrees of freedom of the gaussian interference relay channel the symmetric two user gaussian interference relay channel irc is studied from a generalized degrees of freedom gdof perspective while it is known that the relay does not in crease the dof of the irc such a characterization has not been reported for the gdof yet the focus of this paper is on all cases where the interference link is stronger than the link from the source to the relay this regime basically covers half the space of all possible parameters of the irc by using genie aided approaches new sum capacity upper bounds are derived these bounds are then compared with rates achieved with a novel transmission scheme which is based on a functional decode and forward fdf strategy it is shown that the gdof of the irc is achieved by fdf in the given regime and that a relay can indeed increase the gdof of irc finally the fdf scheme is compared with other schemes like decode and forward as well as compress and forward at low moderate and high signal to noise ratios

signal codes convolutional lattice codes the coded modulation scheme proposed in this paper has a simple construction an integer sequence representing the information is convolved with a fixed continuous valued finite impulse response fir filter to generate the codeword a lattice point due to power constraints the code construction includes a shaping mechanism inspired by precoding techniques such as the tomlinson harashima filter we naturally term these codes convolutional lattice codes or alternatively signal codes due to the signal processing interpretation of the code construction surprisingly properly chosen short fir filters can generate good codes with large minimal distance decoding can be done efficiently by sequential decoding or for better performance by bidirectional sequential decoding error analysis and simulation results indicate that for the additive white gaussian noise awgn channel convolutional lattice codes with computationally reasonable decoders can achieve low error rate close to the channel capacity

algebraic constructions of shannon codes for regular channels the problem of the explicit construction of encoders achieving shannon s capacity and admitting a simple decoding algorithm is considered a solution based on justesen s idea of variable concatenated codes is given for the case of a symmetric memoryless channel with an input alphabet of prime power order under the assumption that the information messages are equiprobable this construction remains good for a nonsymmetric channel provided the encoding rate is smaller than a well defined pseudocapacity in case the channel is regular it is shown that the error probability after decoding is an exponentially decreasing function of the block length for any encoding rate less than the channel capacity

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

the approximate capacity of the many to one and one to many gaussian interference channels recently etkin tse and wang found the capacity region of the two user gaussian interference channel to within 1 bit s hz a natural goal is to apply this approach to the gaussian interference channel with an arbitrary number of users we make progress towards this goal by finding the capacity region of the many to one and one to many gaussian interference channels to within a constant number of bits the result makes use of a deterministic model to provide insight into the gaussian channel the deterministic model makes explicit the dimension of signal level a central theme emerges the use of lattice codes for alignment of interfering signals on the signal level

on error exponents of modulo lattice additive noise channels modulo lattice additive noise mlan channels appear in the analysis of structured binning codes for costa s dirty paper channel and of nested lattice codes for the additive white gaussian noise awgn channel in this paper we derive a new lower bound on the error exponents of the mlan channel with a proper choice of the shaping lattice and the scaling parameter the new lower bound coincides with the random coding lower bound on the error exponents of the awgn channel at the same signal to noise ratio snr in the sphere packing and straight line regions this result implies that at least for rates close to channel capacity 1 writing on dirty paper is as reliable as writing on clean paper and 2 lattice encoding and decoding suffer no loss of error exponents relative to the optimal codes with maximum likelihood decoding for the awgn channel

well posedness of the ferrimagnetic equations in this paper we show the existence of global weak solutions of the ferrimagnetic equations on compact riemannian manifold using the penalty method we also show the uniqueness of the solution and its well posedness by the energy estimates method in lower dimensions in particular when the space dimension is one we can prove that the problem is globally well posed

some exact blow up solutions of landau lifshitz equations without gilbert term in this paper we construct some exact blow up solutions of multi dimensional space time landau lifshitz equations under the radially symmetric coordinates among these solutions two blow up finite energy solutions on s 2 are proposed the question of whether a solution of the inhomogeneous isotropic landau lifshitz equations can develop a finite time singularity on s 2 from smooth initial data with finite energy is not clear our results show that the blow up phenomenon can really happen on this initial data

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

on the diversity multiplexing tradeoff of unconstrained multiple access channels in this paper the optimal diversity multiplexing tradeoff dmt is investigated for the multiple input multiple output fading multiple access channel with no power constraints infinite constellations for k users k 1 m transmit antennas for each user and n receive antennas infinite constellations in general and lattices in particular are shown to attain the optimal dmt of finite constellations for n ge k 1 m 1 i e user limited regime on the other hand for n it is shown that infinite constellations cannot attain the optimal dmt this is in contrast to the point to point case in which the infinite constellations are dmt optimal for any m and n in general this paper shows that when the network is heavily loaded i e k max left 1 n m 1 m right considering the shaping region in the decoding process plays a crucial role in pursuing the optimal dmt by investigating the cases in which the infinite constellations are optimal and suboptimal this paper also gives a geometrical interpretation to the dmt of infinite constellations in multiple access channels

a simple proof for the existence of good pairs of nested lattices this paper provides a simplified proof for the existence of nested lattice codebooks allowing to achieve the capacity of the additive white gaussian noise channel as well as the optimal rate distortion tradeoff for a gaussian source the proof is self contained and relies only on basic probabilistic and geometrical arguments an ensemble of nested lattices that is different and more elementary than the one used in the previous proofs is introduced this ensemble is based on lifting different subcodes of a linear code to the euclidean space using construction a in addition to being simpler the analysis is less sensitive to the assumption that the additive noise is gaussian in particular for additive ergodic noise channels it is shown that the achievable rates of the nested lattice coding scheme depend on the noise distribution only via its power similarly the nested lattice source coding scheme attains the same rate distortion tradeoff for all ergodic sources with the same second moment

on the sum of the k largest eigenvalues of graphs and maximal energy of bipartite graphs abstract let g be a graph of order n also let 1 2 n be the eigenvalues of graph g in this paper we present the following upper bound on the sum of the k n largest eigenvalues of g in terms of the order n and negative inertia the number of negative eigenvalues i 1 k i n 2 1 k k 1 with equality holding if and only if g n k 1 or g p k t n p t p 2 with k 1 where p k t is the complete p partite graph of p t vertices with all partition sets having size t moreover we obtain a sharp upper bound on the energy of bipartite graphs with given order n and rank r we also propose an open problem as follows characterize all connected d regular bipartite graphs of order n with 5 distinct eigenvalues and rank r 2 n 2 4 d n 2 where d n 4 finally we give a partial solution to this problem

uplink downlink duality for integer forcing consider a gaussian multiple input multiple output mimo multiple access channel mac with channel matrix mathbf h and a gaussian mimo broadcast channel bc with channel matrix mathbf h mathsf t for the mimo mac the integer forcing architecture consists of first decoding integer linear combinations of the transmitted codewords which are then solved for the original messages for the mimo bc the integer forcing architecture consists of pre inverting the integer linear combinations at the transmitter so that each receiver can obtain its desired codeword by decoding an integer linear combination in both the cases integer forcing offers higher achievable rates than zero forcing while maintaining a similar implementation complexity this paper establishes an uplink downlink duality relationship for integer forcing i e any sum rate that is achievable via integer forcing on the mimo mac can be achieved via integer forcing on the mimo bc with the same sum power and vice versa using this duality relationship it is shown that integer forcing can operate within a constant gap of the mimo bc sum capacity finally the paper proposes a duality based iterative algorithm for the non convex problem of selecting optimal beamforming and equalization vectors and establishes that it converges to a local optimum

on the well posedness for the nonlinear radial schrodinger equation with spatial variable coefficients abstract in this paper we study the cauchy problem for a generalized nonlinear schrodinger equation with spatial variable coefficients a new weighted sobolev space w 1 q r is introduced and the range of admissible pairs for the strichartz estimates is improved as an application we show the local well posedness result for the initial data in w 1 q r

low density lattice codes low density lattice codes ldlc are novel lattice codes that can be decoded efficiently and approach the capacity of the additive white gaussian noise awgn channel in ldlc a codeword x is generated directly at the n dimensional euclidean space as a linear transformation of a corresponding integer message vector b i e x gb 1 where h g 1 is restricted to be sparse the fact that h is sparse is utilized to develop a linear time iterative decoding scheme which attains as demonstrated by simulations good error performance within 0 5 db from capacity at block length of n 100 000 symbols the paper also discusses convergence results and implementation considerations

approximately achieving gaussian relay network capacity with lattice based qmf codes recently a new relaying strategy quantize map and forward qmf scheme has been demonstrated to approximately achieve within an additive constant number of bits the gaussian relay network capacity universally i e for arbitrary topologies channel gains and snrs this was established using gaussian codebooks for transmission and random mappings at the relays in this paper we develop structured lattice codes that implement the qmf strategy the main result of this paper is that such structured lattice codes can approximately achieve the gaussian relay network capacity universally again within an additive constant in addition we establish a similar result for half duplex networks where we demonstrate that one can approximately achieve the capacity using fixed transmit receive tx rx schedules for the relays with no transmit power optimization across the different tx rx states of the network

lattices over eisenstein integers for compute and forward in this paper we consider the use of lattice codes over eisenstein integers for implementing a compute and forward protocol in wireless networks when channel state information is not available at the transmitter we extend the compute and forward paradigm of nazer and gastpar to decoding eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the poltyrev limit using this result we show that both the outage performance and error correcting performance of the nested lattice codebooks over eisenstein integers surpass those of lattice codebooks over integers considered by nazer and gastpar with no additional computational complexity

eigenvalues and energy in threshold graphs abstract assuming a uniform random model of selecting creation sequences we show that almost every connected threshold graph has more negative than positive eigenvalues we show that no threshold graphs have eigenvalues in 1 0 sequences of equienergetic graphs are given including the striking result that for all n 3 there exist n 1 threshold graphs of order n 2 pairwise noncospectral each equienergetic to k n 2 for all n 8 we find an n vertex hyperenergetic threshold graph