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

on optimal multilayer cyclotomic space time code designs high rate and large diversity product or coding advantage or coding gain or determinant distance or minimum product distance are two of the most important criteria often used for good space time code designs in recent linear lattice based space time code designs more attention is paid to the high rate criterion but less to the large diversity product criterion in this paper we consider these two criteria together for multilayer cyclotomic space time code designs in a previous paper we recently proposed a systematic cyclotomic diagonal space time code design over a general cyclotomic number ring that has infinitely many designs for a fixed number of transmit antennas where diagonal codes correspond to single layer codes in this paper in this paper we first propose a general multilayer cyclotomic space time codes we present a general optimality theorem for these infinitely many cyclotomic diagonal or single layer space time codes over general cyclotomic number rings for a general number of transmit antennas we then present optimal multilayer full rate cyclotomic space time code designs for two and three transmit antennas we also present an optimal two layer cyclotomic space time code design for three and four transmit antennas the optimality here is in the sense that for a fixed mean transmission signal power its diversity product is maximized or equivalently for a fixed diversity product its mean transmission signal power is minimized it should be emphasized that all the optimal multilayer cyclotomic space time codes presented in this paper have the nonvanishing determinant property

stark s conjecture over complex cubic number fields jjsystematic computation of stark units over nontotally real base fields is carried out for the first time since the information provided by stark s conjecture is significantly less in this situation than the information provided over totally real base fields new techniques are required precomputing stark units in relative quadratic extensions where the conjecture is already known to hold and coupling this information with the fincke pohst algorithm applied to certain quadratic forms leads to a significant reduction in search time for finding stark units in larger extensions where the conjecture is still unproven stark s conjecture is verified in each case for these stark units in larger extensions and explicit generating polynomials for abelian extensions over complex cubic base fields including hilbert class fields are obtained from the minimal polynomials of these new stark units

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

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

belief propagation algorithm and the ising model on networks with arbitrary distributions of motifs we generalize the belief propagation algorithm to sparse random networks with arbitrary distributions of motifs triangles loops etc each vertex in these networks belongs to a given set of motifs generalization of the configuration model these networks can be treated as sparse uncorrelated hypergraphs in which hyperedges represent motifs here a hypergraph is a generalization of a graph where a hyperedge can connect any number of vertices these uncorrelated hypergraphs are treelike hypertrees which crucially simplifies the problem and allows us to apply the belief propagation algorithm to these loopy networks with arbitrary motifs as natural examples we consider motifs in the form of finite loops and cliques we apply the belief propagation algorithm to the ferromagnetic ising model with pairwise interactions on the resulting random networks and obtain an exact solution of this model we find an exact critical temperature of the ferromagnetic phase transition and demonstrate that with increasing the clustering coefficient and the loop size the critical temperature increases compared to ordinary treelike complex networks however weak clustering does not change the critical behavior qualitatively our solution also gives the birth point of the giant connected component in these loopy networks

cccp algorithms to minimize the bethe and kikuchi free energies convergent alternatives to belief propagation this article introduces a class of discrete iterative algorithms that are provably convergent alternatives to belief propagation bp and generalized belief propagation gbp our work builds on recent results by yedidia freeman and weiss 2000 who showed that the fixed points of bp and gbp algorithms correspond to extrema of the bethe and kikuchi free energies respectively we obtain two algorithms by applying cccp to the bethe and kikuchi free energies respectively cccp is a procedure introduced here for obtaining discrete iterative algorithms by decomposing a cost function into a concave and a convex part we implement our cccp algorithms on two and three dimensional spin glasses and compare their results to bp and gbp our simulations show that the cccp algorithms are stable and converge very quickly the speed of cccp is similar to that of bp and gbp unlike cccp bp will often not converge for these problems gbp usually but not always converges the results found by cccp applied to the bethe or kikuchi free energies are equivalent or slightly better than those found by bp or gbp respectively when bp and gbp converge note that for these and other problems bp and gbp give very accurate results see yedidia et al 2000 and failure to converge is their major error mode finally we point out that our algorithms have a large range of inference and learning applications

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

inverse determinant sums and connections between fading channel information theory and algebra this work considers inverse determinant sums which arise from the union bound on the error probability as a tool for designing and analyzing algebraic space time block codes a general framework to study these sums is established and the connection between asymptotic growth of inverse determinant sums and the diversity multiplexing gain tradeoff is investigated it is proven that the growth of the inverse determinant sum of a division algebra based space time code is completely determined by the growth of the unit group this reduces the inverse determinant sum analysis to studying certain asymptotic integrals in lie groups using recent methods from ergodic theory a complete classification of the inverse determinant sums of the most well known algebraic space time codes is provided the approach reveals an interesting and tight relation between diversity multiplexing gain tradeoff and point counting in lie groups

majorization of gaussian processes and geometric applications let x i i 1 n denote the decreasing rearrangement of a sequence of real numbers x i i 1 n then for everyi j and every 1 k n the 2 nd order partial distributional derivatives satisfy the inequality frac partial 2 partial x i partial x j left sum limits l 1 k x l right leqq 0 as a consequence we generalize the theorem due to fernique and sudakov a generalization of slepian s lemma is also a consequence of another differential inequality we also derive a new proof and generalizations to volume estimates of intersecting spheres and balls in n proved by gromov

generalized exponential operators in the continuation of the confluent hypergeometric functions abstract use of successive continuation procedures for the confluent hypergeometric functions is developed analyzed and then applied to the evaluation of bessel functions k 0 x k 1 x k iv x as well as fourier series n 1 cos nc k 0 nb a miller formula for k iv x and k iv x is also discussed

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

full diversity high rate space time block codes from division algebras we present some general techniques for constructing full rank minimal delay rate at least one space time block codes stbcs over a variety of signal sets for arbitrary number of transmit antennas using commutative division algebras field extensions as well as using noncommutative division algebras of the rational field spl qopf embedded in matrix rings the first half of the paper deals with constructions using field extensions of spl qopf working with cyclotomic field extensions we construct several families of stbcs over a wide range of signal sets that are of full rank minimal delay and rate at least one appropriate for any number of transmit antennas we study the coding gain and capacity of these codes using transcendental extensions we construct arbitrary rate codes that are full rank for arbitrary number of antennas we also present a method of constructing stbcs using noncyclotomic field extensions in the later half of the paper we discuss two ways of embedding noncommutative division algebras into matrices left regular representation and representation over maximal cyclic subfields the 4 spl times 4 real orthogonal design is obtained by the left regular representation of quaternions alamouti s 1998 code is just a special case of the construction using representation over maximal cyclic subfields and we observe certain algebraic uniqueness characteristics of it also we discuss a general principle for constructing cyclic division algebras using the nth root of a transcendental element and study the capacity of the stbcs obtained from this construction another family of cyclic division algebras discovered by brauer 1933 is discussed and several examples of stbcs derived from each of these constructions are presented

constructing complete tables of quartic fields using kummer theory we explain how to construct tables of quartic fields of discriminant less than or equal to a given bound in an efficient manner using kummer theory instead of the traditional and much less efficient method using the geometry of numbers as an application we describe the computation of quartic fields of discriminant up to 107 the corresponding table being available by anonymous ftp

a note on the cluster variation method kikuchi s cluster variation method cvm is reformulated as the truncation of a mobius inversion an attempt is made to explicate and simplify the various approaches to the cvm this formulation makes apparent the connection of the method with other types of cluster approximation an illustration of the procedure is provided

unsupervised learning of phase transitions from principal component analysis to variational autoencoders we examine unsupervised machine learning techniques to learn features that best describe configurations of the two dimensional ising model and the three dimensional xy model the methods range from principal component analysis over manifold and clustering methods to artificial neural network based variational autoencoders they are applied to monte carlo sampled configurations and have a priori no knowledge about the hamiltonian or the order parameter we find that the most promising algorithms are principal component analysis and variational autoencoders their predicted latent parameters correspond to the known order parameters the latent representations of the models in question are clustered which makes it possible to identify phases without prior knowledge of their existence furthermore we find that the reconstruction loss function can be used as a universal identifier for phase transitions

adaptive estimation of a quadratic functional by model selection we consider the problem of estimating s 2 when s belongs to some separable hilbert space and one observes the gaussian process y t langles t rangle sigmal t for all t epsilon mathbb h where l is some gaussian isonormal process this framework allows us in particular to consider the classical gaussian sequence model for which mathbb h l 2 mathbb n and l t sum lambda geq1 t lambda varepsilon lambda where varepsilon lambda lambda geq1 is a sequence of i i d standard normal variables our approach consists in considering some at most countable families of finite dimensional linear subspaces of mathbb h the models and then using model selection via some conveniently penalized least squares criterion to build new estimators of s 2 we prove a general nonasymptotic risk bound which allows us to show that such penalized estimators are adaptive on a variety of collections of sets for the parameter s depending on the family of models from which they are built in particular in the context of the gaussian sequence model a convenient choice of the family of models allows defining estimators which are adaptive over collections of hyperrectangles ellipsoids l p bodies or besov bodies we take special care to describe the conditions under which the penalized estimator is efficient when the level of noise sigma tends to zero our construction is an alternative to the one by efroimovich and low for hyperrectangles and provides new results otherwise

hard spin mean field theory a systematic derivation and exact correlations in one dimension hard spin mean field theory is an improved mean field approach which has proven to give accurate results especially for frustrated spin systems with relatively little computational effort in this work the previous phenomenological derivation is supplanted by a systematic and generic derivation that opens the possibility for systematic improvements especially for the calculation of long range correlation functions a first level of improvement suffices to recover the exact long range values of the correlation functions in one dimension

explicit bounds for primes in residue classes let e k be an abelian extension of number fields with e q let and n denote the absolute discriminant and degree of e let denote an element of the galois group of e k we prove the following theorems assuming the extended riemann hypothesis 1 there is a degree 1 prime p of k such that p e k satisfying np 1 o 1 log 2n 2 2 there is a degree 1 prime p of k such that p e k generates the same group as satisfying np 1 o 1 log 2 3 for k q there is a prime p such that p e q satisfying p 1 o 1 log 2 in 1 and 2 we can in fact take p to be unramified in k q a special case of this result is the following 4 if gcd m q 1 the least prime p m mod q satisfies p 1 o 1 q log q 2 it follows from our proof that 1 3 also hold for arbitrary galois extensions provided we replace by its conjugacy class our theorems lead to explicit versions of 1 4 including the following the least prime p m mod q is less than 2 q log q 2

almost universal codes for mimo wiretap channels despite several works on secrecy coding for fading and mimo wiretap channels from an error probability perspective the construction of information theoretically secure codes over such channels remains an open problem in this paper we consider a fading wiretap channel model where the transmitter has only partial statistical channel state information our channel model includes static channels i i d block fading channels and ergodic stationary fading with fast decay of large deviations for the eavesdropper s channel we extend the flatness factor criterion from the gaussian wiretap channel to fading and mimo wiretap channels and establish a simple design criterion where the normalized product distance minimum determinant of the lattice and its dual should be maximized simultaneously moreover we propose concrete lattice codes satisfying this design criterion which are built from algebraic number fields with constant root discriminant in the single antenna case and from division algebras centered at such number fields in the multiple antenna case the proposed lattice codes achieve strong secrecy and semantic security for all rates r where c b and c e are bob and eve s channel capacities respectively and kappa is an explicit constant gap furthermore these codes are almost universal in the sense that a fixed code is good for secrecy for a wide range of fading models finally we consider a compound wiretap model with a more restricted uncertainty set and show that rates r are achievable where bar c b is a lower bound for bob s capacity and bar c e is an upper bound for eve s capacity for all the channels in the set

network inference using asynchronously updated kinetic ising model network structures are reconstructed from dynamical data by respectively naive mean field nmf and thouless anderson palmer tap approximations tap approximation adds simple corrections to the nmf approximation taking into account the effect of the focused spin on itself via its influence on other neighboring spins for tap approximation we use two methods to reconstruct the network a iterative method b casting the inference formula to a set of cubic equations and solving it directly we investigate inference of the asymmetric sherrington kirkpatrick as k model using asynchronous update the solutions of the set of cubic equations depend on temperature t in the as k model and a critical temperature t c 2 1 is found the two methods for tap approximation produce the same results when the iterative method is convergent compared to nmf tap is somewhat better at low temperatures but approaches the same performance as temperature increases both nmf and tap approximation reconstruct better for longer data length l but for the degree of improvement tap performs better than nmf

on the convergence of kikuchi s natural iteration method in this article we investigate on the convergence of the natural iteration method a numerical procedure widely employed in the statistical mechanics of lattice systems to minimize kikuchi s cluster variational free energies we discuss a sufficient condition for the convergence based on the coefficients of the cluster entropy expansion depending on the lattice geometry we also show that such a condition is satisfied for many lattices usually studied in applications finally we consider a recently proposed class of methods for the minimization of kikuchi functionals showing that the natural iteration method turns out as a particular instance of that class

hermite and smith normal form algorithms over dedekind domains we show how the usual algorithms valid over euclidean domains such as the hermite normal form the modular hermite normal form and the smith normal form can be extended to dedekind rings in a sequel to this paper we will explain the use of these algorithms for computing in relative extensions of number fields

diagonal lattice space time codes from number fields and asymptotic bounds in this paper we reformulate some constructions of real and complex diagonal lattice space time codes from number fields which have been given explicitly or implicitly by other researchers these constructions establish a connection between good diagonal lattice space time codes and number fields with small absolute values of discriminants we present two tables for diversity products of some lattice space time codes from these constructions the maximal rank of diagonal lattice space time codes with positive diversity product is determined we also discuss the asymptotic problem of lattice space time codes by using an infinite tower of hilbert class field and a tamely ramified class field tower we obtain asymptotically good sequences of lattice space time codes some asymptotic upper bounds are given in the paper as well

stable propagation algorithm for the minimization of the bethe free energy we propose a new propagation algorithm for the minimization of the bethe free energy for a generic lattice model with pair interactions the algorithm turns out to be more stable than belief propagation as it reaches a fixed point also for highly frustrated systems such as spin glasses and faster than the provably convergent double loop algorithms

computing class fields via the artin map based on an explicit representation of the artin map for kummer extensions we present a method to compute arbitrary class fields as in the proofs of the existence theorem the problem is first reduced to the case where the field contains sufficiently many roots of unity using kummer theory and an explicit version of the artin reciprocity law we show how to compute class fields in this case we conclude with several examples

association of normal random variables and slepian s inequality abstract a simple extension of slepian s inequality is given which implies the original inequality and the result that positively correlated normal variables are associated author

analysis of dynamic ising model by a variational approximate method estimation of transfer entropy transfer entropy is an important index for investigating the causal relationship between random variables it is known that a phase transition can be detected by transfer entropy in this work i propose a method for obtaining the transfer entropy by applying the variational approximation method to a dynamic ising model our method is also effective for dynamics that do not satisfy the detailed balance condition our method finds the transfer entropy by solving a transcendental equation when compared with the transfer entropy obtained by monte carlo sampling it was found that they qualitatively matched near the transition point moreover the match was obtained with good accuracy in the region deviated from the transition point

a note on the continuity of solutions of parametric linear programs this note shows that if a parametric linear program min cx ax b1 b2 x 0 has optimal solutions in an interval left underset raise0 3em hbox smash scriptscriptstyle theta bar theta right for then depending on degeneracy the solutions is a continuous vector function or a continuous point to set mapping in the latter case an algorithm is introduced to solve the problem and generate a continuous vector solution in left underset raise0 3em hbox smash scriptscriptstyle theta bar theta right

computing ray class groups conductors and discriminants we use the algorithmic computation of exact sequences of abelian groups to compute the complete structure of z k m for an ideal m of a number field k as well as ray class groups of number fields and conductors and discriminants of the corresponding abelian extensions as an application we give several number fields with discriminants less than previously known ones

ldpc coded mimo multiple access with iterative joint decoding an efficient scheme for the multiple access multiple input multiple output mimo channel is proposed which operates well also in the single user regime as well as in a direct sequence spread spectrum ds cdma setting the design features scalability and is of limited complexity the system employs optimized low density parity check ldpc codes and an efficient iterative belief propagation bp detection which combines linear minimum mean square error lmmse detection and iterative interference cancellation ic this combination is found to be necessary for efficient operation in high system loads spl alpha 1 an asymptotic density evolution de is used to optimize the degree polynomials of the underlining ldpc code and thresholds as close as 0 77 db to the channel capacity are evident for a system load of 2 replacing the lmmse with the complex individually optimal multiuser detector io mud further improves the performance up to 0 14 db from the capacity comparing the thresholds of a good single user ldpc code to the multiuser optimized ldpc code both over the above multiuser channel reveals a surprising 8 db difference emphasizing thus the necessity of optimizing the code the asymptotic analysis of the proposed scheme is verified by simulations of finite systems which reveal meaningful differences between the performances of mimo systems with single and multiple users and demonstrate performance similar to previously reported techniques but with higher system loads and significantly lower receiver complexity

good lattice constellations for both rayleigh fading and gaussian channels recent work on lattices matched to the rayleigh fading channel has shown how to construct good signal constellations with high spectral efficiency we present a new family of lattice constellations based on complex algebraic number fields which have good performance on rayleigh fading channels some of these lattices also present a reasonable packing density and thus may be used at the same time over a gaussian channel conversely we show that particular versions of the best lattice packings d sub 4 e sub 6 e sub 8 k sub 12 spl lambda sub 16 spl lambda sub 24 constructed from totally complex algebraic cyclotomic fields present better performance over the rayleigh fading channel the practical interest in such signal constellations rises from the need to transmit information at high rates over both terrestrial and satellite links some further results in algebraic number theory related to ideals and their factorization are presented and the decoding algorithm used with these lattice constellations are illustrated together with practical results

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

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

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

hermite normal form computation using modulo determinant arithmetic this paper describes a new class of hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation this class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string computational results are also given

a note on cactus trees variational vs recursive approach in this article we deal with the variational approach to cactus trees husimi trees and the more common recursive approach that are in principle equivalent for finite systems we discuss in detail the conditions under which the two methods are equivalent also in the analysis of infinite self similar cactus trees usually investigated to the purpose of approximating ordinary lattice systems such issue is hardly ever considered in the literature we show on significant test models that the phase diagram and the thermodynamic quantities computed by the variational method when they deviates from the exact bulk properties of the cactus system generally provide a better approximation to the behavior of a corresponding ordinary system generalizing a property proved by kikuchi we also show that the numerical algorithm usually employed to perform the free energy minimization in the variational approach is always convergent

on the computation of totally real quartic fields of small discriminant all totally real quartic fields of discriminant less than 106 are computed the method used to generate the fields is derived from delone and faddeev with corrections and improvements a new method for deciding field i8omorphism is used to eliminate redundant examples integral bases and galois groups are given for each field 1 introduction in 3 pp 184 200 delone and faddeev give a number geometric method for computing generating equations for all totally real quartic fields of discriminant not exceeding a fixed upper bound l e r 0 this method is well suited for implementation on a computer unfortunately the original paper contains many mistakes and the proofs are sometimes hard to understand in the present paper we give a corrected version of the delone faddeev algorithm we describe its implementation and we give a table of all totally real quartic fields of discriminant less than 1 000 000 including their integral bases and the galois groups of their normal closures a different method for computing totally real quartic fields was given by god win 6 and for general algorithms for computing fields of small discriminants see martinet 8 and pohst 9 10

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

separation of transparent layers using focus consider situations where the depth at each point in the scene is multi valued due to the presence of a virtual image semi reflected by a transparent surface the semi reflected image is linearly superimposed on the image of an object that is behind the transparent surface a novel approach is proposed for the separation of the superimposed layers focusing on either of the layers yields initial separation but crosstalk remains the separation is enhanced by mutual blurring of the perturbing components in the images however this blurring requires the estimation of the defocus blur kernels we thus propose a method for self calibration of the blur kernels given the raw images the kernels are sought to minimize the mutual information of the recovered layers autofocusing and depth estimation in the presence of semi reflections are also considered experimental results are presented

two efficient algorithms for the computation of ideal sums in quadratic orders this paper deals with two different asymptotically fast algorithms for the computation of ideal sums in quadratic orders if the class number of the quadratic number field is equal to 1 these algorithms can be used to calculate the gcd in the quadratic order we show that the calculation of an ideal sum in a fixed quadratic order can be done as fast as in z up to a constant factor i e in o n logn where n bounds the size of the operands and n denotes an upper bound for the multiplication time of n bit integers using schonhage strassen s asymptotically fast multiplication for n bit integers we achieve n o n log n log log n

algebraic lattice constellations bounds on performance in this work we give a bound on performance of any full diversity lattice constellation constructed from algebraic number fields we show that most of the already available constructions are almost optimal in the sense that any further improvement of the minimum product distance would lead to a negligible coding gain furthermore we discuss constructions minimum product distance and bounds for full diversity complex rotated z i sup n lattices for any dimension n which avoid the need of component interleaving

universal space time trellis codes this article gives practical examples of space time trellis codes performing as predicted by root and varaiya s 1968 compound channel theorem specifically 32 state and 64 state 2 spl times 2 space time trellis codes are presented that provide a bit error rate ber of 10 sup 5 on all 2 spl times 2 matrix channels with an excess mutual information mi within 8 of the excess mi required by standard trellis codes of the same complexity operating only in additive white gaussian noise awgn not surprisingly the universal space time trellis coded modulations st tcms provide average bit and frame error rates in quasi static rayleigh fading qrf that are comparable to those achieved by st tcms designed specifically for qrf as well as standard tcms followed by the alamouti 1998 space time block code however all of these other schemes require more excess mi in the worst case and some have a significantly wider variation in the required excess mi the article also compares the universal and quasi static rayleigh fading design approaches analytically and bounds the worst case distance of a trellis code on a 2 spl times 2 channel using the distances of the code on singular and unitary channels this bound is extended to the more general n sub t spl times n sub r scenario

elliptically contoured distributions given two covariance matricesr ands for a given elliptically contoured distribution we show how simple inequalities between the matrix elements imply thate r f e s f e g whenx xi1 i2 in is a multiindex vector and f x mathop min limits i 1 mathop max limits i 2 mathop min limits i 3 max x i 1 i n orf x is the indicator function of sets such as mathop cap limits i 1 mathop cup limits i 2 mathop cap limits i 3 cup x i 1 i n mathop limits lambda i 1 i n of which the well known slepian s inequality n 1 is a special case

cluster variation method in statistical physics and probabilistic graphical models the cluster variation method cvm is a hierarchy of approximate variational techniques for discrete ising like models in equilibrium statistical mechanics improving on the mean field approximation and the bethe peierls approximation which can be regarded as the lowest level of the cvm in recent years it has been applied both in statistical physics and to inference and optimization problems formulated in terms of probabilistic graphical models the foundations of the cvm are briefly reviewed and the relations with similar techniques are discussed the main properties of the method are considered with emphasis on its exactness for particular models and on its asymptotic properties the problem of the minimization of the variational free energy which arises in the cvm is also addressed and recent results about both provably convergent and message passing algorithms are discussed

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

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

constellations matched to the rayleigh fading channel we introduce a new technique for designing signal sets matched to the rayleigh fading channel in particular we look for n dimensional n spl ges 2 lattices whose structure provides nth order diversity our approach is based on a geometric formulation of the design problem which in turn can be solved by using a number geometric approach specifically a suitable upper bound on the pairwise error probability makes the design problem tantamount to the determination of what is called a critical lattice of the body s x x sub 1 spl middot spl middot spl middot x sub n spl isin r sup n spl pi sub i 1 sup n x sub i spl les 1 the lattices among which we search for an optimal solution are the standard embeddings in r sup n of the number ring of some totally real number field of degree n over q simulation results confirm that this approach yields lattices with considerable coding gains

on three new cluster variation approximations abstract we discuss in some detail three new cluster variation approximations we used in a previous paper in connection with pade approximants to investigate critical behaviour of classical spin models on various lattices the approximations are applied to the ising model on the simple cubic face centered cubic and semi infinite simple cubic lattices the natural iteration equations and estimates for the critical temperatures and for the range of validity of the approximations are given

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

random k satisfiability problem from an analytic solution to an efficient algorithm we study the problem of satisfiability of randomly chosen clauses each with k boolean variables using the cavity method at zero temperature we find the phase diagram for the k 3 case we show the existence of an intermediate phase in the satisfiable region where the proliferation of metastable states is at the origin of the slowdown of search algorithms the fundamental order parameter introduced in the cavity method which consists of surveys of local magnetic fields in the various possible states of the system can be computed for one given sample these surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems one such algorithm is shown here for the k 3 satisfiability problem with very good performances

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

computing the hilbert class field of real quadratic fields using the units appearing in stark s conjectures on the values of l functions at s 0 we give a complete algorithm for computing an explicit generator of the hilbert class field of a real quadratic field

computing the multiplicative group of residue class rings let k be a global field with maximal order 0k and let m0 be an ideal of 0k we present algorithms for the computation of the multiplicative group 0k m0 of the residue class ring 0k m0 and the discrete logarithm therein based on the explicit representation of the group of principal units we show how these algorithms can be combined with other methods in order to obtain more efficient algorithms they are applied to the computation of the ray class group clkm modulo m m0m where m denotes a formal product of real infinite places and also to the computation of conductors of ideal class groups and of discriminants and genera of class fields

approximately universal codes over slow fading channels performance of reliable communication over a coherent slow fading multiple input multiple output mimo channel at high signal to noise ratio snr is succinctly captured as a fundamental tradeoff between diversity and multiplexing gains this paper studies the problem of designing codes that optimally tradeoff the diversity and multiplexing gains the main contribution is a precise characterization of codes that are universally tradeoff optimal i e they optimally tradeoff the diversity and multiplexing gains for every statistical characterization of the fading channel this characterization is referred to as approximate universality the approximation is in the connection between error probability and outage capacity with diversity and multiplexing gains respectively the characterization of approximate universality is then used to construct new coding schemes as well as to show optimality of several schemes proposed in the space time coding literature

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

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

asymptotically fast triangularization of matrices over rings this paper considers the problems of triangularizing and diagonalizing matrices over rings with particular emphasis on the integral case it begins with a description of fast algorithms for the computation of hermite and smith normal forms of integer matrices then it shows how to apply fast matrix multiplication techniques to the problem of triangularizing a matrix over a ring using elementary column operations these general results lead to an algorithm for triangularizing integer matrices that has a faster running time than the known hermite normal form algorithms the triangular matrix that is computed has small entries like the hermite normal form and will suffice for many applications

tables of octic fields with a quartic subfield we describe the computation of extended tables of degree 8 fields with a quartic subfield using class field theory in particular we find the minimum discriminants for all signatures and for all the possible galois groups we also discuss some phenomena and statistics discovered while making the tables such as the occurrence of 11 non isomorphic number fields having the same discriminant or several pairs of non isomorphic number fields having the same dedekind zeta function