on the solution of stein s equation and fisher s information matrix of an armax process the main goal of this paper consists in expressing the solution of a stein equation in terms of the fisher information matrix fim of a scalar armax process a condition for expressing the fim in terms of a solution to a stein equation is also set forth such interconnections can be derived when a companion matrix with eigenvalues equal to the roots of an appropriate polynomial associated with the armax process is inserted in the stein equation the case of algebraic multiplicity greater than or equal to one is studied the fim and the corresponding solution to stein s equation are presented as solutions to systems of linear equations the interconnections are obtained by using the common particular solution of these systems the kernels of the structured coefficient matrices are described as well as some right inverses this enables us to find a solution to the newly obtained linear system of equations

unequal error protection for convolutional codes in this correspondence unequal error correcting capabilities of convolutional codes are studied for errors in the information symbols and code symbols the free input and output distances respectively serve as unequal counterparts to the free distance when communication takes place close to or above the channel capacity the error bursts tend to be long and the free distance is not any longer useful as the measure of the error correcting capability thus the active burst distance for a given output and the active burst distance for a given input are introduced as unequal counterparts to the active burst distance and improved estimates of the unequal error correcting capabilities of convolutional codes are obtained and illustrated by examples finally it is shown how to obtain unequal error protection for both information and code symbols using woven convolutional codes

symbolic computation of fisher information matrices for parametrized state space systems the asymptotic fisher information matrix fim has several applications in linear systems theory and statistical parameter estimation it occurs in relation to the cramer rao lower bound for the covariance of unbiased estimators it is explicitly used in the method of scoring and it determines the asymptotic convergence properties of various system identification methods it defines the fisher metric on manifolds of systems and it can be used to analyze questions on identifiability of parametrized model classes for many of these applications exact symbolic computation of the fim can be of great use in this paper two different methods are described for the symbolic computation of the asymptotic fim the first method applies to parametrized mimo state space systems driven by stationary gaussian white noise and proceeds via the solution of discrete time lyapunov and sylvester equations for which a method based on faddeev sequences is used this approach also leads to new short proofs of certain well known results on the structure of the fim for siso arma systems the second method applies to parametrized siso state space systems and uses an extended faddeev algorithm for both algorithms the concept of a faddeev reachability matrix and the solution of discrete time sylvester equations in controller companion form are central issues the methods are illustrated by two worked examples the first concerns three different parametrizations of the class of stable ar n systems the second concerns a model for an industrial mixing process in which the value of exact computation to answer identifiability questions becomes particularly clear

optical orthogonal codes with unequal auto and cross correlation constraints an optical orthogonal code ooc is a collection of binary sequences with good auto and cross correlation properties they were defined by salehi and others as a means of obtaining code division multiple access on optical networks up to now all work on oocs have assumed that the constraint placed on the autocorrelation and that placed on the cross correlation are the same we consider codes for which the two constraints are not equal specifically we develop bounds on the size of such oocs and demonstrate constriction techniques for building them the results demonstrate that a significant increase in the code size is possible by letting the autocorrelation constraint exceed the cross correlation constraint these results suggest that for a given performance requirement the optimal ooc may be one with unequal constraints this paper also views oocs with unequal auto and cross correlation constraints as constant weight unequal error protection uep codes with two levels of protection the bounds derived are interpreted from this viewpoint

a unified method for evaluating the error correction radius of reliability based soft decision algorithms for linear block codes this paper presents a unified method for evaluating the error correction radii of many reliability based soft decision decoding algorithms for binary linear block codes based on this unified method these decoding algorithms as well as their potential improvements can be compared directly the error correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft decision decoder to the transmitted signal sequence it is shown that this problem can be formulated as a constrained optimization problem although no general closed form expression is possible a simple algorithm that always converges to the optimum solution of this optimization problem is presented based on these results the error correction radii of some well known reliability based soft decision decoding algorithms are revisited and compared

tailbiting codes bounds and search results tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied the notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower bounded the asymptotic behavior of the derived bound is investigated furthermore for regular tailbiting codes the product state complexity is lower bounded tables of new tailbiting trellis representations of linear block codes of rates 1 2 1 3 and 1 4 are presented almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time varying trellises which are optimal five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters four of them are also superior to any previously known nonlinear code with the same parameters also more than 40 other quasi cyclic codes have been found that improve the parameter set of previously known quasi cyclic codes

tensor sylvester matrices and the fisher information matrix of varmax processes the purpose of this paper is to develop compact expressions for the fisher information matrix fim of a gaussian stationary vector autoregressive and moving average process with exogenous or input variables a vector armax or varmax process we develop a representation of the fim based on multiple sylvester matrices an extension of this representation yields another one but in terms of tensor sylvester matrices in order to obtain the results presented in this paper the approach used in a klein g melard p spreij on the resultant property of the fisher information matrix of a vector arma process linear algebra appl 403 2005 291 313 is extended

on minima of a functional of the gradient necessary conditions in this paper we propose a new method for texture segmentation based on the use of texture feature detectors derived from a decorrelation procedure of a modified version of a pseudo wigner distribution pwd the decorrelation procedure is accomplished by a cascade recursive least squared crls principal component pc neural network the goal is to obtain a more eacient analysis of images by combining the advantages of using a high resolution joint representation given by the pwd with an eaective adaptive principal component analysis pca through the use of feedforward neural networks 1999 elsevier science b v all rights reserved

fast generalized minimum distance decoding of algebraic geometry and reed solomon codes generalized minimum distance gmd decoding is a standard soft decoding method for block codes we derive an efficient general gmd decoding scheme for linear block codes in the framework of error correcting pairs special attention is paid to reed solomon rs codes and one point algebraic geometry ag codes for rs codes of length n and minimum hamming distance d the gmd decoding complexity turns out to be in the order o nd where the complexity is counted as the number of multiplications in the field of concern for ag codes the gmd decoding complexity is highly dependent on the curve in consideration it is shown that we can find all relevant error erasure locating functions with complexity o o sub 1 nd where o sub 1 is the size of the first nongap in the function space associated with the code a full gmd decoding procedure for a one point ag code can be performed with complexity o dn sup 2

a bounded distance decoding algorithm for the leech lattice with generalizations an algorithm is given that decodes the leech lattice with not much more than twice the complexity of soft decision decoding of the golay code the algorithm has the same effective minimum distance as maximum likelihood decoding and increases the effective error coefficient by less than a factor or two the algorithm can be recognized as a member of the class of multistage algorithms that are applicable to hierarchical constructions it is readily generalized to lattices that can be expressed in terms of binary code formulas and in particular to construction b lattices

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

the bezoutian state space realizations and fisher s information matrix of an arma process in this paper we derive some properties of the bezout matrix and relate the fisher information matrix for a stationary arma process to the bezoutian some properties are explained via realizations in state space form of the derivatives of the white noise process with respect to the parameters a factorization of the fisher information matrix as a product in factors which involve the bezout matrix of the associated ar and ma polynomials is derived from this factorization we can characterize singularity of the fisher information matrix

coding for the binary symmetric broadcast channel with two receivers block coding for the binary symmetric broadcast channel with two receivers is investigated a graph theoretic approach to the construction of a class of block codes with unequal error protection for two different sets of messages is presented a code in this class is a direct sum of two component codes each set of messages is encoded based on one component code the codes in this class are easy to implement decoding of these codes is presented and lower bounds on the achievable rates of these codes are derived the bounds are tighter than the katsman s bounds

calculation of the fisher information matrix for multidimensional data sets data sets that are acquired in many practical systems can be described as the output of a multidimensional linear separable denominator system with gaussian measurement noise an important example is nuclear magnetic resonance nmr spectroscopy in nmr spectroscopy high accuracy parameter estimation is of central importance a classical result on the crame spl acute r rao lower bound states that the inverse of the fisher information matrix fim provides a lower bound for the covariance of any unbiased estimator of the parameter vector the calculation of the fim is therefore of central importance for an assessment of the accuracy with which parameters can be estimated it is shown how the fim can be expressed using the matrices that determine the system that generates the data set for uniformly sampled data it is shown how the fim can be expressed through the solutions of lyapunov equations the novel techniques are demonstrated with an example arising from nmr spectroscopy

minimizing non convex multiple integrals a density result we consider variational problems of the form where is a bounded open set in r n f r n r is a possibly non convex lower semicontinuous function with p growth at infinity for some 1 p u 0 is any function in w 1 p assuming that the convex envelope f of f is affine on each connected component of the set f f we prove the existence of solutions to p for every continuous function g such that i g has no strict local minima and ii every convergent sequence of extremum points of g eventually belongs to an interval where g is constant thus showing that the set of continuous functions g that yield existence to p is dense in the space of continuous functions on r

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

decoding the golay codes we introduce exceptionally simple decoding algorithms for the two extended golay codes the algorithms are based on recent methods of conway and curtis of finding the unique blocks containing five points in either the 5 8 24 steiner system or the 5 6 12 steiner system these decoding methods are simple enough to enable decoding extended golay codes by hand both of the methods involve relations between the extended golay codes and other self dual codes proofs are given explaining these relationships and why the decoding methods work the decoding algorithms are explained and illustrated with many examples 3 chap 12 has facts about the mathieu group and some details about decoding the golay codes

bounded distance decoding algorithms decision regions and pseudo nearest neighbors for a code c bounded distance decoding algorithms perform as optimal algorithms within the balls b c centered at the codewords c spl isin c with radius equal to half the minimum euclidean distance of the code thus distinct bounded distance algorithms vary in performance due to their different behavior outside the balls b c we investigate this issue by analyzing the decision regions of some known e g gmd and some new bounded distance algorithms presented in this work in particular we show that there are three distinct types of nearest neighbors and classify them according to their influence on the decision region simulation results and computer generated images of the decision regions are provided to illustrate the analytical results for block and lattice codes on additive white gaussian noise awgn channels

on random rotations diversity and minimum mse decoding of lattices we establish a simple relation between high diversity multidimensional rotations obtained from totally complex cyclotomic fields and the discrete fourier transform the diversity distribution of an hadamard like random rotation is derived analytically it is shown that a random multidimensional rotation exhibits an excellent diversity distribution and can be combined to quadrature amplitude modulation qam constellations to combat channel fading we also describe a mean square error mse universal lattice decoder suitable for large dimensions up to 1024 the mse criterion treats the lattice structure as intersymbol interference the universal decoder is applied to both gaussian and rayleigh fading channels to decode dense lattice sphere packings and rotated cubic constellations respectively

a uniqueness result for a class of non strictly convex variational problems abstract let be a smooth domain in r 2 we prove that if g 0 0 is convex with g 0 g t whenever t 0 then there exists an unique minimizer u c 0 1 of the functional u g u d x d y among all lipschitz continuous functions that assume the same value of u on

the leech lattice and the golay code bounded distance decoding and multilevel constructions multilevel constructions of the binary golay code and the leech lattice are described both constructions are based upon the projection of the golay code and the leech lattice onto the 6 3 4 hexacode over gf 4 however unlike the previously reported constructions the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates based upon the multilevel structure of the golay code and the leech lattice efficient bounded distance decoding algorithms are devised the bounded distance decoder for the binary golay code requires at most 431 operations as compared to 651 operations for the best known maximum likelihood decoder efficient bounded distance decoding of the leech lattice is achieved by means of partitioning it into four cosets of q sub 24 beyond the conventional partition into two h sub 24 cosets the complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case as compared to about 3600 operations for the best known in maximum likelihood decoder it is shown that the proposed algorithms decode correctly at least up to the guaranteed error correction radius of the maximum likelihood decoder thus the loss in coding gain is due primarily to an increase in the effective error coefficient which is calculated exactly for both algorithms furthermore the performance of the leech lattice decoder on the awgn channel is evaluated experimentally by means of a comprehensive computer simulation the results show a loss in coding gain of less than 0 1 db relative to the maximum likelihood decoder for ber ranging from 10 sup 1 to 10 sup 7

maximum likelihood soft decoding of binary block codes and decoders for the golay codes maximum likelihood soft decision decoding of linear block codes is addressed a binary multiple check generalization of the wagner rule is presented and two methods for its implementation one of which resembles the suboptimal forney chase algorithms are described besides efficient soft decoding of small codes the generalized rule enables utilization of subspaces of a wide variety thereby yielding maximum likelihood decoders with substantially reduced computational complexity for some larger binary codes more sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the 24 12 golay code yielding a computational gain factor of about 2 with respect to previous methods a ternary single check version of the wagner rule is applied for efficient soft decoding of the 12 6 ternary golay code

geometrical and performance analysis of gmd and chase decoding algorithms the overall number of nearest neighbors in bounded distance decoding bdd algorithms is given by n sub 0 eff n sub 0 n sub bdd where nbdd denotes the number of additional non codeword neighbors that are generated during the suboptimal decoding process we identify and enumerate the nearest neighbors associated with the original generalized minimum distance gmd and chase 1972 decoding algorithms after careful examination of the decision regions of these algorithms we derive an approximated probability ratio between the error contribution of a noncodeword neighbor one of n sub bdd points and a codeword nearest neighbor for chase algorithm 1 it is shown that the contribution to the error probability of a noncodeword nearest neighbor is a factor of 2 sup d 1 less than the contribution of a codeword while for chase algorithm 2 the factor is 2 sup d 2 1 d being the minimum hamming distance of the code for chase algorithm 3 and gmd a recursive procedure for calculating this ratio which turns out to be nonexponential in d is presented this procedure can also be used for specifically identifying the error patterns associated with chase algorithm 3 and gmd utilizing the probability ratio we propose an improved approximated upper bound on the probability of error based on the union bound approach simulation results are given to demonstrate and support the analytical derivations

computer search for binary cyclic uep codes of odd length up to 65 exhaustive computation by a computer was used to find the unequal error protection capabilities of all binary cyclic codes of odd length up to 65 that have minimum distances of at least 3 for those codes for which upper bounds can only be computed on their unequal error protection capabilities an analytic method developed by v n dynkin and v a togonidze 1976 is used to show that the upper bounds meet the exact unequal error protection capabilities

cyclic unequal error protection codes constructed from cyclic codes of composite length the unequal error correction capabilities of binary cyclic codes of composite length are investigated under certain conditions direct sums of concatenated codes have unequal error correction capabilities by a modified hartmann and tzeng 1973 algorithm it is shown that a binary cyclic code of composite length is equivalent to the direct sum of concatenated codes with this some binary cyclic unequal error protection uep codes are constructed finally the authors present a class of two level uep cyclic direct sum codes which provide error correction capabilities higher than those guaranteed by the blokh zyablov 1974 constructions

on minima of radially symmetric functionals of the gradient where b is the unit ball of ir and the map u g r v is lower semicontinuous but not necessarily convex problems of this kind arise in domains as different as nonlinear elasticity fluid dynamics and shape optimization and either for the problem of the existence of solutions or for the properties of the relaxed problem are considered in l 7 in particular the very same problem is considered in 7 our results present the following features a no smoothness on g or h is required g is either a normal integrand or a lower semicontinuous function b the case h 0 is allowed in this case the assumption on g reduces for the existence of solutions to g being lower semicontinuous and growing at infinity as is to be expected for the uniqueness in addition to g being strictly increasing as is also to be expected c the case h au is allowed for a 0 our theorems yield at once existence and uniqueness of solutions with no further assumptions on g besides lower semicontinuity and growth at infinity

construction of the exact fisher information matrix of gaussian time series models by means of matrix differential rules the fisher information matrix is of fundamental importance for the analysis of parameter estimation of time series models in this paper the exact information matrix of a multivariate gaussian time series model expressed in state space form is derived a computationally efficient procedure is used by applying matrix differential rules for the derivatives of a matrix function j j with respect to its vector argument an algorithm is given it is sketched for the general state space structure without specifying a parametrization it is then detailed for the vector autoregressive moving average varma model with a given parametrization where explicit recurrent relations are developed

existence of solutions for a class of non convex minimum problems p celada1 s perrotta2 g treu3 1 dipartimento di scienze matematiche universita degli studi di trieste p le europa 1 i 34127 trieste italia e mail celada univ trieste it 2 dipartimento di matematica pura ed applicata g vitali universita degli studi di modena via campi 213 b i 41100 modena italia e mail perrotta c220 unimo it 3 dipartimento di matematica ed informatica universita degli studi di udine v le delle scienze 206 i 33100 udine italia e mail treu dimi uniud it

a new reed solomon code decoding algorithm based on newton s interpolation a reed solomon code decoding algorithm based on newton s interpolation is presented this algorithm has as main application fast generalized minimum distance decoding of reed solomon codes it uses a modified berlekamp massey algorithm to perform all necessary generalized minimum distance decoding steps in only one run with a time domain form of the new decoder the overall asymptotic generalized minimum distance decoding complexity becomes o dn with n the length and d the distance of the code including the calculation of all error locations and values this asymptotic complexity is optimal other applications are the possibility of fast decoding of reed solomon codes with adaptive redundancy and a general parallel decoding algorithm with zero delay

existence of scalar minimizers for nonconvex simple integrals of sum type abstract one usually asks as sufficient condition to guarantee the existence of minimizersx a b r in ac for the integral ba h x x dt withh having superlinear growth at and being any lower semicontinuous function bounded below thath be convex that is h h at all in r the purpose of this paper is to show that the conditionh 0 h 0 is actually enough to guarantee the existence of minimizers

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

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

error trellises for convolutional codes i construction an error trellis is a directed graph that represents all the sequences belonging to the coset which contains the symbol by symbol detected version of a given received sequence a modular construction of error trellises for an n k convolutional code over gf q is presented the trellis is designed on the basis of partitioning the scalar check matrix of the code into submatrices of l rows accompanied with a corresponding segmentation of the syndrome the value of the design parameter l is an essentially unconstrained multiple of n k for all the cosets of the code the sections of the error trellis are drawn from a collection of only q sup l modules the module for each section is determined by the value of the associated syndrome segment in case the construction is based on a basic polynomial check matrix either canonical or noncanonical then the error trellis is minimal in the sense that spl sigma spl les spl mu where spl sigma is the dimension of the state space of the trellis and spl mu is the constraint length of a canonical generator matrix for the code for basic check matrices with delay free columns the inequality reduces to spl sigma spl mu

the nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

asymptotic optimality of the gmd and chase decoding algorithms the generalized minimum distance gmd and chase 1972 decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white gaussian noise awgn channel we compute the limitation of the ratio between the probability of decoding error for the gmd or any one of the chase decoding algorithms and that of the maximum likelihood ml decoding when the signal to noise ratio snr approaches infinity if the minimum hamming distance of the code is greater than 2 the limitation is shown to be equal to 1 and thus the gmd and chase decoding algorithms are asymptotically optimum

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

computation of the fisher information matrix for siso models closed form expressions and an algorithm for obtaining the fisher information matrix of gaussian single input single output siso time series models are presented it enables the computation of the asymptotic covariance matrix of maximum likelihood estimators of the parameters the procedure makes use of the autocovariance function of one or more autoregressive processes under certain conditions the siso model can be a special case of a vector autoregressive moving average arma model for which there is a method to evaluate the fisher information matrix that method is compared with the procedure described in the paper

an improvement to generalized minimum distance decoding a novel acceptance criterion that is less stringent than previous criteria is developed the criterion accepts the codeword that is closest to the received vector for many cases where previous criteria fail to accept any codeword as a result the performance of generalized minimum distance gmd decoding is better if the new criterion is used for m ary signaling the weights used in gmd decoding are generalized to permit each of the possible m symbol values to have a different weight

nonlinear unequal error protection codes are sometimes better than linear ones the parameters of some binary nonlinear two level unequal error protection uep codes are evaluated and compared with nonexistence bounds for linear uep codes the major points is that for uep codes linearity is an essential restriction the results demonstrate that at least in certain cases nonlinear codes perform better than linear ones this is in contrast to the situation for ordinary codes when in most cases the best possible result can be obtained even with the restriction of linearity

minimal and canonical rational generator matrices for convolutional codes a full rank k spl times n matrix g d over the rational functions f d generates a rate r k n convolutional code c g d is minimal if it can be realized with as few memory elements as any encoder for c and g d is canonical if it has a minimal realization in controller canonical form we show that g d is minimal if and only if for all rational input sequences u d the span of u d g d covers the span of u d alternatively g d is minimal if and only if g d is globally zero free or globally invertible we show that g d is canonical if and only if g d is minimal and also globally orthogonal in the valuation theoretic sense of monna 1970

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

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

decoding by embedding correct decoding radius and dmt optimality the closest vector problem cvp and shortest nonzero vector problem svp are the core algorithmic problems on euclidean lattices they are central to the applications of lattices in many problems of communications and cryptography kannan s embedding technique is a powerful technique for solving the approximate cvp yet its remarkable practical performance is not well understood in this paper the embedding technique is analyzed from a bounded distance decoding bdd viewpoint we present two complementary analyses of the embedding technique we establish a reduction from bdd to hermite svp via unique svp which can be used along with any hermite svp solver including among others the lenstra lenstra and lovasz lll algorithm and show that in the special case of lll it performs at least as well as babai s nearest plane algorithm lll aided successive interference cancellation the former analysis helps us to explain the folklore practical observation that unique svp is easier than standard approximate svp it is proven that when the lll algorithm is employed the embedding technique can solve the cvp provided that the noise norm is smaller than a decoding radius 1 2 where 1 is the minimum distance of the lattice and o 2n 4 this substantially improves the previously best known correct decoding bound o 2n focusing on the applications of bdd to decoding of multiple input multiple output systems we also prove that bdd of the regularized lattice is optimal in terms of the diversity multiplexing gain tradeoff and propose practical variants of embedding decoding which require no knowledge of the minimum distance of the lattice and or further improve the error performance

even more efficient bounded distance decoding of the hexacode the golay code and the leech lattice we present a new bounded distance decoding algorithm for the hexacode which requires at most 34 real operations in the worst case as compared to 57 such operations in the best previously known decoder the new algorithm is then employed for bounded distance decoding of the leech lattice and the golay code the error correction radius of the resulting decoders is equal to that of a maximum likelihood decoder the resulting decoding complexity is at most 331 real operations for the leech lattice and at most 121 operations for the golay code for all the three codes the hexacode the golay code and the leech lattice the proposed decoders are considerably more efficient than any decoder presently known

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

further exploration of convolutional encoders for unequal error protection and new uep convolutional codes in this paper the unequal error protection uep capability of convolutional encoders in terms of the separation vector is studied from an algebraic viewpoint a simple procedure is presented for constructing a generator matrix which is basic and has the largest separation vector for every convolutional code such a generator matrix would be desirable since the corresponding encoder not only achieves uep optimality but also avoids undesired catastrophic error propagation in addition canonical generator matrices which are both basic and reduced are even more preferable for encoding since they attain the lowest complexity for viterbi decoding however the direct transformation from a uep optimal generator matrix to a canonical generator matrix may come with an unexpected loss of the separation vector we also propose a specific type of transformation matrix that reduces the external degrees of the generator matrices from which canonical generator matrices can be constructed that include the mitigated degradation of the separation vector finally beneficial uep convolutional codes that achieve the maximum free distances for the given code parameters are provided

encoding and decoding binary product lattices a binary product lattice is generated from two binary component lattices of lower dimensions by employing the kronecker product this work focuses on codes carved from binary product lattices defined as such an intriguing problem is that of effectively mapping independent data sequences onto a selected subset of lattice points a novel approach is disclosed yielding an explicit connection between source bits and lattice points decoding methods typically used for binary product codes do not apply for product lattices several alternative decoding approaches are discussed in particular a provably bounded distance decoder is presented it relies on the fact that a product lattice code point may be regarded as a two dimensional array whose rows and columns are points in the component lattices the obtained results are compared with classical lattices known in the art

the fisher information matrix for linear systems estimation of parameters of linear systems is a problem often encountered in applications the cramer rao lower bound gives a lower bound on the variance of any unbiased parameter estimation method and therefore provides an important tool in the assessment of a parameter estimation method and for experimental design here we study the calculation of the fisher information matrix the inverse of the cramer rao lower bound from a system theoretic point of view a number of results appear in the literature that deal with the case where the stationary data is given as the output of a linear system driven by gaussian noise the non stationary situation where the data is the output of a linear system with gaussian measurement noise is rarely considered despite its importance in applications a general description will be given for fisher information for such data in terms of a derivative system for a uniformly sampled data set of impulse response type a closed form expression can be given for the fisher information using the solution of a lyapunov equation

minimization of functionals of the gradient by baire s theorem we give sufficient conditions for the existence of solutions of the minimum problem cal p u 0 qquad hbox minimize quad int omega g du x dx quad u in u 0 w 0 1 p omega bbb r based on the structure of the epigraph of the lower convex envelope of g which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space no convexity conditions are required on g and no assumptions are made on the boundary datum u 0 in w 0 1 p omega bbb r

on algebraic soft decision decoding algorithms for bch codes three algebraic soft decision decoding algorithms are presented for binary bose chaudhuri hocquengham bch codes two of these algorithms are based on the bounded distance bd 1 generalized minimum distance gmd decoding presented by berlekamp 1984 and the other is based on chase 1972 decoding a simple algebraic algorithm is first introduced and it forms a common basis for the decoding algorithms presented next efficient bd 1 gmd and bd 2 gmd decoding algorithms are presented it is shown that for binary bch codes with odd designed minimum distance d and length n both the bd 1 gmd and the bd 2 gmd decoding algorithms can be performed with complexity o nd the error performance of these decoding algorithms is shown to be significantly superior to that of conventional gmd decoding by computer simulation finally an efficient algorithm is presented for chase decoding of binary bch codes like a one pass gmd decoding algorithm this algorithm produces all necessary error locator polynomials for chase decoding in one run

on the reliability order based decoding algorithms for binary linear block codes in this correspondence we consider the decoding of binary block codes over the additive white gaussian noise awgn channel with binary phase shift keying bpsk signaling by a reliability order based decoding algorithm robda we mean a soft decision decoding algorithm which decodes to the best most likely codeword of the form that is the sum of the hard decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions examples of robdas include many well known decoding algorithms such as the generalized minimum distance gmd decoding algorithm chase decoding algorithms and the reliability based decoding algorithms proposed by fossorier and lin it is known that the squared error correction radii of robdas can be computed from the minimal squared euclidean distances mseds between the all one sequence and the polyhedra corresponding to the error patterns for the computation of such mseds we give a new method which is more compact than the one proposed by fossorier and lin these results are further used to show that any bounded distance robda is asymptotically optimal the ratio between the probability of decoding error of a bounded distance robda and that of the maximum likelihood ml decoding approaches 1 when the signal to noise ratio snr approaches infinity provided that the minimum hamming distance of the code is greater than 2

invariance properties of binary linear codes over a memoryless channel with discrete input this work studies certain properties of the probability density function pdf of the bit log likelihood ratio llr for binary linear block codes over a memoryless channel with discrete input and discrete or continuous output we prove that under a set of mild conditions the pdf of the bit llr of a specific bit position is independent of the transmitted codeword it is also shown that the pdf of a given bit llr when the corresponding bit takes the values of zero and one are symmetric with respect to each other reflection of one another with respect to the vertical axis for the case of channels with binary input a sufficient condition for two bit positions to have the same pdf is presented

computation of the exact information matrix of gaussian dynamic regression time series models in this paper the computation of the exact fisher information matrix of a large class of gaussian time series models is considered this class which is often called the single input single output siso model includes dynamic regression with autocorrelated errors and the transfer function model with autoregressive moving average errors the method is based on a combination of two computational procedures recursions for the covariance matrix of the derivatives of the state vector with respect to the parameters and the fast kalman filter recursions used in the evaluation of the likelihood function it is much faster than existing procedures an expression for the asymptotic information matrix is also given

practical encoders and decoders for euclidean codes from barnes wall lattices we address the application of barnes wall bw lattice codes for communication over additive white gaussian noise awgn channels we introduce construction a prime of complex bw lattices that makes new connection between linear codes over polynomial rings and lattices we show that construction a prime of bw lattices is equivalent to the multilevel construction from reed muller codes proposed by forney to decode the bw lattice code we adapt the low complexity sequential bw lattice decoder sbwd proposed by micciancio and nicolosi first we study the error performance of sbwd for decoding the infinite lattice and demonstrate that it is powerful in making correct decisions well beyond the packing radius subsequently we use the sbwd to decode lattice codes through a novel noise trimming technique where the received vector is appropriately scaled before applying the sbwd we show that the noise trimming technique is most effective for decoding bw lattice codes in smaller dimensions while the gain diminishes for decoding codes in larger dimensions

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

cyclic concatenated codes with constacyclic outer codes the known construction of cyclic concatenated codes is based on the fact that the inner is a cyclic minimal code the outer code is cyclic and the lengths of the inner and outer codes are relatively prime it is shown that if the outer code is a suitably chosen constacyclic code the overall concatenated code is always cyclic regardless of the length of the outer code moreover it follows that any cyclic code of composite length is a direct sum of cyclic concatenated codes with inner cyclic minimal codes and outer constacyclic codes this description of cyclic codes of composite length leads to the introduction of the concept of a poor code length pcl all but low rate codes of this length have a poor minimum distance a pcl is directly related to the existence of irreducible binomials in the binary case the first few pcls are 9 25 27 45 49 75 81 and 99 arbitrarily long binary cyclic codes that are better than binary bch codes of primitive length are constructed

lattice codes and generalized minimum distance decoding for ofdm systems lattice coding of orthogonal frequency division multiplexing ofdm systems is considered mapping of multilevel construction lattices to ofdm blocks is shown and a methodology for probabilistic analysis of multistage generalized minimum distance gmd decoding of the received ofdm blocks is shown as a case study transmission of points from a 128 dimensional barnes wall lattice is considered tight approximations to the system error rate are obtained and verified by simulation it appears that gmd decoding of lattice encoded ofdm provides high coding gain at low complexity

generalized minimum distance decoding of euclidean space codes and lattices it is shown that multistage generalized minimum distance gmd decoding of euclidean space codes and lattices can provide an excellent tradeoff between performance and complexity we introduce a reliability metric for gaussian channels that is easily computed from an inner product and prove that a multistage gmd decoder using this metric is a bounded distance decoder up to the true packing radius the effective error coefficient of multistage gmd decoding is determined two simple modifications in the gmd decoding algorithm that drastically reduce this error coefficient are proposed it is shown that with these modifications gmd decoding achieves the error coefficient of maximum likelihood decoding for block codes and for generalized construction a lattices multistage gmd decoding of the lattices d sub 4 e sub 8 k sub 12 bw sub 16 and spl lambda sub 24 is investigated in detail for k sub 12 bw sub 16 and spl lambda sub 24 the gmd decoders have considerably lower complexity than the best known maximum likelihood or bounded distance decoding algorithms and appear to be the most practically attractive decoders available for high dimensional codes and lattices spl ges 64 dimensions maximum likelihood decoding becomes infeasible while gmd decoding algorithms remain quite practical as an example we devise a multistage gmd decoder for a 128 dimensional sphere packing with a nominal coding gain of 8 98 db that attains an effective error coefficient of 1365760 this decoder requires only about 400 real operations in addition to algebraic errors and erasures decoding of certain bch and hamming codes it therefore appears to be practically feasible to implement algebraic multistage gmd decoders for high dimensional sphere packings and thus achieve high effective coding gains

active distances for convolutional codes a family of active distance measures for general convolutional codes is defined these distances are generalizations of the extended distances introduced by thommesen and justesen 1983 for unit memory convolutional codes it is shown that the error correcting capability of a convolutional code is determined by the active distances the ensemble of periodically time varying convolutional codes is defined and lower bounds on the active distances are derived for this ensemble the active distances are very useful in the analysis of concatenated convolutional encoders

a linear algebra approach to minimal convolutional encoders the authors review the work of g d forney jr on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest an example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices however this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form thus this encoding matrix is not minimal according to forney s definition of a minimal encoder to resolve this difficulty the following three minimality criteria are introduced minimal basic encoding matrix minimal encoding matrix and minimal encoder it is shown that all minimal basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal basic several equivalent conditions are given for an encoding matrix to be minimal it is proven that the constraint lengths of two equivalent minimal basic encoding matrices are equal one by one up to a rearrangement all results are proven using only elementary linear algebra

woven convolutional codes i encoder properties encoders for convolutional codes with large free distances can be constructed by combining several less powerful convolutional encoders this paper is devoted to constructions in which the constituent convolutional codes are woven together in a manner that resembles the structure of a fabric the general construction is called twill and it is described together with two special cases viz woven convolutional encoders with outer warp and with inner warp the woven convolutional encoders inherit many of their structural properties such as minimality and catastrophicity from their constituent encoders for all three types of woven convolutional codes upper and lower bounds on their free distances as well as lower bounds on the active distances of their encoders are derived

on fisher s information matrix of an armax process and sylvester s resultant matrices we establish a relation between fisher s information matrix of a stationary autoregressive moving average process with an exogenous component and two sylvester s resultant matrices

maximum likelihood decoding of the leech lattice an algorithm for maximum likelihood decoding of the leech lattice is presented the algorithm involves projecting the points of the leech lattice directly onto the codewords of the 6 3 4 quaternary code the hexacode projection on the hexacode induces a partition of the leech lattice into four cosets of a certain sublattice 24 such a partition into cosets enables maximum likelihood decoding of the leech lattice with 3595 real operations in the worst case and only 2955 operations on the average this is about half the worst case and the average complexity of the best previously known algorithm

encoding and decoding for the minimization of message symbol error rates in linear block codes given any fixed linear block code the error rates for the message symbols depend both on the encoding function and on the decoding map this research shows how to optimize the choice of a generator matrix and decoding map simultaneously to minimize the error rates for all message symbols the model used assumes that the distribution of messages is flat and that the distribution of error vectors defining the channel is independent of the message transmitted in addition it is shown that with proper choice of coset leaders standard array decoding is optimal in this circumstance the results generalize previously known results on unequal error protection and are sufficiently general to apply when a code is used for error detection only

the concatenated structure of cyclic and abelian codes it is shown that a linear two dimensional cyclic code of length nn can be factorized into a direct sum of concatenated codes with cyclic inner and outer codes and conversely thai a two dimensional cyclic code can be constructed in this way this result is extended and it appears that the abelian codes are obtainable by taking a direct sum of several concatenations of cyclic codes codes are constructed that are better than any previously known in particular low rate cyclic codes superior to the duals of high rate bch codes are constructed

on unequal error protection of convolutional codes from an algebraic perspective in this paper convolutional codes are studied for unequal error protection uep from an algebraic theoretical viewpoint we first show that for every convolutional code there exists at least one optimal generator matrix with respect to uep the uep optimality of convolutional encoders is then combined with several algebraic properties e g systematic basic canonical and minimal to establish the fundamentals of convolutional codes for uep in addition a generic lower bound on the length of a uep convolutional code is proposed good uep codes with their lengths equal to the derived lower bound are obtained by computer search

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

a bounded distance decoding algorithm for binary linear block codes achieving the minimum effective error coefficient a new bounded distance bd decoding algorithm is presented for binary linear n k d block codes on additive white gaussian noise channels the algorithm is based on the generalized minimum distance gmd decoding algorithm of forney 1989 using the acceptance criterion of taipale and pursley gmd tp proposed in 1991 it is shown that the gmd tp decoding algorithm is a bd decoding algorithm with effective error coefficient sup n sub d it is also shown that the decision regions of gmd tp are good inner approximations of those of full gmd decoding and therefore full gmd decoding is bd and has an effective error coefficient that is well approximated by sup n sub d moreover by adding a d erasure correction step to gmd decoding the effective error coefficient can be reduced to a sub d the number of minimum weight codewords which is the same as the effective error coefficient of maximum likelihood decoding the decoding algorithm is mainly based on algebraic errors and erasures decoding and therefore has polynomial rather than exponential complexity

bounded distance 1 soft decision reed solomon decoding we present a new reed solomon decoding algorithm which embodies several refinements of an earlier algorithm some portions of this new decoding algorithm operate on symbols of length lgq bits other portions operate on somewhat longer symbols in the worst case the total number of calculations required by the new decoding algorithm is proportional to nr where n is the code s block length and r is its redundancy this worst case workload is very similar to prior algorithms but in many applications average case workload and error correcting performance are both much better the input to the new algorithm consists of n received symbols from gf q and n nonnegative real numbers each of which is the reliability of the corresponding received symbol any conceivable errata pattern has a score equal to the sum of the reliabilities of its locations with nonzero errata values a max likelihood decoder would find the minimum score over all possible errata patterns our new decoding algorithm finds the minimum score only over a subset of these possible errata patterns the errata within any candidate errata pattern may be partitioned into errors and erasures depending on whether the corresponding reliabilities are above or below an erasure threshold different candidate errata patterns may have different thresholds each chosen to minimize its corresponding errata count which is defined as 2 spl middot number of errors number of erasures the new algorithm finds an errata pattern with minimum score among all errata patterns for which errata count spl les r 1 where r is the redundancy of the rs code conventional algorithms also require that the erasure threshold be set a priori the new algorithm obtains the best answer over all possible settings of the erasure threshold conventional cyclic rs codes have length n q 1 and their locations correspond to the nonzero elements of gf q the new algorithm also applies very naturally to rs codes which have been doubly extended by the inclusion of 0 and spl infin as additional locations

two topics on linear unequal error protection codes bounds on their length and cyclic code classes it is possible for a linear block code to provide more protection for selected positions in the input message words than is guaranteed by the minimum distance of the code linear codes having this property are called linear unequal error protection luep codes bounds on the length of a luep code that ensures a given unequal error protection are derived a majority decoding method for certain classes of cyclic binary uep codes is treated a list of short i e of length less than 16 binary luep codes of optimal i e minimal length and a list of all cyclic binary uep codes of length less than 40 are included

a block hankel generalized confluent vandermonde matrix abstract vandermonde matrices are well known they have a number of interesting properties and play a role in lagrange interpolation problems partial fraction expansions and finding solutions to linear ordinary differential equations to mention just a few applications usually one takes these matrices square q q say in which case the i th column is given by u z i where we write u z 1 z z q 1 if all the z i i 1 q are different the vandermonde matrix is non singular otherwise not the latter case obviously takes place when all z i are the same z say in which case one could speak of a confluent vandermonde matrix non singularity is obtained if one considers the matrix v z whose i th column i 1 q is given by the i 1 th derivative u i 1 z we will consider generalizations of the confluent vandermonde matrix v z by considering matrices obtained by using as building blocks the matrices m z u z w z with u z as above and w z 1 z z r 1 together with its derivatives m k z specifically we will look at matrices whose ij th block is given by m i j z where the indices i j by convention have initial value zero these in general non square matrices exhibit a block hankel structure we will answer a number of elementary questions for this matrix what is the rank what is the null space can the latter be parametrized in a simple way does it depend on z what are left or right inverses it turns out that answers can be obtained by factorizing the matrix into a product of other matrix polynomials having a simple structure the answers depend on the size of the matrix m z and the number of derivatives m k z that is involved the results are obtained by mostly elementary methods no specific knowledge of the theory of matrix polynomials is needed

on the resultant property of the fisher information matrix of a vector arma process a matrix is called a multiple resultant matrix associated to two matrix polynomials when it becomes singular if and only if the two matrix polynomials have at least one common eigenvalue in this paper a new multiple resultant matrix is introduced it concerns the fisher information matrix fim of a stationary vector autoregressive and moving average time series process varma the two matrix polynomials are the autoregressive and the moving average matrix polynomials of the varma process in order to show that the fim is a multiple resultant matrix two new representations of the fim are derived to construct such representations appropriate matrix differential rules are applied the newly obtained representations are expressed in terms of the multiple sylvester matrix and the tensor sylvester matrix the representation of the fim expressed by the tensor sylvester matrix is used to prove that the fim becomes singular if and only if the autoregressive and moving average matrix polynomials have at least one common eigenvalue it then follows that the fim and the tensor sylvester matrix have equivalent singularity conditions in a simple numerical example it is shown however that the fim fails to detect common eigenvalues due to some kind of numerical instability whereas the tensor sylvester matrix reveals it clearly proving the usefulness of the results derived in this paper

matrix differential calculus applied to multiple stationary time series and an extended whittle formula for information matrices abstract the purpose of this paper is to set forth easily implementable expressions for the fisher information matrix fim of a gaussian stationary vector autoregressive and moving average process with exogenous or input variables a vector armax or varmax process the entries of the fim are represented as circular integral expressions and can be computed by applying cauchy s residue theorem an extension of the whittle formula for the fim of multiple time series processes is developed for varmax processes it will be shown that the extended whittle formula yields the fim when a bivariate structure consisting of the varmax process and the exogenous input process is considered consequently the equivalence between a frequency and time domain representation of the fim of varmax processes is established in order to obtain the results presented in this paper the differentiation techniques developed and used in a klein p spreij an explicit expression for the fisher information matrix of a multiple time series process linear algebra appl 417 2006 140 149 are applied