simultaneous partial inverses and decoding interleaved reed solomon codes this paper introduces the simultaneous partial inverse problem spi for polynomials and develops its application to decoding interleaved reed solomon codes beyond half the minimum distance while closely related both to standard key equations and to well known pade approximation problems the spi problem stands out in several respects first the spi problem has a unique solution up to a scale factor which satisfies a natural degree bound second the spi problem can be transformed monomialized into an equivalent spi problem where all moduli are monomials third the spi problem can be solved by an efficient algorithm of the berlekamp massey type fourth decoding interleaved reed solomon codes or subfield evaluation codes beyond half the minimum distance can be analyzed in terms of a partial inverse condition for the error pattern if that condition is satisfied then the true error locator polynomial is the unique solution of a standard key equation and can be computed in many different ways including the well known multi sequence berlekamp massey algorithm and the spi algorithm of this paper two of the best performance bounds from the literature the schmidt sidorenko bossert bound and the roth vontobel bound are generalized to hold for the partial inverse condition and thus to apply to several different decoding algorithms

computing m dfts over gf q with one dft over gf q sup m over the field of complex numbers it is well known that if the input is real then it is possible to compute two real dfts with one complex dft the authors extend the result to finite fields and show how to compute m dfts over gf q with one dft over gf q sup m

estimation of stochastic volatility models via monte carlo maximum likelihood this paper discusses the monte carlo maximum likelihood method of estimating stochastic volatility sv models the basic sv model can be expressed as a linear state space model with log chi square disturbances the likelihood function can be approximated arbitrarily accurately by decomposing it into a gaussian part constructed by the kalman filter and a remainder function whose expectation is evaluated by simulation no modifications of this estimation procedure are required when the basic sv model is extended in a number of directions likely to arise in applied empirical research this compares favorably with alternative approaches the finite sample performance of the new estimator is shown to be comparable to the monte carlo markov chain mcmc method

likelihood analysis of non gaussian measurement time series summary in this paper we provide methods for estimating non gaussian time series models these techniques rely on markov chain monte carlo to carry out simulation smoothing and bayesian posterior analysis of parameters and on importance sampling to estimate the likelihood function for classical inference the time series structure of the models is used to ensure that our simulation algorithms are efficient

fft algorithm for binary extension finite fields and its application to reed solomon codes recently a new polynomial basis over binary extension fields was proposed such that the fast fourier transform fft over such fields can be computed in the complexity of order mathcal o n lg n where n is the number of points evaluated in fft in this paper we reformulate this fft algorithm such that it can be easier understood and be extended to develop frequency domain decoding algorithms for n 2 m k systematic reed solomon rs codes over mathbb f 2 m m in mathbb z with n k a power of two first the basis of syndrome polynomials is reformulated in the decoding procedure so that the new transforms can be applied to the decoding procedure a fast extended euclidean algorithm is developed to determine the error locator polynomial the computational complexity of the proposed decoding algorithm is mathcal o n lg n k n k lg 2 n k improving upon the best currently available decoding complexity mathcal o n lg 2 n lg lg n and reaching the best known complexity bound that was established by justesen in 1976 however justesen s approach is only for the codes over some specific fields which can apply cooley tukey ffts as revealed by the computer simulations the proposed decoding algorithm is 50 times faster than the conventional one for the 2 16 2 15 rs code over mathbb f 2 16

on abelian subgroups ofp groups abstract this is a continuation of our note j algebra 186 1996 120 131 we generalize and present simplified proofs almost all elementary lemmas from section 8 of the odd order paper in many places we use hall s enumeration principle and other simple combinatorial arguments a number of related results are proved as well some open questions are posed

simplified procedure for decoding nonsystematic reed solomon codes over gf 2 m using euclid s algorithm and the fast fourier transform gao s algorithm similar to shiozaki s algorithm for decoding nonsystematic reed solomon rs codes which is a subclass of the redundant residue polynomial codes is considered here this algorithm computes the message polynomial directly without explicitly finding the error locator polynomial and errorevaluator polynomial in this letter a simplified decoding algorithm to correct both errors and erasures is used in conjunction with gao s algorithm for efficiently decoding rs codes in addition we show that the extended gao algorithm similar to the shiozaki truong cheung reed algorithm significantly reduces the decoding complexity

decoding of redundant residue polynomial codes using euclid s algorithm a decoding method is proposed for the redundant residue polynomial codes a class that includes reed solomon codes the method is based on properties of euclid s algorithm the advantage of the method is that the computation of both the error locator polynomial and the error evaluator polynomial needed in conventional decoding methods can be avoided the method is suitable for decoding concatenated codes whose outer codes are redundant residue polynomial codes since they are easily decoded by ignoring erasures detected in the inner codes

the wishart autoregressive process of multivariate stochastic volatility abstract the wishart autoregressive war process is a dynamic model for time series of multivariate stochastic volatility the war naturally accommodates the positivity and symmetry of volatility matrices and provides closed form non linear forecasts the estimation of the war is straighforward as it relies on standard methods such as the method of moments and maximum likelihood for illustration the war is applied to a sequence of intraday realized volatility covolatility matrices from the toronto stock market tsx

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

duhamel hollmann like discrete fourier transform algorithm with the smallest multiplicative complexity over a finite field the new method for the discrete fourier transform computation over a finite field is introduced this method is a nontrivial generalization of the duhamel hollmann algorithm with replacement of the toeplitz convolution calculation by the normalized cyclic convolution calculation both algorithms have the smallest multiplicative complexity

time series analysis of non gaussian observations based on state space models from both classical and bayesian perspectives the analysis of non gaussian time series by using state space models is considered from both classical and bayesian perspectives the treatment in both cases is based on simulation using importance sampling and antithetic variables markov chain monte carlo methods are not employed non gaussian disturbances for the state equation as well as for the observation equation are considered methods for estimating conditional and posterior means of functions of the state vector given the observations and the mean square errors of their estimates are developed these methods are extended to cover the estimation of conditional and posterior densities and distribution functions the choice of importance sampling densities and antithetic variables is discussed the techniques work well in practice and are computationally efficient their use is illustrated by applying them to a univariate discrete time series a series with outliers and a volatility series

a multi move sampler for estimating non gaussian time series models comments on shephard pitt 1997 this note points out a problem in the multi move sampler as proposed by shephard pitt 1997 and provides an alternative correct formulation

error correction in polynomial remainder codes with non pairwise coprime moduli and robust chinese remainder theorem for polynomials this paper investigates polynomial remainder codes with non pairwise coprime moduli we first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small namely the robust chinese remainder theorem crt for polynomials it basically says that a polynomial can be reconstructed from its erroneous residues such that the degree of the reconstruction error is upper bounded by tau whenever the degrees of all residue errors are upper bounded by tau where a sufficient condition for tau and a reconstruction algorithm are obtained by relaxing the constraint that all residue errors have small degrees another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues we finally obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result some errors with small degrees can be also corrected in the residues with this newly obtained result improvements in uncorrected error probability and burst error correction capability in data transmission are illustrated

the structure of dynamic correlations in multivariate stochastic volatility models this paper proposes two types of stochastic correlation structures for multivariate stochastic volatility msv models namely the constant correlation cc msv and dynamic correlation dc msv models from which the stochastic covariance structures can easily be obtained both structures can be used for purposes of determining optimal portfolio and risk management strategies through the use of correlation matrices and for calculating value at risk var forecasts and optimal capital charges under the basel accord through the use of covariance matrices a technique is developed to estimate the dc msv model using the markov chain monte carlo mcmc procedure and simulated data show that the estimation method works well various multivariate conditional volatility and msv models are compared via simulation including an evaluation of alternative var estimators the dc msv model is also estimated using three sets of empirical data namely nikkei 225 index hang seng index and straits times index returns and significant dynamic correlations are found the dynamic conditional correlation dcc model is also estimated and is found to be far less sensitive to the covariation in the shocks to the indexes the correlation process for the dcc model also appears to have a unit root and hence constant conditional correlations in the long run in contrast the estimates arising from the dc msv model indicate that the dynamic correlation process is stationary

fast transform for decoding both errors and erasures of reed solomon codes over gf 2 sup m for 8 spl les m spl les 10 in this letter it is shown that a fast prime factor discrete fourier transform dft algorithm can be modified to compute fourier like transforms of long sequences of 2 sup m 1 points over gf 2 sup m where 8 spl les m spl les 10 using these transforms together with the berlekamp massey algorithm the complexity of the transform domain decoder for correcting both errors and erasures of the reed solomon codes of block length 2 sup m 1 over gf 2 sup m for 8 spl les m spl les 10 is reduced substantially from the previous time domain decoder a computer simulation verifies these new results

on algorithms and complexities of cyclotomic fast fourier transforms over arbitrary finite fields discrete fourier transforms over finite fields are significant due to their widespread applications in cryptography and error control codes which in turn are used in all digital communication and storage systems cyclotomic fast fourier transforms cffts are of great interest due to their low multiplicative complexities however all existing cffts are for characteristic 2 fields and the computational complexities of cffts have not been analyzed theoretically this paper addresses both problems for cffts and has three main contributions to this end first we propose an efficient bilinear algorithm to compute toeplitz matrix vector products tmvps which has a lower computational complexity than existing algorithms and works on all finite fields as well as the real and complex fields second we propose an efficient algorithm for cyclic convolutions over arbitrary finite fields which is essential in deriving efficient cffts over arbitrary finite fields finally we derive bounds on the additive and multiplicative complexities of cffts over arbitrary finite fields our results confirm that cffts have the smallest multiplicative complexities among all known algorithms although their high additive complexities render them asymptotically suboptimal cffts remain valuable since they have the smallest overall complexities for dfts of up to thousands of symbols in most cases

markov chain monte carlo for dynamic generalised linear models summary this paper presents a new methodological approach for carrying out bayesian inference about dynamic models for exponential family observations the approach is simulationbased and involves the use of markov chain monte carlo techniques a metropolishastings algorithm is combined with the gibbs sampler in repeated use of an adjusted version of normal dynamic linear models different alternative schemes based on sampling from the system disturbances and state parameters separately and in a block are derived and compared the approach is fully bayesian in obtaining posterior samples with state parameters and unknown hyperparameters illustrations with real datasets with sparse counts and missing values are presented extensions to accommodate more general evolution forms and distributions for observations and disturbances are outlined

stochastic volatility with leverage fast and efficient likelihood inference this paper is concerned with the bayesian analysis of stochastic volatility sv models with leverage specifically the paper shows how the often used kim et al 1998 stochastic volatility likelihood inference and comparison with arch models review of economic studies 65 361 393 method that was developed for sv models without leverage can be extended to models with leverage the approach relies on the novel idea of approximating the joint distribution of the outcome and volatility innovations by a suitably constructed ten component mixture of bivariate normal distributions the resulting posterior distribution is summarized by mcmc methods and the small approximation error in working with the mixture approximation is corrected by a reweighting procedure the overall procedure is fast and highly efficient we illustrate the ideas on daily returns of the tokyo stock price index finally extensions of the method are described for superposition models where the log volatility is made up of a linear combination of heterogenous and independent autoregressions and heavy tailed error distributions student and log normal

a semiparametric stochastic volatility model abstract in this paper the correlation structure in the classical leverage stochastic volatility sv model is generalized based on a linear spline in the new model the correlation between the return and volatility innovations is time varying and depends nonparametrically on the type of news arrived to the market theoretical properties of the proposed model are examined the model estimation and comparison are conducted by bayesian methods the performance of the estimates are examined in simulations the new model is fitted to daily and weekly us data and compared with the classical sv and garch models in terms of their in sample and out of sample performances empirical results suggest evidence in favor of the proposed model in particular the new model finds strong evidence of time varying leverage effect in individual stocks when the classical model fails to identify the leverage effect

on elements of order p in powerful p groups abstract we investigate the set 1 p of elements of order at most p in a powerful p group p and prove that 1 p p p p as a corollary we obtain a necessary and sufficient condition for 1 p to be of exponent p we give an example to show that for p 2 there is a powerful 2 group such that 1 p is not of exponent 2

alternate proofs of two theorems of philip hall on finite p groups and some related results abstract new very detailed proofs of theorems 2 5 and 2 64 from the seminal paper of philip hall p hall on a theorem of frobenius proc london math soc ser 2 40 1936 468 501 are given a number of generalizations of these theorems are proved for example we show that if g is a p group of order p k p 1 3 k 2 and exponent p k with k g g then either g is of maximal class or g possesses a normal subgroup h of order p p and exponent p such that g h is of maximal class counting theorems play important role in this note

fast prime factor discrete fourier transform algorithms over gf 2m for 8 m 10 in this paper it is shown that winograd s algorithm for computing convolutions and a fast prime factor discrete fourier transform dft algorithm can be modified to compute fourier like transforms of long sequences of 2 m 1 points over gf 2 m for 8

polynomial identities in graded group rings restricted lie algebras and p adic analytic groups let g be any finitely generated group and let k be a field of characteristic p 0 it is shown that the graded group ring gr kg satisfies a nontrivial polynomial identity if and only if the pro p completion of g is p adic analytic i e can be given the structure of a lie group over the p adic field q p the proof applies theorems of lazard quillen and passman as well as results on engel lie algebras and on dimension subgroups in positive characteristic

reprojecting partially observed systems with application to interest rate diffusions abstract we introduce reprojection as a general purpose technique for characterizing the dynamic response of a partially observed nonlinear system to its observable history reprojection is the third step of a procedure wherein first data are summarized by projection onto a hermite series representation of the unconstrained transition density for observables second system parameters are estimated by minimum chi squared where the chi squared criterion is a quadratic form in the expected score of the projection and third the constraints on dynamics implied by the nonlinear system are imposed by projecting a long simulation of the estimated system onto a hermite series representation of the constrained transition density for observables the constrained transition density can be used to study the response of the system to its observable history we utilize the technique to assess the dynamics of several diffusion models for the short term interest rate that have been proposed and to compare them to a ne

partial inverses mod m x and reverse berlekamp massey decoding this semi tutorial paper introduces the partial inverse problem for polynomials and develops its application to decoding reed solomon codes and some related codes the most natural algorithm to solve the partial inverse problem is very similar to but more general than the berlekamp massey algorithm two additional algorithms are obtained as easy variations of the basic algorithm the first variation is entirely new while the second variation may be viewed as a version of the euclidean algorithm decoding reed solomon codes and some related codes can be reduced to the partial inverse problem both via the standard key equation and more naturally via an alternative key equation with a new converse shortened and singly extended reed solomon codes are automatically included using the properties of the partial inverse problem two further key equations with attractive properties are obtained the paper also points out a variety of options for interpolation

a future simplification of procedure for decoding nonsystematic reed solomon codes using the berlekamp massey algorithm it is well known that the euclidean algorithm can be used o find the systematic errata locator polynomial and the errata evaluator polynomial simultaneously in berlekamp s key equation that is needed to decode a reed solomon rs codes in this paper a simplified decoding algorithm to correct both errors and erasures is used in conjunction with the euclidean algorithm for efficiently decoding nonsystematic rs codes in fact this decoding algorithm is an appropriate modification to the algorithm developed by shiozaki and gao based on the ideas presented above a fast algorithm described from blahut s classic book is derivated and proved in this paper to correct erasures as well as errors by replacing the euclidean algorithm by the berlekamp massey bm algorithm these facts lead to significantly reduce the decoding complexity of the proposed rs decoder in addition computer simulations show that this simple and fast decoding technique reduces the decoding time when compared with existing efficient algorithms including the new euclidean algorithm based decoding approach proposed in this paper

dimension subgroups nilpotency indices and the number of generators of ideals in p group algebras abstract various problems in modular p group algebras are solved through extensive study of dimension subgroups as an application it is shown that if g is an infinite res p group p 2 and k is a field of characteristic p then the least upper bound on the numbers of generators of ideals in kg equals the minimal index of a cyclic subgroup of g

stochastic volatility in asset prices estimation with simulated maximum likelihood abstract the stochastic volatility model is used to estimate daily asset price dynamics the model is estimated by integrating latent volatility out of the joint density of prices and volatility to obtain the marginal density of prices due to high number of dimensions of the integral no conventional integration technique is applicable a monte carlo method called simulated maximum likelihood is used to obtain the marginal density where the latent variable is simulated conditional on available information the model is estimated by 2022 observations from the s p 500 index for comparison arch type models are estimated with the same data

faster polynomial multiplication over finite fields polynomials over finite fields play a central role in algorithms for cryptography error correcting codes and computer algebra the complexity of multiplying such polynomials is still a major open problem let p be a prime and let mp n denote the bit complexity of multiplying two polynomials in fp x of degree less than n for n large compared to p we establish the bound mp n o n log n 8log n log p where log n min k n log k log n 1 stands for the iterated logarithm this improves on the previously best known bound mp n o n log n log log n log p which essentially goes back to the 1970s

a new structured design method for convolutions over finite fields part i the structure of bilinear cyclic convolution algorithms is explored over finite fields the algorithms derived are valid for any length not divisible by the field characteristic and are based upon the small length polynomial multiplication algorithms the multiplicative complexity of these algorithms is small and depends on the field of constants the linear transformation matrices a b premultiplication and c postmultiplication defining the algorithm have block structures which are related to one another the rows of a and b and the columns of c are maximal length recurrent sequences because of the highly regular structure of a b and c the algorithms can be very easily designed even for large lengths the application of these algorithms to the decoding of reed solomon codes is also examined

mastrovito multiplier for all trinomials an efficient algorithm for the multiplication in gf 2 sup m was introduced by mastrovito the space complexity of the mastrovito multiplier for the irreducible trinomial x sup m x 1 was given as m sup 2 1 xor and m sup 2 and gales in this paper we describe an architecture based on a new formulation of the multiplication matrix and show that the mastrovito multiplier for the generating trinomial x sup m x sup n 1 where m spl ne 2n also requires m sup 2 1 xor and m sup 2 and gates however m sup 2 x sup m 2 xor gates are sufficient when the generating trinomial is of the form x sup m x sup m 2 1 for an even m we also calculate the time complexity of the proposed mastrovito multiplier and give design examples for the irreducible trinomials x sup 7 x sup 4 1 and x sup 6 x sup 3 1

circular shift linear network coding we study a class of linear network coding lnc schemes called circular shift lnc whose encoding operations consist of only circular shifts and bit wise additions formulated as a special vector linear code over gf 2 an l dimensional circular shift linear code of degree delta restricts its local encoding kernels to be the summation of at most delta cyclic permutation matrices of size l we show that on a general network for a certain block length l every scalar linear solution over gf 2 l 1 can induce an l dimensional circular shift linear solution with 1 bit redundancy per edge transmission consequently specific to a multicast network such a circular shift linear solution of an arbitrary degree delta can be efficiently constructed which has an interesting complexity tradeoff between encoding and decoding with different choices of delta by further proving that circular shift lnc is insufficient to achieve the exact capacity of certain multicast networks we show the optimality of the efficiently constructed circular shift linear solution in the sense that its 1 bit redundancy is inevitable finally both theoretical and numerical analysis imply that with increasing l a randomly constructed circular shift linear code has linear solvability behavior comparable to a randomly constructed permutation based linear code but has shorter overheads

on the upper exponent of a finite p group abstract let p be a prime and for a finite p group g let 0 g g and i g 1 i 1 g for i n a theorem is proved stating that if exp g q and cl g c then the length of the i series cannot be bounded by a function of q alone an upper bound in terms of q and c is given which is shown to be attained in p groups of arbitrarily large class

on leverage in a stochastic volatility model this paper is concerned with the specification for modelling financial leverage effect in the context of stochastic volatility sv models two alternative specifications co exist in the literature one is the euler approximation to the well known continuous time sv model with leverage effect and the other is the discrete time sv model of jacquier et al j econometrics 122 2004 185 using a gaussian nonlinear state space form with uncorrelated measurement and transition errors i show that it is easy to interpret the leverage effect in the conventional model whereas it is not clear how to obtain and interpret the leverage effect in the model of jacquier et al empirical comparisons of these two models via bayesian markov chain monte carlo mcmc methods further reveal that the specification of jacquier et al is inferior simulation experiments are conducted to study the sampling properties of bayes mcmc for the conventional model

cyclotomic ffts with reduced additive complexities based on a novel common subexpression elimination algorithm in this paper we first propose a novel common subexpression elimination cse algorithm for matrix vector multiplications over characteristic 2 fields as opposed to previously proposed cse algorithms which usually focus on complexity savings due to recurrences of subexpressions our cse algorithm achieves two types of complexity reductions differential savings and recurrence savings by taking advantage of the cancellation property of characteristic 2 fields using our cse algorithm we reduce the additive complexities of cyclotomic fast fourier transforms cffts using a weighted sum of the numbers of multiplications and additions as a metric our cffts achieve smaller total complexities than previously proposed cffts and other ffts requiring both fewer multiplications and fewer additions in many cases

bayesian semiparametric modeling of realized covariance matrices this paper introduces several new bayesian nonparametric models suitable for capturing the unknown conditional distribution of realized covariance rcov matrices existing dynamic wishart models are extended to countably infinite mixture models of wishart and inverse wishart distributions in addition to mixture models with constant weights we propose models with time varying weights to capture time dependence in the unknown distribution each of our models can be combined with returns to provide a coherent joint model of returns and rcov the extensive forecast results show the new models provide very significant improvements in density forecasts for rcov and returns and competitive point forecasts of rcov

a simple algorithm for decoding reed solomon codes and its relation to the welch berlekamp algorithm a simple and natural gao algorithm for decoding algebraic codes is described its relation to the welch berlekamp and euclidean algorithms is given

a self organizing state space model a self organizing filter and smoother for the general nonlinear non gaussian state space model is proposed an expanded state space model is defined by augmenting the state vector with the unknown parameters of the original state space model the state of the augmented state space model and hence the state and the parameters of the original state space model are estimated simultaneously by either a non gaussian filter smoother or a monte carlo filter smoother in contrast to maximum likelihood estimation of model parameters in ordinary state space modeling for which the recursive filter computation has to be done many times model parameter estimation in the proposed self organizing filter smoother is achieved with only two passes of the recursive filter and smoother operations examples such as automatic tuning of dispersion and the shape parameters adaptation to changes of the amplitude of a signal in seismic data state estimation for a nonlinear state space model with unknown parameters and seasonal adjustment with a nonlinear model with changing variance parameters are shown to exemplify the usefulness of the proposed method

additive fast fourier transforms over finite fields we present new additive fast fourier transform fft algorithms based on taylor expansions over finite fields of characteristic two the new algorithms improve previous approaches by wang and zhu 1988 cantor 1989 and von zur gathen and gerhard 1996

novel polynomial basis with fast fourier transform and its application to reed solomon erasure codes in this paper we present a fast fourier transform algorithm over extension binary fields where the polynomial is represented in a non standard basis the proposed fourier like transform requires mathcal o h lg h field operations where h is the number of evaluation points based on the proposed fourier like algorithm we then develop the encoding decoding algorithms for n 2 m k reed solomon erasure codes the proposed encoding erasure decoding algorithm requires mathcal o n lg n in both additive and multiplicative complexities as the complexity leading factor is small the proposed algorithms are advantageous in practical applications finally the approaches to convert the basis between the monomial basis and the new basis are proposed

a fast algorithm for the syndrome calculation in algebraic decoding of reed solomon codes in this paper fedorenko and trifonov s procedure is applied to evaluate the syndrome of the received word in time domain reed solomon decoders this application leads to a substantial reduction of the computational complexity of the syndrome polynomial for correcting both errors and erasures moreover simulation results for this new syndrome method are given

the simulation smoother for time series models summary recently suggested procedures for simulating from the posterior density of states given a gaussian state space time series are refined and extended we introduce and study the simulation smoother which draws from the multivariate posterior distribution of the disturbances of the model so avoiding the degeneracies inherent in state samplers the technique is important in gibbs sampling with non gaussian time series models and for performing bayesian analysis of gaussian time series