numerical analysis for a new non conforming linear finite element on quadrilaterals starting with a short introduction of a new non conforming linear quadrilateral p1 finite element which has been recently proposed by park a study on locking phenomena in finite element methods ph d thesis seoul national university february 2002 and park and sheen p1 nonconforming quadrilateral finite element methods for second order elliptic problems siam j numer anal 41 2 2003 624 640 we examine in detail the numerical behaviour of this element with special emphasis on the treatment of dirichlet boundary conditions efficient matrix assembly and solver aspects furthermore we compare the numerical characteristics of p1 with other low order finite elements also regarding its use for the incompressible navier stokes equations several test examples show the efficiency and reliability of the proposed methods for elliptic second order problems

unique decoding of general ag codes a unique decoding algorithm for general ag codes namely multipoint evaluation codes on algebraic curves is presented it is a natural generalization of the previous decoding algorithm which was only for one point ag codes as such it retains the same advantages of fast speed regular structure and direct message recovery upon this generalization we add a technique from the guruswami sudan list decoding that boosts the decoding speed significantly compared with other known decoding algorithms for general ag codes it has a similar decoding performance and allows streamlined practical implementation by its simple and regular structure

locally testable and locally correctable codes approaching the gilbert varshamov bound one of the most important open problems in the theory of error correcting codes is to determine the tradeoff between the rate r and minimum distance delta of a binary code the best known tradeoff is the gilbert varshamov bound and says that for every delta in 0 1 2 there are codes with minimum distance delta and rate r r mathsf gv delta 0 for a certain simple function r mathsf gv cdot in this paper we show that the gilbert varshamov bound can be achieved by codes which support local error detection and error correction algorithms specifically we show the following results 1 local testing for all delta in 0 1 2 and all r there exist codes with length n rate r and minimum distance delta that are locally testable with mathsf quasipolylog n query complexity 2 local correction for all epsilon 0 for all delta sufficiently large and all r there exist codes with length n rate r and minimum distance delta that are locally correctable from delta 2 o 1 fraction errors with o n epsilon query complexity furthermore these codes have an efficient randomized construction and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity our results on locally correctable codes also immediately give locally decodable codes with the same parameters our local testing result is obtained by combining thommesen s random concatenation technique and the best known locally testable codes by kopparty et al our local correction result which is significantly more involved also uses random concatenation along with a number of further ideas the guruswami sudan indyk list decoding strategy for concatenated codes alon edmonds luby distance amplification and the local list decodability local list recoverability and local testability of reed muller codes curiously our final local correction algorithms go via local list decoding and local testing algorithms this seems to be the first time local testability is used in the construction of a locally correctable code

progressive algebraic soft decision decoding of reed solomon codes the algebraic soft decision decoding asd algorithm is a polynomial time soft decoding algorithm for reed solomon rs codes it outperforms both the algebraic hard decision decoding ahd and the conventional unique decoding algorithms but with a high computational cost this paper proposes a progressive asd pasd algorithm that enables the conventional asd algorithm to perform decoding with an adjustable designed factorization output list size ols the ols is enlarged progressively leading to an incremental computation for the interpolation and an enhanced error correction capability multiple factorizations are performed in order to find out the intended message polynomial which will be validated by a cyclic redundant check crc code the incremental interpolation constraints are introduced to characterize the progressive decoding the validity analysis of the algorithm shows the pasd algorithm is a natural and computationally saving generalization of the asd algorithm delivering the same interpolation solution the average decoding complexity of the algorithm is further theoretically characterized revealing its dependence on the channel condition the simulation results further validate the analysis by showing that the average decoding complexity can be converged to the minimal level in a good channel condition finally performance evaluation shows the pasd algorithm preserves the error correction capability of the asd algorithm

improved two point codes on hermitian curves one point codes on the hermitian curve produce long codes with excellent parameters feng and rao introduced a modified construction that improves the parameters while still using one point divisors a separate improvement of the parameters was introduced by matthews considering the classical construction but with two point divisors those two approaches are combined to describe an elementary construction of two point improved codes upon analysis of their minimum distance and redundancy it is observed that they improve on the previous constructions for a large range of designed distances

an absolutely stabilized finite element method for the stokes problem on presente une formulation a elements finis absolument stabilisee pour le probleme de stokes on etablit des estimations d erreurs optimales dans la norme l 2 pour l approximation des champs de vitesses et de pressions

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

algebraic decoding using special divisors the basic algorithm for decoding of algebraic geometric codes corrects up to d sub c 1 2 g 2 errors where d sub c denotes the designed minimum distance of a code and g denotes the genus of a curve the modified algorithm improves on this but applies to a restricted class of codes an extended modified algorithm that applies to all codes is formulated it will correct up to d sub c 1 2 s errors s is called the clifford defect of a curve for curves with g or 1 this defect satisfies 0

fast chase decoding algorithms and architectures for reed solomon codes chase decoding is a prevalent soft decision decoding method for algebraic codes where an efficient bounded distance decoder is available essentially it repeatedly applies bounded distance decoding upon combinatorially flipping certain least reliable bits or patterns for nonbinary case in this paper we devise a one pass chase decoding algorithm of reed solomon codes such that the desired error locator polynomial by flipping an error pattern is obtained in one pass when operated in parallel through utilizing the preceding results this is effectively achieved through cumulative interpolation and linear feedback shift register lfsr synthesis techniques furthermore through converting the algorithm into the transform domain exhaustive root search for each error locator polynomial is circumvented computationally the new algorithm exhibits linear complexity in attempting to determine a candidate codeword associated with each additionally flipped symbol pattern it compares favorably to quadratic complexity by straightforwardly utilizing hard decision decoding where and denote the code length and minimum distance respectively we also reveal a corrected algorithm for one pass generalized minimum distance gmd decoding for reed solomon codes from koetter s original work in addition we devise a highly efficient one pass chase decoding algorithm for binary bose chaudhuri hocquenghem bch codes by taking advantage of a key characteristic of the berlekamp algorithm finally we present a systolic very large scale integration vlsi decoder architecture through slightly compromising the proposed one pass chase decoding algorithm it takes clock cycles to complete one iteration by flipping one error pattern both circuit and memory complexities are dictated by the dimension of flipping patterns in particular they are independent of the code length or the minimum distance rendering it highly attractive for various applications

complexity of decoding positive rate primitive reed solomon codes it has been proved that the maximum likelihood decoding problem of reed solomon codes is np hard however the length of the code in the proof is at most polylogarithmic in the size of the alphabet for the complexity of maximum likelihood decoding of the primitive reed solomon code whose length is one less than the size of alphabet the only known result states that it is at least as hard as the discrete logarithm in some cases where the information rate unfortunately goes to zero in this paper it is proved under a well known cryptography hardness assumption that 1 there does not exist a randomized polynomial time maximum likelihood decoder for the reed solomon code family q k q q where k x is any function in z z computable in time xo 1 satisfying x k x x x 2 there does not exist a randomized polynomial time bounded distance decoder for primitive reed solomon codes at distance 2 3 of the minimum distance for any constant 0 1 3 in particular this rules out the possibility of a polynomial time algorithm for maximum likelihood decoding problem of primitive reed solomon codes of any rate under the assumption

framework for the a posteriori error analysis of nonconforming finite elements this paper establishes a unified framework for the a posteriori error analysis of a large class of nonconforming finite element methods the theory assures reliability and efficiency of explicit residual error estimates up to data oscillations under the conditions h1 h2 and applies to several nonconforming finite elements the crouzeix raviart triangle element the han parallelogram element the nonconforming rotated nr parallelogram element of rannacher and turek the constrained nr parallelogram element of hu and shi the p 1 element on parallelograms due to park and sheen and the dssy parallelogram element the theory is extended to include 1 irregular meshes with at most one hanging node per edge

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

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

on semigroups generated by two consecutive integers and improved hermitian codes analysis of the berlekamp massey sakata algorithm for decoding one point codes leads to two methods for improving code rate one method due to feng and rao removes parity checks that may be recovered by their majority voting algorithm the second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic in this work formulae are given for the redundancies of hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance the results proceed from an analysis of numerical semigroups generated by two consecutive integers

on a decoding algorithm for codes on maximal curves a decoding algorithm for algebraic geometric codes that was given by a n skorobogatov and s g vladut preprint inst problems of information transmission 1988 is considered the author gives a modified algorithm with improved performance which he obtains by applying the above algorithm a number of times in parallel he proves the existence of the decoding algorithm on maximal curves by showing the existence of certain divisors however he has so far been unable to give an efficient procedure of finding these divisors

on determining deep holes of generalized reed solomon codes for a linear code deep holes are defined to be vectors that are further away from codewords than all other vectors the problem of deciding whether a received word is a deep hole for generalized reed solomon grs codes is proved to be co np complete by guruswami and vardy for the extended reed solomon codes rs q mathbb f q k a conjecture was made to classify deep holes by cheng and murray since then efforts have been made to prove the conjecture or its various forms in this paper we classify deep holes completely for grs codes rs p d k where p is a prime d k geqslant p 1 2 our techniques are built on the idea of deep hole trees and several results concerning the erdos heilbronn conjecture

projection finite element methods for semiconductor device equations abstract in this paper a class of nonstandard finite element methods which we call projection finite element methods is introduced to numerically solve the stationary drift diffusion semiconductor device equations in two and three space dimensions the methods are based on the use of nonconforming finite elements and the projection of coefficients into finite element spaces produce symmetric and positive definite systems of algebraic equations allow to design optimal order multigrid methods for the solution of the linear systems and yield error estimates of high order numerical results are presented to show the performance of the methods

limits to list decoding reed solomon codes in this paper we prove the following two results that expose some combinatorial limitations to list decoding reed solomon codes 1 given n distinct elements alpha1 alphan from a field f and n subsets s1 sn of f each of size at most l the list decoding algorithm of guruswami and sudan can in polynomial time output all polynomials p of degree at most k that satisfy p alphai isinsi for every i as long as l 0 agreement of k is trivial to achieve such a bound was known earlier only for a nonexplicit center finding explicit bad list decoding configurations is of significant interest for example the best known rate versus distance tradeoff due to xing is based on a bad list decoding configuration for algebraic geometric codes which is unfortunately not explicitly known

analysis of mixed finite elements methods for the stokes problem a unified approach we develop a method for the analysis of mixed finite element methods for the stokes problem in the velocity pressure formulation a technical macroelement condition which is sufficient for the classical babuska brezzi inequality to be valid is introduced using this condition we are able to verify the stability and optimal order of convergence of several known mixed finite element methods

a non conforming piecewise quadratic finite element on triangles it is shown in this paper that non conforming finite elements on the triangle using second degree polynomials can be easily built and used indeed they appear as an enriched version of the standard piecewise quadratic six node element this work is divided into two parts in the first we present the basic properties of the element namely how it can be built and basic error estimates we also show that this element exhibits a very peculiar regularity property in the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials

p _1 nonconforming finite elements on triangulations into triangles and quadrilaterals the p 1 nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected lipschitz domains an explicit a priori analysis for the combination of the park sheen and the crouzeix raviart nonconforming finite element methods is given for second order elliptic pdes with inhomogeneous dirichlet boundary conditions

a three dimensional quadratic nonconforming element we define a second degree nonconforming element on tetrahedra we build a basis for the opproximation space derived from this element we prove a discrete regularity property similar to the one that holds for the corresponding two dimensional element

stabilized mixed methods for the stokes problem the solution of the stokes problem is approximated by three stabilized mixed methods one due to hughes balestra and franca and the other two being variants of this procedure in each case the bilinear form associated with the saddle point problem of the standard mixed formulation is modified to become coercive over the finite element space error estimates are derived for each procedure

deterministic factorization of sparse polynomials with bounded individual degree in this article we study the problem of deterministic factorization of sparse polynomials we show that if f f x1 x2 xn is a polynomial with s monomials with individual degrees of its variables bounded by d then f can be deterministically factored in time spoly d log n prior to our work the only efficient factoring algorithms known for this class of polynomials were randomized and other than for the cases of d 1 and d 2 only exponential time deterministic factoring algorithms were known a crucial ingredient in our proof is a quasi polynomial sparsity bound for factors of sparse polynomials of bounded individual degree in particular we show that if f is an s sparse polynomial in n variables with individual degrees of its variables bounded by d then the sparsity of each factor of f is bounded by s 9 d2 log n this is the first non trivial bound on factor sparsity for d 2 our sparsity bound uses techniques from convex geometry such as the theory of newton polytopes and an approximate version of the classical caratheodory s theorem our work addresses and partially answers a question of von zur gathen and kaltofen 1985 who asked whether a quasi polynomial bound holds for the sparsity of factors of sparse polynomials

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

improved geometric goppa codes i basic theory in this paper we present a construction of improved geometric goppa codes which for the case of r 2g are often more efficient than the current geometric goppa codes derived from some varieties which include algebraic curves hyperplanes surfaces and other varieties for the special case of a plane in a three dimensional projective space the improved geometric goppa codes are reduced to linear multilevel codes for these improved geometric goppa codes a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given

a fourth order time splitting laguerre hermite pseudospectral method for bose einstein condensates a fourth order time splitting laguerre hermite pseudospectral method is introduced for bose einstein condensates becs in three dimensions with cylindrical symmetry the method is explicit time reversible and time transverse invariant it conserves the position density and is spectral accurate in space and fourth order accurate in time moreover the new method has two other important advantages i it reduces a three dimensional 3 d problem with cylindrical symmetry to an effective two dimensional 2 d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain the method is applied to vector gross pitaevskii equations vgpes for multicomponent becs extensive numerical tests are presented for the one dimensional 1 d gpe the 2 d gpe with radial symmetry the 3 d gpe with cylindrical symmetry as well as 3 d vgpes for two component becs to show the efficiency and accuracy of the new numerical method

on rational interpolation based list decoding and list decoding binary goppa codes we derive the wu list decoding algorithm for generalized reed solomon grs codes by using grobner bases over modules and the euclidean algorithm as the initial algorithm instead of the berlekamp massey algorithm we present a novel method for constructing the interpolation polynomial fast we give a new application of the wu list decoder by decoding irreducible binary goppa codes up to the binary johnson radius finally we point out a connection between the governing equations of the wu algorithm and the guruswami sudan algorithm immediately leading to equality in the decoding range and a duality in the choice of parameters needed for decoding both in the case of grs codes and in the case of goppa codes

on the decoding of algebraic geometric codes over f sub q for q or 16 it is proved that for algebraic geometric codes on a curve over f sub q for q or 37 or on a curve of sufficiently large genus over f sub q for q or 16 there exists a polynomial decoding algorithm up to d 1 2 errors d being the designed minimum distance

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

subspace polynomials and limits to list decoding of reed solomon codes we show combinatorial limitations on efficient list decoding of reed solomon codes beyond the johnson guraswami sudan bounds in particular we show that for arbitrarily large fields fn fn n for any 0 1 and k n 1 existence there exists a received word wn fn fn that agrees with a super polynomial number of distinct degree k polynomials on n points each 2 explicit there exists a polynomial time constructible received word w n fn fn that agrees with a superpolynomial number of distinct degree k polynomials on 2 log n k points each in both cases our results improve upon the previous state of the art which was n points of agreement for the existence case proved by justesen and hoholdt and 2n points of agreement for the explicit case proved by guruswami and rudra furthermore for close to 1 our bound approaches the guruswami sudan bound which is n k and implies limitations on extending their efficient reed solomon list decoding algorithm to larger decoding radius our proof is based on some remarkable properties of sub space polynomials using similar ideas we then present a family of low rate codes that are efficiently list decodable beyond the johnson bound this leads to an optimal list decoding algorithm for the family of matrix codes

partial geometric codes we present a theory of partial geometric goppa 1981 codes which can be applied to curves for which the genus is not explicitly known that is often the case if the curve is nonsmooth or nonplanar the only algebraic geometry required is familiarity with the basic concepts yet we are able to give a full account of all one point codes including their error processors we are also able to give a completely explicit construction for the initial code in a good sequence of codes based on a set of curves introduced by garcia and stichtenoth acad sci paris vol 322 i p 1067 70 1996 for which the genus has not been determined

a subspace of the dssy nonconforming quadrilateral finite element space for the stokes equations in this paper we propose a subspace of the dssy nonconforming quadrilateral finite element space the product of this space together with the piecewise constant space can be used for approximating the velocity and pressure variables respectively in solving stokes problems more precisely this space consists of the p 1 nonconforming quadrilateral finite element space augmented by macro bubble functions based on the dssy nonconforming quadrilateral space under a hood taylor type assumption on meshes it is shown that the pair satisfies the discrete inf sup condition using a boundedness estimate of an interpolation operator based on edge integrals numerical results are presented

analysis of a chebyshev type pseudo spectral scheme for the nonlinear schrdinger equation in this paper we derive several error estimates that are pertinent to the study of chebyshev type spectral approximations on the real line the results are applied to construct a stable and accurate pseudo spectral chebyshev scheme for the nonlinear schrdinger equation the new technique has several computational advantages as compared to fourier and hermite type spectral schemes described in the literature see e g 1 3 similar to hermite type methods we do not require domain truncation and or use of artificial boundary conditions at the same time the computational complexity is comparable to the best fourier type spectral methods described in the literature

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

improved probabilistic decoding of interleaved reed solomon codes and folded hermitian codes probabilistic simultaneous polynomial reconstruction algorithm of bleichenbacher kiayias and yung is extended to the polynomials whose degrees are allowed to be distinct specifically for a finite field f positive integers n r t and distinct elements z 1 z 2 z email protected f we present a probabilistic algorithm which can recover polynomials p 1 p 2 p email protected f x of degree less than k 1 k 2 k r respectively for a given instance i 1 n satisfying p l z i y i l for all email protected 1 2 r and for all email protected email protected 1 2 n such that i t with probability at least 1 n t f and with time complexity at most o rn 4 if t max k 1 k 2 k r email protected j 1 rk jr 1 next by using this algorithm we present a probabilistic decoder for interleaved reed solomon codes it is observed that interleaved reed solomon codes over f with rate r can be decoded up to burst error rate rr 1 1 r probabilistically for an interleaving parameter r then it is proved that q folded hermitian codes over f q 2 q with rate r can be decoded up to error rate qq 1 1 r probabilistically

construction and decoding of a class of algebraic geometry codes a class of codes derived from algebraic plane curves is constructed the concepts and results from algebraic geometry that were used are explained in detail no further knowledge of algebraic geometry is needed parameters generator and parity check matrices are given the main result is a decoding algorithm which turns out to be a generalization of the peterson algorithm for decoding bch decoder codes

hard problems of algebraic geometry codes the minimum distance is one of the most important combinatorial characterizations of a code the maximum likelihood decoding problem is one of the most important algorithmic problems of a code while these problems are known to be hard for general linear codes the techniques used to prove their hardness often rely on the construction of artificial codes in general much less is known about the hardness of the specific classes of natural linear codes in this correspondence we show that both problems are np hard for algebraic geometry codes we achieve this by reducing a well known np complete problem to these problems using a randomized algorithm the family of codes in the reductions is based on elliptic curves they have positive rates but the alphabet sizes are exponential in the block lengths

a nonconforming finite element method for the stationary smagorinsky model abstract in this paper we focus on a low order nonconforming finite element method fem for the stationary smagorinsky model the velocity and pressure are approximated by the constrained nonconforming rotated q1 element cn q 1 r o t and piecewise constant element respectively optimal error estimates of the velocity in the broken h1 norm and l2 norm and the pressure in the l2 norm are derived by some nonlinear analysis techniques and aubin nitsche duality argument the supercloseness and superconvergent results are also obtained under some reasonable regularity assumptions finally a numerical example is implemented to confirm our theoretical analysis

unique decoding of plane ag codes via interpolation we present a unique decoding algorithm of algebraic geometry ag codes on plane curves hermitian codes in particular from an interpolation point of view the algorithm successfully corrects errors of weight up to half of the order bound on the minimum distance of the ag code it is the first decoding algorithm to combine some features of the interpolation based list decoding with the performance of the syndrome decoding with the majority voting scheme the regular structure of the algorithm allows a straightforward parallel implementation

the minimum distance of codes in an array coming from telescopic semigroups the concept of an error correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound this gives a unified point of view which explains several improvements on the minimum distance of algebraic geometric codes moreover it is explained in terms of linear algebra and the theory of semigroups only

a generalization of the berlekamp massey algorithm for multisequence shift register synthesis with applications to decoding cyclic codes a generalization of the berlekamp massey algorithm is presented for synthesizing minimum length linear feedback shift registers for generating prescribed multiple sequences a more general problem is first considered that of finding the smallest initial set of linearly dependent columns in a matrix over an arbitrary field which includes the multisequence problem as a special case a simple iterative algorithm the fundamental iterative algorithm fia is presented for solving this problem the generalized algorithm is then derived through a refinement of the fia application of this generalized algorithm to decoding cyclic codes up to the hartmann tzeng ht bound and roos bound making use of multiple syndrome sequences is considered conditions for guaranteeing that the connection polynomial of the shortest linear feedback shift register obtained by the algorithm will be the error locator polynomial are determined with respect to decoding up to the ht bound and special cases of the roos bound

a unifying theory of a posteriori error control for nonconforming finite element methods residual based a posteriori error estimates were derived within one unifying framework for lowest order conforming nonconforming and mixed finite element schemes in carstensen numer math 100 617 637 2005 therein the key assumption is that the conforming first order finite element space v h c annulates the linear and bounded residual l written v h c subseteq rm ker ell that excludes particular nonconforming finite element methods ncfems on parallelograms in that v h c not subset rm ker ell the present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates also for mortar and discontinuous galerkin methods the key assumption is the existence of some bounded linear operator pi v h c rightarrow v h nc with some elementary properties it is conjectured that the more general hypothesis h1 h3 can be established for all known ncfems applications on various nonstandard finite element schemes for the laplace stokes and navier lame equations illustrate the presented unifying theory of a posteriori error control for ncfem

list decoding of crisscross errors in this paper list decoding of crisscross errors in arrays over finite fields is considered for this purpose the so called cover metric is used where the cover of a matrix is a set of rows and columns which contains all non zero elements of the matrix a johnson like upper bound on the maximum list size in the cover metric is derived showing that the list of codewords has polynomial size up to a certain radius furthermore a simple list decoding algorithm for a known optimal code construction is presented which decodes errors in the cover metric up to our upper bound these results reveal significant differences between the cover metric and the rank metric and show that the cover metric is more suitable for correcting crisscross errors

vlsi architectures for soft decision decoding of reed solomon codes soft decision decoding of reed solomon codes delivers significant coding gains over classical minimum distance decoding in this paper we present architectures for polynomial interpolation and factorization the two main steps of the soft decoding algorithm we introduce an algorithmic transformation for reducing the iterations required in generating the interpolation polynomial and present efficient architectures by sharing computations we also describe algorithmic transformations for further reducing the interpolation and factorization latency an area efficient folded pipelined version of the interpolation architecture is also described finally we present an example of a reed solomon soft decoder utilizing the presented architectures having a 250 mbps throughput

a parametric approach to list decoding of reed solomon codes using interpolation in this paper we present a minimal list decoding algorithm for reed solomon rs codes minimal list decoding for a code c refers to list decoding with radius l where l is the minimum of the distances between the received word r and any codeword in c we consider the problem of determining the value of l as well as determining all the codewords at distance l our approach involves a parametrization of interpolating polynomials of a minimal grobner basis g we present two efficient ways to compute g we also show that so called re encoding can be used to further reduce the complexity we then demonstrate how our parametric approach can be solved by a computationally feasible rational curve fitting solution from a recent paper by wu besides we present an algorithm to compute the minimum multiplicity as well as the optimal values of the parameters associated with this multiplicity which results in overall savings in both memory and computation

extractor codes we study error correcting codes for highly noisy channels for example every received signal in the channel may originate from some half of the symbols in the alphabet our main conceptual contribution is an equivalence between error correcting codes for such channels and extractors our main technical contribution is a new explicit error correcting code based on trevisan s extractor that can handle such channels and even noisier ones our new code has polynomial time encoding and polynomial time soft decision decoding we note that reed solomon codes cannot handle such channels and our study exposes some limitations on list decoding of reed solomon codes another advantage of our equivalence is that when the johnson bound is restated in terms of extractors it becomes the well known leftover hash lemma this yields a new proof of the johnson bound which applies to large alphabets and soft decoding our explicit codes are useful in several applications first they yield algorithms to extract many hardcore bits using few auxiliary random bits second they are the key tool in a recent scheme to compactly store a set of elements in a way that membership in the set can be determined by looking at only one bit of the representation finally they are the basis for the recent construction of high noise almost optimal rate list decodable codes over large alphabets

on krylov subspace approximations to the matrix exponential operator krylov subspace methods for approximating the action of matrix exponentials are analyzed in this paper we derive error bounds via a functional calculus of arnoldi and lanczos methods that reduces the study of krylov subspace approximations of functions of matrices to that of linear systems of equations as a side result we obtain error bounds for galerkin type krylov methods for linear equations namely the biconjugate gradient method and the full orthogonalization method for krylov approximations to matrix exponentials we show superlinear error decay from relatively small iteration numbers onwards depending on the geometry of the numerical range the spectrum or the pseudospectrum the convergence to exp tau a v is faster than that of corresponding krylov methods for the solution of linear equations i tau a x v which usually arise in the numerical solution of stiff ordinary differential equations odes we therefore propose a new class of time integration methods for large systems of nonlinear differential equations which use krylov approximations to the exponential function of the jacobian instead of solving linear or nonlinear systems of equations in every time step

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

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

syndrome generation and error location search for the decoding of algebraic geometry codes on plane garcia stichtenoth curves in this paper the basic function field of an asymptotically optimal tower of function fields by garcia and stichtenoth is studied where bases of one point linear systems can be explicitly constructed a representation of finite gf q 2 rational points on the associated plane curve is derived in this paper this representation is exploited to extend the use of horner s loops and the mechanism of chien search in the decoding of reed solomon codes for syndrome generation and error location search respectively to algebraic geometry codes built on the plane garcia stichtenoth curve

algebraic soft decision decoding of reed solomon codes using bit level soft information the performance of algebraic soft decision decoding of reed solomon codes using bit level soft information is investigated optimal multiplicity assignment strategies for algebraic soft decision decoding sdd with infinite cost are first studied over erasure channels and the binary symmetric channel the corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived the multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic sdd over a mixed error and bit level erasure channel the bit level decoding region of the proposed multiplicity assignment strategy is shown to be significantly larger than that of conventional berlekamp massey decoding as an application a bit level generalized minimum distance decoding algorithm is proposed the proposed decoding compares favorably with many other reed solomon sdd algorithms over various channels moreover owing to the simplicity of the proposed bit level generalized minimum distance decoding its performance can be tightly bounded using order statistics

a stabilized finite element method based on two local gauss integrations for the stokes equations this paper considers a stabilized method based on the difference between a consistent and an under integrated mass matrix of the pressure for the stokes equations approximated by the lowest equal order finite element pairs i e the p 1 p 1 and q 1 q 1 pairs this method only offsets the discrete pressure space by the residual of the simple and symmetry term at element level in order to circumvent the inf sup condition optimal error estimates are obtained by applying the standard galerkin technique finally the numerical illustrations agree completely with the theoretical expectations

performance of sphere decoding of block codes a sphere decoder searches for the closest lattice point within a certain search radius the search radius provides a tradeoff between performance and complexity we focus on analyzing the performance of sphere decoding of linear block codes we analyze the performance of soft decision sphere decoding on awgn channels and a variety of modulation schemes a hard decision sphere decoder is a bounded distance decoder with the corresponding decoding radius we analyze the performance of hard decision sphere decoding on binary and q ary symmetric channels an upper bound on the performance of maximum likelihood decoding of linear codes defined over fq e g reed solomon codes and transmitted over q ary symmetric channels is derived and used in the analysis we then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes the tradeoff between the performance and complexity of a sphere decoder is then discussed

efficient interpolation in the guruswami sudan algorithm a novel algorithm is proposed for the interpolation step of the guruswami sudan list decoding algorithm the proposed method is based on the binary exponentiation algorithm and can be considered as an extension of the lee o sullivan method the algorithm is shown to achieve both asymptotical and practical performance gain compared to the case of iterative interpolation algorithm further complexity reduction is achieved by employing the reencoding transformation the key contribution of the paper which enables the complexity reduction is a novel randomized ideal multiplication algorithm

linearized polynomial interpolation and its applications in this paper we first propose an interpolation algorithm in a well ordered free module of a linearized polynomial ring and then use this algorithm to decode several important families of codes gabidulin codes kotter and kschischang kk codes and mahdavifar and vardy mv codes our decoding algorithm for gabidulin codes is different from the polynomial reconstruction algorithm by loidreau when applied to decode kk codes our interpolation algorithm is equivalent to the sudan style list 1 decoding algorithm proposed by kotter and kschischang for kk codes the interpolation approach is also capable of solving the interpolation problem for the list decoding of mv codes proposed by mahdavifar and vardy and has a lower complexity than gaussian elimination an interpolator for list decoding of mv codes has also been implemented in hardware and the synthesis results show that it leads to better throughput and efficiency than gaussian elimination

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

a cheapest nonconforming rectangular finite element for the stationary stokes problem abstract we introduce a least degrees of freedom nonconforming finite element pair to approximate the stokes equations based on quadrilateral meshes the finite element space for the velocity field is composed of the p 1 nonconforming quadrilateral element plus only one additional global dssy douglas santos sheen ye or rt rannacher turek type of bubble function the pressure field is approximated by the piecewise constant function we show that the discrete problem is nonsingular and the element pair satisfies a weak discrete inf sup condition several numerical examples are shown to confirm the efficiency and reliability of the proposed method

list decoding of generalized reed solomon codes over commutative rings we show that the list decoding procedure of guruswami and sudan may be used to decode generalized reed solomon rs codes defined over commutative rings with identity an algorithm for performing the first of the two phases of this decoding procedure is also given in particular we show that for generalized rs codes over chain rings the probability of picking an incorrect codeword from a given list can be reduced by increasing the size of the code alphabet while keeping the length and minimum hamming distance of the code constant

cryptanalysis of mceliece cryptosystem based on algebraic geometry codes and their subcodes we give polynomial time attacks on the mceliece public key cryptosystem based either on algebraic geometry ag codes or on small co dimensional subcodes of ag codes these attacks consist in the blind reconstruction either of an error correcting pair ecp or an error correcting array eca from the single data of an arbitrary generator matrix of a code an ecp provides a decoding algorithm that corrects up to d 1 g 2 errors where d denotes the designed distance and g denotes the genus of the corresponding curve while with an eca the decoding algorithm corrects up to d 1 2 errors roughly speaking for a public code of length n over mathbb f q these attacks run in o n 4 log n operations in mathbb f q for the reconstruction of an ecp and o n 5 operations for the reconstruction of an eca a probabilistic shortcut allows to reduce the complexities respectively to o n 3 varepsilon log n and o n 4 varepsilon compared with the previous known attack due to faure and minder our attack is efficient on codes from curves of arbitrary genus furthermore we investigate how far these methods apply to subcodes of ag codes

uniform error estimates of finite difference methods for the nonlinear schrodinger equation with wave operator we establish uniform error estimates of finite difference methods for the nonlinear schrodinger equation nls perturbed by the wave operator nlsw with a perturbation strength described by a dimensionless parameter varepsilon varepsilon in 0 1 when varepsilon to0 nlsw collapses to the standard nls in the small perturbation parameter regime i e 0 varepsilon ll1 the solution of nlsw is perturbed from that of nls with a function oscillating in time with o varepsilon 2 wavelength at o varepsilon 4 and o varepsilon 2 amplitudes for well prepared and ill prepared initial data respectively this high oscillation of the solution in time brings significant difficulties in establishing error estimates uniformly in varepsilon of the standard finite difference methods for nlsw such as the conservative crank nicolson finite difference cnfd method and the semi implicit finite difference sifd method we obtain error bounds uniformly in varepsilon at the order of o h 2 tau and o h 2 tau 2 3 with time step tau and mesh size h for well prepared and ill prepared initial data respectively for both cnfd and sifd in the l 2 norm and discrete semi h 1 norm our error bounds are valid for general nonlinearity in nlsw and for one two and three dimensions to derive these uniform error bounds we combine varepsilon dependent error estimates of nlsw varepsilon dependent error bounds between the numerical approximate solutions of nlsw and the solution of nls together with error bounds between the solutions of nlsw and nls other key techniques in the analysis include the energy method cut off of the nonlinearity and a posterior bound of the numerical solutions by using the inverse inequality and discrete semi h 1 norm estimate finally numerical results are reported to confirm our error estimates of the numerical methods and show that the convergence rates are sharp in the respective parameter regimes

reconstructing algebraic functions from mixed data we consider a variant of the traditional task of explicitly reconstructing algebraic functions from black box representations in the traditional setting for such problems one is given access to an unknown function f that is represented by a black box or an oracle which can be queried for the value of f at any input given a guarantee that this unknown function f is some nice algebraic function say a polynomial in its input of degree bound d the goal of the reconstruction problem is to explicitly determine the coefficients of the unknown polynomial all work on polynomial interpolation especially sparse ones are or may be presented in such a setting the work of kaltofen and trager computing with polynomials given by black boxes for their evaluations greatest common divisors factorization separation of numerators and denominators in proc 29th ann ieee symp on foundations of computer science 1988 pp 296 305 for instance highlights the utility of this setting by performing numerous manipulations on polynomials presented as black boxes the variant considered in this paper differs from the traditional setting in that our black boxes represent several algebraic functions f1 fk where at each input x the box arbitrarily chooses a subset of f1 x fk x to output and we do not know which subset it outputs we show how to reconstruct the functions f1 fk from the black box provided the black box outputs according to these functions often this allows us to group the sample points into sets such that for each set all outputs to points in the set are from the same algebraic function our methods are robust in the presence of a small fraction of arbitrary errors in the black box our model and techniques can be applied in the areas of computer vision machine learning curve fitting and polynomial approximation self correcting programs and bivariate polynomial factorization

combinatorial bounds for list decoding informally an error correcting code has nice list decodability properties if every hamming ball of large radius has a small number of codewords in it we report linear codes with nontrivial list decodability i e codes of large rate that are nicely list decodable and codes of large distance that are not nicely list decodable specifically on the positive side we show that there exist codes of rate r and block length n that have at most c codewords in every hamming ball of radius h sup 1 1 r 1 c spl middot n this answers the main open question from the work of elias 1957 this result also has consequences for the construction of concatenated codes of good rate that are list decodable from a large fraction of errors improving previous results of guruswami and sudan see ieee trans inform theory vol 45 p 1757 67 sept 1999 and proc 32nd acm symp theory of computing stoc portland or p 181 190 may 2000 in this vein specifically for every spl epsi 0 we present a polynomial time constructible asymptotically good family of binary codes of rate spl omega spl epsi sup 4 that can be list decoded in polynomial time from up to a fraction 1 2 spl epsi of errors using lists of size o spl epsi sup 2 on the negative side we show that for every spl delta and c there exists spl tau 0 and an infinite family of linear codes c sub i sub i such that if n sub i denotes the block length of c sub i then c sub i has minimum distance at least spl delta spl middot n sub i and contains more than c sub 1 spl middot n sub i sup c codewords in some hamming ball of radius spl tau spl middot n sub i while this result is still far from known bounds on the list decodability of linear codes it is the first to bound the radius for list decodability by a polynomial sized list away from the minimum distance of the code

syndrome decoding of reed solomon codes beyond half the minimum distance based on shift register synthesis in this paper a new approach for decoding low rate reed solomon codes beyond half the minimum distance is considered and analyzed the maximum error correcting radius coincides with the error correcting radius of the sudan algorithm published in 1997 however unlike the sudan algorithm the approach described here is not a list decoding algorithm and is not based on polynomial interpolation the algorithm in this paper is rather syndrome based like classical algebraic decoding algorithms the computational complexity of the new algorithm is of the same order as the complexity of the well known berlekamp massey algorithm to decode errors beyond half the minimum distance the new decoder is allowed to fail for some high weight error patterns with a very small probability

analysis of two low order equal order finite element pairs for stokes equations over quadrilaterals abstract two quadrilateral low order equal order finite element schemes are analyzed for stokes equations both of these schemes adopt the quadrilateral p 1 nonconforming finite element to approximate the pressure over a coarser mesh the velocity spaces are constructed over a finer mesh where the standard q 1 conforming element space and the quadrilateral p 1 nonconforming element space are selected respectively the stability assertion is given for each pair moreover the superconvergence property of the pressure is obtained over uniform rectangular meshes all the analyses above are verified by numerical tests

generation of matrices for determining minimum distance and decoding of algebraic geometric codes newton s identities have played a significant role in decoding and minimum distance determination of cyclic and bch codes the present paper carries the notion over from cyclic codes to algebraic geometric ag codes and introduces newton s identities for ag codes also for the purpose of minimum distance determination and decoding

high order time splitting hermite and fourier spectral methods in this paper we are concerned with the numerical solution of the time dependent gross pitaevskii equation gpe involving a quasi harmonic potential primarily we consider discretisations that are based on spectral methods in space and higher order exponential operator splitting methods in time the resulting methods are favourable in view of accuracy and efficiency moreover geometric properties of the equation such as particle number and energy conservation are well captured regarding the spatial discretisation of the gpe we consider two approaches in the unbounded domain we employ a spectral decomposition of the solution into hermite basis functions on the other hand restricting the equation to a sufficiently large bounded domain fourier techniques are applicable for the time integration of the gpe we study various exponential operator splitting methods of convergence orders two four and six our main objective is to provide accuracy and efficiency comparisons of exponential operator splitting fourier and hermite pseudospectral methods for the time evolution of the gpe furthermore we illustrate the effectiveness of higher order time splitting methods compared to standard integrators in a long term integration

decoding of codes defined by a single point on a curve a decoding algorithm for certain codes from algebraic curves is presented the dual code which is used for decoding is formed by evaluating rational functions having poles at a single point a theoretical foundation is developed from which an improved bound for the minimum distance and results on decoding up to that bound are derived the decoding is done by a computationally efficient berlekamp massey type algorithm which is intrinsic to the curve

a relationship between stabilized finite element methods and the galerkin method with bubble functions abstract a relation between stabilized finite element methods and the galerkin method employing interpolations with bubble functions is established for the advective diffusive model and for the linearized compressible navier stokes equations the bubble functions are shown to help in stabilizing the advective operator without recourse to upwinding or any other numerical strategy in particular for the advective diffusive model the galerkin method employing piecewise linears with bubble functions is shown to be equivalent to the streamline upwind petrov galerkin supg method in the diffusive limit

majority coset decoding a majority coset decoding mcd procedure that can be applied to an arbitrary geometric code is discussed in general the basic algorithm for decoding of algebraic geometric codes does not correct up to the designed minimum distance in mcd a reduction step is added to the basic algorithm in case the basic algorithm fails a majority scheme is used to obtain an additional syndrome for the error vector thus a strictly smaller cost containing the error vector is obtained in this way the basic algorithm is applied to a decreasing chain of cosets and after finitely many steps the coset will be small enough for successful application of the basic algorithm

a code decomposition approach for decoding cyclic and algebraic geometric codes an approach based on code decomposition and partial transform for constructing algorithms for decoding cyclic codes and algebraic geometric ag codes is introduced a general decoding procedure applicable to arbitrary linear codes and their subfield subcodes is then developed in particular we have developed algorithms for error and erasure decoding of cyclic codes up to their actual minimum distance error and erasure decoding of ag codes up to their designed minimum distance and decoding of subfield subcodes of ag codes geometric bch codes moreover a generalization of geometric bch codes is introduced it is shown by an example that for a simple class of ag codes the so called one point codes this generalization brings a better estimate of their minimum distance it is also shown that the algorithms developed in this paper can be applied to decode these codes up to their estimated minimum distance

a generalized laguerre spectral petrov galerkin method for the time fractional subdiffusion equation on the semi infinite domain abstract in this paper a generalized laguerre spectral petrov galerkin method is proposed to solve a class of time fractional subdiffusion equations on the semi infinite domain we use the generalized associated laguerre functions of the first kind and the generalized laguerre functions as basis functions in time and space directions separately the respective projection error estimates can be obtained as well we derive the projection error estimates of the solutions in time space directions the approximation results of the fully discrete spectral scheme are introduced in this paper some numerical results are presented to illustrate the efficiency of this method

iterative soft input soft output decoding of reed solomon codes by adapting the parity check matrix an iterative algorithm is presented for soft input soft output siso decoding of reed solomon rs codes the proposed iterative algorithm uses the sum product algorithm spa in conjunction with a binary parity check matrix of the rs code the novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the spa is applied at each iteration the proposed algorithm can be geometrically interpreted as a two stage gradient descent with an adaptive potential function this adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and therefore significantly improves the performance simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding hdd and compare favorably with other popular soft decision decoding methods

high rate codes with sublinear time decoding locally decodable codes are error correcting codes that admit efficient decoding algorithms any bit of the original message can be recovered by looking at only a small number of locations of a corrupted codeword the tradeoff between the rate of a code and the locality efficiency of its decoding algorithms has been well studied and it has widely been suspected that nontrivial locality must come at the price of low rate a particular setting of potential interest in practice is codes of constant rate for such codes decoding algorithms with locality o k were known only for codes of rate 1 where k is the length of the message furthermore for codes of rate 1 2 no nontrivial locality had been achieved in this article we construct a new family of locally decodable codes that have very efficient local decoding algorithms and at the same time have rate approaching 1 we show that for every 0 and 0 for infinitely many k there exists a code c which encodes messages of length k with rate 1 and is locally decodable from a constant fraction of errors using o k queries and time these codes which we call multiplicity codes are based on evaluating multivariate polynomials and their derivatives multiplicity codes extend traditional multivariate polynomial codes they inherit the local decodability of these codes and at the same time achieve better tradeoffs and flexibility in the rate and minimum distance

p1 nonconforming quadrilateral finite element methods for second order elliptic problems a p1 nonconforming quadrilateral finite element is introduced for second order elliptic problems in two dimensions unlike the usual quadrilateral nonconforming finite elements which contain quadratic polynomials or polynomials of degree greater than 2 our element consists of only piecewise linear polynomials that are continuous at the midpoints of edges one of the benefits of using our element is convenience in using rectangular or quadrilateral meshes with the least degrees of freedom among the nonconforming quadrilateral elements an optimal rate of convergence is obtained also a nonparametric reference scheme is introduced in order to systematically compute stiffness and mass matrices on each quadrilateral an extension of the p1 nonconforming element to three dimensions is also given finally several numerical results are reported to confirm the effective nature of the proposed new element

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

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

pseudospectral laguerre approximation of transport fragmentation equations we develop a laguerre type pseudo spectral scheme for solving transport fragmentation equations the method converges rapidly for certain type of fragmentation problems and works under mild restrictions on the growth rate of the coefficients of the equation rigorous stability and convergence analyses are provided numerical simulations illustrate the performance of the scheme

stable cheapest nonconforming finite elements for the stokes equations we introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the stokes equations one pair has each component of its velocity field to be approximated by the p 1 nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two dimensional subspaces removed one removed space is due to the integral mean zero property and the other space consists of global checker board patterns the other pair consists of the velocity space as the p 1 nonconforming quadrilateral element enriched by a globally one dimensional macro bubble function space based on d s s y douglas santos sheen ye nonconforming finite element space the pressure field is approximated by the piecewise constant function with mean zero space eliminated we show that two element pairs satisfy the discrete inf sup condition uniformly and we investigate the relationship between them several numerical examples are shown to confirm the efficiency and reliability of the proposed methods