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

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

on the list and bounded distance decodability of reed solomon codes for an error correcting code and a distance bound the list decoding problem is to compute all the codewords within a given distance to a received message the bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance or to output the empty set if there is not obviously the bounded distance decoding problem is not as hard as the list decoding problem for a reed solomon code n k q a simple counting argument shows that for any integer 0 0 we show that the discrete logarithm problem over bf f q h can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the reed solomon code q g h q with radius q g these results show that the decoding problems for the reed solomon code are at least as hard as the discrete logarithm problem over certain finite fields for the list decoding problem of reed solomon codes although the infeasible radius that we obtain is much larger than the radius which is known to be feasible it is the first nontrivial bound our result on the bounded distance decodability of reed solomon codes is also the first of its kind the main tools for obtaining these results are an interesting connection between the problem of list decoding of reed solomon code the problem of a discrete logarithm over finite fields and a generalization of katz s theorem on representations of elements in an extension finite field by products of distinct linear factors

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

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

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

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

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

bounds on list decoding of rank metric codes so far there is no polynomial time list decoding algorithm beyond half the minimum distance for gabidulin codes these codes can be seen as the rank metric equivalent of reed solomon codes in this paper we provide bounds on the list size of rank metric codes in order to understand whether polynomial time list decoding is possible or whether it works only with exponential time complexity three bounds on the list size are proven the first one is a lower exponential bound for gabidulin codes and shows that for these codes no polynomial time list decoding beyond the johnson radius exists second an exponential upper bound is derived which holds for any rank metric code of length n and minimum rank distance d the third bound proves that there exists a rank metric code over bbfqm of length n m such that the list size is exponential in the length for any radius greater than half the minimum rank distance this implies that there cannot exist a polynomial upper bound depending only on n and d similar to the johnson bound in hamming metric all three rank metric bounds reveal significant differences to bounds for codes in hamming metric

the integrable configuration of the disordered n 3 discrete self trapping equation abstract we consider the discrete self trapping equation with three degrees of freedom containing disorder in the system parameters using su 3 notations a system of eight real first order ordinary differential equations is derived for the integrable configuration an open chain with a single anharmonicity placed at the central site we can deduce from this system a second order oscillator equation for the probability amplitude of the central site excitation using the corresponding anharmonic potential for a study of the oscillations of the central oscillator amplitude the dynamics of the chain can be mapped on the disorder nonlinearity parameter plane we discuss the localization properties of the excitation energy transfer on the trimer the interplay between nonlinearity and disorder leads to some interesting effects such as suppression and amplification of localization

an interface capturing method with a continuous function the thinc method with multi dimensional reconstruction an interface capturing method with a continuous function is proposed within the framework of the volume of fluid vof method being different from the traditional vof methods that require a geometrical reconstruction and identify the interface by a discontinuous heaviside function the present method makes use of the hyperbolic tangent function known as one of the sigmoid type functions in the tangent of hyperbola interface capturing thinc method f xiao y honma k kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer methods fluids 48 2005 1023 1040 to retrieve the interface in an algebraic way from the volume fraction data of multi component materials instead of the 1d reconstruction in the original thinc method a multi dimensional hyperbolic tangent function is employed in the present new approach the present scheme resolves moving interface with geometric faithfulness and compact thickness and has at least the following advantages 1 the geometric reconstruction is not required in constructing piecewise approximate functions 2 besides a piecewise linear interface curved quadratic surface can be easily constructed as well and 3 the continuous multi dimensional hyperbolic tangent function allows the direct calculations of derivatives and normal vectors numerical benchmark tests including transport of moving interface and incompressible interfacial flows are presented to validate the numerical accuracy for interface capturing and to show the capability for practical problems such as a stationary circular droplet a drop oscillation a shear induced drop deformation and a rising bubble

an eulerian interface sharpening algorithm for compressible two phase flow the algebraic thinc approach article history we describe a novel interface sharpening approach for efficient numerical resolution of a compressible homogeneous two phase flow governed by a quasi conservative five equation model of allaire et al 2001 1 the algorithm uses a semi discrete wave propagation method to find approximate solution of this model numerically in the algorithm in regions near the interfaces where two different fluid components are present within a cell the thinc tangent of hyperbola for interface capturing scheme is used as a basis for the reconstruction of a sub grid discontinuity of volume fractions at each cell edge and it is complemented by a homogeneous equilibrium consistent technique that is derived to ensure a consistent modeling of the other interpolated physical variables in the model in regions away from the interfaces where the flow is single phase standard reconstruction scheme such as muscl or weno can be used for obtaining high order interpolated states these reconstructions are then used as the initial data for riemann problems and the resulting fluctuations form the basis for the spatial discretization time integration of the algorithm is done by employing a strong stability preserving runge kutta method numerical results are shown for sample problems with the mie gruneisen equation of state for characterizing the materials of interests in both one and two space dimensions that demonstrate the feasibility of the proposed method for interface sharpening of compressible two phase flow to demonstrate the competitiveness of our approach we have also included results obtained using the anti diffusion interface sharpening method

an interface capturing method for the simulation of multi phase compressible flows a novel finite volume interface contact capturing method is presented for simulation of multi component compressible flows with high density ratios and strong shocks in addition the materials on the two sides of interfaces can have significantly different equations of state material boundaries are identified through an interface function which is solved in concert with the governing equations on the same mesh for long simulations the method relies on an interface compression technique that constrains the thickness of the diffused interface to a few grid cells throughout the simulation this is done in the spirit of shock capturing schemes for which numerical dissipation effectively preserves a sharp but mesh representable shock profile for contact capturing the formulation is modified so that interface representations remain sharp like captured shocks countering their tendency to diffuse via the same numerical diffusion needed for shock capturing special techniques for accurate and robust computation of interface normals and derivatives of the interface function are developed the interface compression method is coupled to a shock capturing compressible flow solver in a way that avoids the spurious oscillations that typically develop at material boundaries convergence to weak solutions of the governing equations is proved for the new contact capturing approach comparisons with exact riemann problems for model one dimensional multi material flows show that the interface compression technique is accurate the method employs cartesian product stencils and therefore there is no inherent obstacles in multiple dimensions examples of two and three dimensional flows are also presented including a demonstration with significantly disparate equations of state a shock induced collapse of three dimensional van der waal s bubbles air in a stiffened equation of state liquid water adjacent to a mie gruneisen equation of state wall copper

a unified view on known algebraic decoding algorithms and new decoding concepts known properties of cyclic codes are used to give a unified description of many classical decoding algorithms for reed solomon codes up to half the minimum distance this description allows also simplified proofs for these decoders further a novel decoding algorithm is derived using these properties directly and variants of a new error erasure decoding algorithm are given for decoding beyond half the minimum distance a basis of all solutions for decoding is derived this basis allows to use side information in order to decode beyond half the minimum distance other methods where this basis can be used are power decoding also known as virtual syndrome extension where additional equations are generated by taking powers of the received symbols and interleaved reed solomon codes the extended euclidean algorithm which calculates the greatest common divisor plays an essential role in many presented methods

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

nonlinear codes from algebraic curves improving the tsfasman vladut zink bound in the present paper we construct a class of nonlinear codes by making use of higher order derivatives of certain functions of algebraic curves it turns out that the asymptotic bound derived from the goppa geometry codes can be improved for the entire interval 0 1 in particular the tsfasman vladut zink tvz bound is ameliorated for the entire interval 0 1

on the average complexity of reed solomon list decoders the number of monomials required to interpolate a received word in an algebraic list decoder for reed solomon codes depends on the instantaneous channel error and not only on the decoder design parameters the implications of this fact are that the decoder should be able to exhibit lower decoding complexity for low weight errors and consequently enjoy a better average case decoding complexity and a higher decoding throughput on the analytical side this paper studies the dependence of interpolation costs on instantaneous errors in both hard and soft decision decoders on the algorithmic side it provides an efficient interpolation algorithm based on the state of the art interpolation algorithm that enjoys reduced running times for reduced interpolation costs

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

grobner basis solutions of constrained interpolation problems this paper extends the previous work of the authors on recursive grobner basis techniques in coding theory pade approximation partial realization interpolation and modelling discrete time behaviours we present a general algorithm applicable to a wide range of constrained interpolation problems in coding theory and systems theory including list decoding and m pade approximation

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

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 robust and accurate approach to computing compressible multiphase flow stratified flow model and ausm up scheme in this paper we propose a new approach to compute compressible multifluid equations firstly a single pressure compressible multifluid model based on the stratified flow model is proposed the stratified flow model which defines different fluids in separated regions is shown to be amenable to the finite volume method we can apply the conservation law to each subregion and obtain a set of balance equations secondly the ausm scheme which is originally designed for the compressible gas flow is extended to solve compressible liquid flows by introducing additional dissipation terms into the numerical flux the new scheme called ausm up can be applied to both liquid and gas flows thirdly the contribution to the numerical flux due to interactions between different phases is taken into account and solved by the exact riemann solver we will show that the proposed approach yields an accurate and robust method for computing compressible multiphase flows involving discontinuities such as shock waves and fluid interfaces several one dimensional test problems are used to demonstrate the capability of our method including the ransom s water faucet problem and the air water shock tube problem finally several two dimensional problems will show the capability to capture enormous details and complicated wave patterns in flows having large disparities in the fluid density and velocities such as interactions between water shock wave and air bubble between air shock wave and water column s and underwater explosion

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

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

a fourier analytic approach to reed muller decoding we present a fourier analytic approach to list decoding reed muller codes over arbitrary finite fields we use this to show that quadratic forms over any field are locally list decodable up to their minimum distance the analogous statement for linear polynomials was proved in the celebrated works of goldreich previously tight bounds for quadratic polynomials were known only for q 2 and 3 the best bound known for other fields was the johnson radius departing from previous work on reed muller decoding which relies on some form of self corrector our work applies ideas from fourier analysis of boolean functions to low degree polynomials over finite fields in conjunction with results about the weight distribution we believe that the techniques used here could find other applications we present some applications to testing and learning

computations of two fluid models based on a simple and robust hybrid primitive variable riemann solver with ausmd this paper is to continue our previous work in 2008 on solving a two fluid model for compressible liquid gas flows we proposed a pressure velocity based diffusion term original derived from ausmd scheme of wada and liou in 1997 to enhance its robustness the proposed ausmd schemes have been applied to gas and liquid fluids universally to capture fluid discontinuities such as the fluid interfaces and shock waves accurately for the ransom s faucet problem air water shock tube problems and 2d shock water liquid interaction problems however the proposed scheme failed at computing liquid gas interfaces in problems under large ratios of pressure density and volume of fraction the numerical instability has been remedied by chang and liou in 2007 using the exact riemann solver to enhance the accuracy and stability of numerical flux across the liquid gas interface here instead of the exact riemann solver we propose a simple ausmd type primitive variable riemann solver pvrs which can successfully solve 1d stiffened water air shock tube and 2d shock gas interaction problems under large ratios of pressure density and volume of fraction without the expensive cost of tedious computer time in addition the proposed approach is shown to deliver a good resolution of the shock front rarefaction and cavitation inside the evolution of high speed droplet impact on the wall

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

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

short note revisit to the thinc scheme a simple algebraic vof algorithm this short note presents an improved multi dimensional algebraic vof method to capture moving interfaces the interface jump in the thinc tangent of hyperbola for interface capturing scheme is adaptively scaled to a proper thickness according to the interface orientation the numerical accuracy in computing multi dimensional moving interfaces is significantly improved without any geometrical reconstruction the proposed method is extremely simple and easy to use and its numerical accuracy is superior to other existing methods of its kind and comparable to the conventional plic piecewise linear interface calculation type vof schemes

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

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

a finite volume hllc based scheme for compressible interfacial flows with surface tension shock waves are often used in experiments to create a shear flow across liquid droplets to study secondary atomization similar behavior occurs inside of supersonic combustors scramjets under startup conditions but it is challenging to study these conditions experimentally in order to investigate this phenomenon further a numerical approach is developed to simulate compressible multiphase flows under the effects of surface tension forces the flow field is solved via the compressible multicomponent euler equations i e the five equation model discretized with the finite volume method on a uniform cartesian grid the solver utilizes a total variation diminishing tvd third order rungekutta method for time marching and second order tvd spatial reconstruction surface tension is incorporated using the continuum surface force csf model fluxes are upwinded with a modified hartenlaxvan leer contact hllc approximate riemann solver an interface compression scheme is employed to counter numerical diffusion of the interface the present work includes modifications to both the hllc solver and the interface compression scheme to account for capillary force terms and the associated pressure jump across the gasliquid interface a simple method for numerically computing the interface curvature is developed and an acoustic scaling of the surface tension coefficient is proposed for the non dimensionalization of the model the model captures the surface tension induced pressure jump exactly if the exact curvature is known and is further verified with an oscillating elliptical droplet and mach 1 47 and 3 shock droplet interaction problems the general characteristics of secondary atomization at a range of weber numbers are also captured in a series of simulations

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

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

stopping sets of hermitian codes combinatorial structures called stopping sets are useful in analyzing the performance of a linear code when coupled with an iterative decoding algorithm over an erasure channel in this paper we consider stopping sets of hermitian codes

how to solve compressible multifluid equations a simple robust and accurate method solving multifluid equations of compressible multiphase flows has proven to be extremely demanding because of some peculiar mathematical properties such as nonhyperbolicity nonconservative form and stiffness due to disparity in fluid properties and flow scales occurring typically in this paper we first consider the mathematical issues concerning nonhyperbolicity and nonconservative form their effects on the stability and convergence of numerical solutions are the theme of our presentation we shall present solutions for a range of problems selected to illuminate these numerical issues to this end we present a new numerical method that is simple to implement for a general class of fluids and yet is capable of robustly and accurately calculating phenomena involving material and shock discontinuities and interactions between them additionally the paper is completed with a new information for ensuring hyperbolicity under an interfacial pressure representation

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

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 simple interface sharpening technique with a hyperbolic tangent function applied to compressible two fluid modeling a simple interface sharpening technique based on hyperbolic tangent interpolation which was proposed in the previous study f xiao y honma k kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer methods fluids 48 2005 1023 1040 is applied to the compressible two fluid modeling the implementation of this scheme is very simple the interpolation of the volume fraction in the monotonicity upwind scheme for conservation law muscl solver is just replaced by the hyperbolic tangent interpolation while the muscl interpolations for other variables are maintained this technique is limited for the region near the interface to prevent the spurious oscillations of a minor phase the one dimensional and two dimensional problems are solved and the results are compared with those of the original muscl solver the results show that the interface is significantly sharpened with this technique and its sharpness is well controlled by one parameter in addition the robustness of the scheme does not change with sharpening the interface in the range we investigated an interface sharpening technique is proposed for compressible two fluid modeling this method simply uses the hyperbolic tangent interpolation for volume fraction interface thickness is well controlled by one parameter in the scheme

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

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

how to prevent pressure oscillations in multicomponent flow calculations a new numerical scheme for computing multicomponent flows is presented it is able to handle strong shocks is pressure oscillation free through the contact discontinuities and guarantees the posi tivity of mass fractions some numerical experiments on shock tubes indicate the convergence to the correct weak solution comparisons with classical finite volume schemes are also proposed

on the list decodability of rank metric codes let k n m in mathbb z be integers such that k leq n leq m let mathrm g n k in mathbb f q m n be a delsarte gabidulin code recently wachter zeh proved that codes belonging to this family cannot be efficiently list decoded for any radius tau providing tau is large enough this achievement essentially relies on proving a lower bound for the list size of some specific words in mathbb f q m n some years later raviv and wachter zeh improved this bound in a special case i e when n mid m as a consequence they were able to detect infinite families of delsarte gabidulin codes that cannot be efficiently list decoded at all in this article we determine similar lower bounds for maximum rank distance codes belonging to a wider class of examples containing generalized gabidulin codes generalized twisted gabidulin codes and examples recently described by trombetti and zhou by exploiting arguments such as those used by raviv and wachter zeh when n mid m we also show infinite families of generalized gabidulin codes that cannot be list decoded efficiently at any radius greater than or equal to left lfloor frac d 1 2 right rfloor 1 where d is its minimum distance nonetheless in all the examples belonging to above mentioned class we detect infinite families that cannot be list decoded efficiently at any radius greater than or equal to left lfloor frac d 1 2 right rfloor 2 where d is its minimum distance in particular this leads to show infinite families of gabidulin codes with underlying parameters not already covered by the result of raviv and wachter zeh having this decodability defect finally relying on the properties of a set of subspace trinomials recently presented by mcguire and mueller we are able to prove our main result that is any rank metric code of mathbb f q m n of order q kn with n dividing m such that 4n 3 is a square in mathbb z and containing mathrm g n 2 is not efficiently list decodable at some values of the radius tau

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

some gabidulin codes cannot be list decoded efficiently at any radius gabidulin codes can be seen as the rank metric equivalent of reed solomon codes it was recently proved using subspace polynomials that gabidulin codes cannot be list decoded beyond the so called johnson radius in another result cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials in this paper these subspace codes are used to prove two bounds on the list size in decoding certain gabidulin codes the first bound is an existential one showing that exponentially sized lists exist for codes with specific parameters the second bound presents exponentially sized lists explicitly for a different set of parameters both bounds rule out the possibility of efficiently list decoding several families of gabidulin codes for any radius beyond half the minimum distance such a result was known so far only for non linear rank metric codes and not for gabidulin codes using a standard operation called lifting identical results also follow for an important class of constant dimension subspace codes

the four element discrete nonlinear schrodinger equation non integrability and arnold diffusion we study the integrability of the discrete nonlinear schrodinger equation with four elements using the two element system as the underlying integrable subsystem we treat the coupling to the additional oscillators perturbationally a symmetry reduction method is used to map the coupled system of four elements on a three degree of freedom hamiltonian system in the reduced phase space we analyse the geometrical structure of the phase space of the resulting reduced system focusing on the homoclinic manifold by means of wiggins generalized melnikov method we prove non integrability through the existence of chaotic dynamics in demonstrating the presence of arnold diffusion for the coupled dynamics

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

hamiltonian hopf bifurcations in the discrete nonlinear schrodinger trimer oscillatory instabilities quasi periodic solutions and a new type of self trapping transition oscillatory instabilities in hamiltonian anharmonic lattices are known to appear through hamiltonian hopf bifurcations of certain time periodic solutions of multibreather type here we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear the three site discrete nonlinear schrodinger model with periodic boundary conditions the stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes we numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types close to the lower amplitude threshold stable two frequency quasi periodic solutions exist surrounding the unstable stationary solution and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small by contrast close to the higher amplitude threshold all two frequency quasi periodic solutions are detached from the unstable stationary solution and the resulting dynamics is of population inversion type involving also the originally unexcited site

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

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 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

an adaptive interface sharpening methodology for compressible multiphase flows abstract a numerical methodology is developed to combine the advantages of adaptive mesh refinement amr and interface sharpening technique a five equation compressible multiphase model with capillary and viscous effects is considered the solver employs a wave propagation method along with the tangent of hyperbola for interface capturing thinc scheme to calculate interface normal and curvature an implicit filtering method is introduced which transforms the sharpened volume fraction variable to a variant with smoothed distribution the accuracy and performance of our method is assessed through its application to multiple compressible interface problems ranging from high mach number shock interface interaction to gravity driven flows with viscosity and surface tension effects the results obtained for one dimensional shock tube and tin air interaction problems are shown to compare well with analytical data the flow patterns predicted for shock bubble interaction and under water explosion match those from the landmark experimental and numerical studies furthermore the trends and values predicted for spike position in the rayleigh taylor instability and bubble s center location in bubble rising are consistent with those found in literature particularly it is shown that the coupled amr thinc method remarkably prevents excessive interface smearing and captures delicate interfacial features such as shear induced instabilities encountered in shock bubble interaction

the discrete self trapping equation and the painleve property from the discrete self trapping dst equation for three degrees of freedom rewritten in terms of density matrix one can derive an autonomous system of real first order ordinary differential equations by using su 3 notation performing a painleve analysis it is found that depending on the parameters of the system integrable and non integrable cases can be distinguished

robust pcps of proximity shorter pcps and applications to coding we continue the study of the trade off between the length of probabilistically checkable proofs pcps and their query complexity establishing the following main results which refer to proofs of satisfiability of circuits of size n 1 we present pcps of length exp o log log n 2 cdot n that can be verified by making o log log n boolean queries 2 for every epsilon 0 we present pcps of length exp log epsilon n cdot n that can be verified by making a constant number of boolean queries in both cases false assertions are rejected with constant probability which may be set to be arbitrarily close to 1 the multiplicative overhead on the length of the proof introduced by transforming a proof into a probabilistically checkable one is just quasi polylogarithmic in the first case of query complexity o log log n and is 2 log n epsilon for any epsilon 0 in the second case of constant query complexity our techniques include the introduction of a new variant of pcps that we call robust pcps of proximity these new pcps facilitate proof composition which is a central ingredient in the construction of pcp systems a related notion and its composition properties were discovered independently by dinur and reingold our main technical contribution is a construction of a length efficient robust pcp of proximity while the new construction uses many of the standard techniques used in pcp constructions it does differ from previous constructions in fundamental ways and in particular does not use the parallelization step of arora et al j acm 45 1998 pp 501 555 the alternative approach may be of independent interest we also obtain analogous quantitative results for locally testable codes in addition we introduce a relaxed notion of locally decodable codes and present such codes mapping k information bits to codewords of length k 1 epsilon for any epsilon 0

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

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

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

subspace polynomials and cyclic subspace codes subspace codes have received an increasing interest recently due to their application in error correction for random network coding in particular cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms in this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits we further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials we present some constructions of such codes which are cyclic and analyze their parameters