efficient root finding algorithm with application to list decoding of algebraic geometric codes a list decoding for an error correcting code is a decoding algorithm that generates a list of codewords within a hamming distance t from the received vector where t can be greater than the error correction bound in previous work by m shokrollahi and h wasserman see ibid vol 45 p 432 7 march 1999 a list decoding procedure for reed solomon codes was generalized to algebraic geometric codes recent work by v guruswami and m sudan see ibid vol 45 p 1757 67 sept 1999 gives improved list decodings for reed solomon codes and algebraic geometric codes that work for all rates and have many applications however these list decoding algorithms are rather complicated r roth and g ruckenstein see ibid vol 46 p 246 57 jan 2000 proposed an efficient implementation of the list decoding of reed solomon codes in this correspondence extending roth and ruckenstein s fast algorithm for finding roots of univariate polynomials over polynomial rings i e the reconstruct algorithm we present an efficient algorithm for finding the roots of univariate polynomials over function fields based on the extended algorithm we give an efficient list decoding algorithm for algebraic geometric codes

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

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

global well posedness for a boussinesq navier stokes system with critical dissipation abstract in this paper we study a fractional diffusion boussinesq model which couples a navier stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature we establish global well posedness results with rough initial data

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

incremental numerical recipes for the high efficient inversion of the confluent vandermonde matrices in the article we propose an incremental algorithm for calculating the inversion of the confluent vandermonde matrix and triangular factorization of this inversion we implemented all the incremental operations i e adding deleting and changing the single matrix parameter to avoid repeating the same calculations again and again besides contrary to other works in this field the article derives an explicit analytic formula for the calculation of the triangular factorization of the confluent vandermonde matrix inversion additionally we propose a solution to these two problems with the use of a system of linear recursive equations the results of this article do not require any symbolic calculations therefore they can be performed by a numerical algorithm implemented in any general purpose programming language

exponential error bounds for algebraic soft decision decoding of reed solomon codes algebraic soft decision decoding of reed solomon codes is a promising technique for exploiting reliability information in the decoding process while the algorithmic aspects of the decoding algorithm are reasonably well understood and in particular complexity is polynomially bounded in the length of the code the performance analysis has relied almost entirely on simulation results analytical exponential error bounds that can be used to tightly bound the performance of reed solomon codes under algebraic soft decision decoding are presented in this paper the analysis is used in a number of examples and several extensions and consequences of the results are presented

list decoding of algebraic geometric codes we generalize sudan s see j compl vol 13 p 180 93 1997 results for reed solomon codes to the class of algebraic geometric codes designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional error correction bound d 1 2 d being the minimum distance of the code our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields for the latter problem we design a polynomial time algorithm and show that the resulting overall list decoding algorithm runs in polynomial time under some mild conditions several examples are included

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

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

inversion of confluent vandermonde matrices abstract this paper is concerned with the inversion of confluent vandermonde matrices a novel and simple recursive algorithm for inverting confluent vandermonde matrices is presented the algorithm is suitable for classroom use in both numerical as well as symbolic computation examples are included to illustrate the proposed algorithm

linear diophantine equations over polynomials and soft decoding of reed solomon codes this paper generalizes the classical knuth schoumlnhage algorithm computing the greatest common divisor gcd of two polynomials for solving arbitrary linear diophantine systems over polynomials in time quasi linear in the maximal degree as an application the following weighted curve fitting problem is considered given a set of points in the plane find an algebraic curve satisfying certain degree conditions that goes through each point the prescribed number of times the main motivation for this problem comes from coding theory namely it is ultimately related to the list decoding of reed solomon codes this paper presents a new fast algorithm for the weighted curve fitting problem based on the explicit construction of a groebner basis this gives another fast algorithm for the soft decoding of reed solomon codes different from the procedure proposed by feng which works in time w r o 1 nlog2n where r is the rate of the code and w is the maximal weight assigned to a vertical line

on the bounded sum of digits discrete logarithm problem in finite fields in this paper we study the bounded sum of digits discrete logarithm problem in finite fields our results are concerned primarily with fields fqn where n q 1 the fields are called kummer extensions of fq it is known that we can efficiently construct an element g with order exponential in n let s q bullet be the function from integers to the sum of digits in their q ary expansions we first present an algorithm that given ge 0 leq e qn finds e in random polynomial time provided that sq e n we then show that the problem is solvable in random polynomial time for most of the exponent e with sq e 1 32 n by exploring an interesting connection between the discrete logarithm problem and the problem of list decoding of reed solomon codes and applying the guruswami sudan algorithm as far as we are aware our algorithm is the first one which can solve discrete logarithms of 2 log 1 epsilon q n many instances in polynomial time for infinite many constant characteristic fields fqn furthermore since every finite field has an extension of reasonable degree which is a kummer extension our result reveals an unexpected property of the discrete logarithm problem namely the bounded sum of digits discrete logarithm problem in any given finite field becomes polynomial time solvable in certain low degree extensions as a side result we obtain a sharper lower bound on the number of congruent polynomials generated by linear factors than the one based on the stothers mason abc theorem we also prove that in the field fqq 1 the bounded sum of digits discrete logarithm with respect to g can be computed in random time o f w log4 qq 1 where f is a subexponential function and w is the bound on the q ary sum of digits of the exponent hence the problem is fixed parameter tractable these results are shown to be generalized to artin schreier extension fpp where p is a prime

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

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

reed solomon list decoding from a system theoretic perspective in this paper the sudan guruswami approach to list decoding of reed solomon rs codes is cast in a system theoretic framework with the data a set of trajectories or time series is associated which is then modeled as a so called behavior in this way a connection is made with the behavioral approach to system theory it is shown how a polynomial representation of the modeling behavior gives rise to the bivariate interpolating polynomials of the sudan guruswami approach the concept of weighted row reduced is introduced and used to achieve minimality two decoding methods are derived and a parametrization of all bivariate interpolating polynomials is given

on the global well posedness of 2 d inhomogeneous incompressible navier stokes system with variable viscous coefficient abstract given solenoidal vector u 0 h 2 h 1 r 2 0 1 l 2 r 2 and 0 l w 1 r r 2 with a positive lower bound for 0 1 2 and 2 r 2 1 2 we prove that 2 d incompressible inhomogeneous navier stokes system 1 1 has a unique global solution provided that the viscous coefficient 0 is close enough to 1 in the l norm compared to the size of and the norms of the initial data with smoother initial data we can prove the propagation of regularities for such solutions furthermore for 1 p 4 if 0 1 u 0 belongs to the critical besov spaces b p 1 2 p r 2 b p 1 1 2 p l 2 r 2 and the b p 1 2 p r 2 norm of 0 1 is sufficiently small compared to the exponential of u 0 l 2 2 u 0 b p 1 1 2 p we prove the global well posedness of 1 1 in the scaling invariant spaces finally for initial data in the almost critical besov spaces we prove the global well posedness of 1 1 under the assumption that the l norm of 0 1 is sufficiently small

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

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

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

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

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

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

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

global well posedness and asymptotic behavior for the 2d boussinesq system with variable viscosity abstract in this paper we investigate the global existence and uniqueness of strong solutions to 2d boussinesq system with variable kinematic viscosity depending on the temperature comparing with the previous results given by abidi and zhang 4 who considered the critical case 1 we weaken the dissipation effect in the temperature equation to the supercritical case 0 1 furthermore we obtain the algebraic decay estimate for u l 2

an average weight distance enumerator for binary expansions of reed solomon codes an average hamming weight enumerator is derived for the codewords at each hamming distance from a received pattern in the set of all possible binary expansions of a reed solomon code since these codes may be decoded by list decoders such as those studied by sudan 1997 the enumerator can be used to estimate the average number of codewords in the list returned by such a decoder

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

on the global well posedness of 2 d boussinesq system with variable viscosity abstract in this paper we investigate the global well posedness of 2 d boussinesq system which has variable kinematic viscosity and with thermal conductivity of d with general initial data provided that the viscosity coefficient is sufficiently close to some positive constant in l norm

global regularity for the two dimensional anisotropic boussinesq equations with vertical dissipation this paper establishes the global in time existence of classical solutions to the two dimensional anisotropic boussinesq equations with vertical dissipation when only vertical dissipation is present there is no direct control on the horizontal derivatives and the global regularity problem is very challenging to solve this problem we bound the derivatives in terms of the l infty norm of the vertical velocity v and prove that v l r with 2 leqq r infty does not grow faster than sqrt r log r at any time as r increases a delicate interpolation inequality connecting v l infty and v l r then yields the desired global regularity

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

a pointwise inequality for fractional laplacians abstract the fractional laplacian is an operator appearing in several evolution models where diffusion coming from a levy process is present but also in the analysis of fluid interphases we provide an extension of a pointwise inequality that plays a role in their study we begin recalling two scenarios where it has been used after stating the results for fractional laplace beltrami and dirichlet neumann operators we provide a sketch of their proofs unravelling the underlying principle to such inequalities

the importance of the unsteady kutta condition when modelling gust aerofoil interaction abstract the kutta condition is applied to aerofoils with sharp trailing edges to allow for viscous effects to be considered within a simplified system of equations that are inviscid this paper discusses in detail the inclusion of an unsteady kutta condition at a sharp trailing edge during gust aerofoil interaction and illustrates how the analytic solution for the far field noise generated by this interaction changes if the unsteady kutta condition is neglected or more precisely if the unsteady pressure is permitted to be singular at the trailing edge the analytic solution both with and without the unsteady kutta condition is compared with numerical results that have no imposed unsteady kutta condition importantly the results agree well only when the unsteady kutta condition is neglected in the analytic solution this paper highlights where the far field acoustics are most affected by neglecting the unsteady kutta condition for a variety of singularities that can occur in the unsteady pressure at the trailing edge and shows that results permitting different behaviour in the unsteady surface pressure at the trailing edge could give significantly different far field noise predictions even though the surface pressure elsewhere along the aerofoil surface agrees with benchmark solutions

maximum likelihood decoding of reed solomon codes is np hard maximum likelihood decoding is one of the central algorithmic problems in coding theory it has been known for over 25 years that maximum likelihood decoding of general linear codes is np hard nevertheless it was so far unknown whether maximum likelihood decoding remains hard for any specific family of codes with nontrivial algebraic structure in this paper we prove that maximum likelihood decoding is np hard for the family of reed solomon codes we moreover show that maximum likelihood decoding of reed solomon codes remains hard even with unlimited preprocessing thereby strengthening a result of bruck and naor

algebraic soft decision decoding of hermitian codes an algebraic soft decision decoder for hermitian codes is presented we apply koetter and vardy s soft decision decoding framework now well established for reed solmon codes to hermitian codes first we provide an algebraic foundation for soft decision decoding then we present an interpolation algorithm to find the q polynomial that plays a key role in the decoding with some simulation results we compare performances of the algebraic soft decision decoders for hermitian codes and reed solmon codes favorable to the former

iterative soft decision decoding of hermitian codes this paper proposes an iterative soft decision decoding algorithm for one of the most popular algebraic geometric ag codes hermitian codes the algorithm is designed by integrating the two most powerful soft decision decoding algorithms the adaptive belief propagation abp algorithm and the koetter vardy kv list decoding algorithm the abp algorithm performs iterative decoding based on an adapted parity check matrix of a hermitian code to enhance the reliability of the soft received information with the enhanced reliability the kv algorithm performs soft decision list decoding to obtain the original message since the matrix adaptation relies on bit reliabilities regrouping of the unreliable bits is introduced to assist the abp decoding a complexity reducing abp kv decoding approach is proposed based on assessing the soft information provided by the abp algorithm and determining whether the following kv decoding steps should be carried out geometric interpretation of the abp algorithm is presented demonstrating the necessity of performing matrix adaptation our performance analysis shows the proposed iterative decoding algorithm outperforms both the existing decoding approaches for hermitian codes and the abp kv decoding of reed solomon rs codes

on the geometry of hermitian one point codes abstract here we describe the algebraic geometry of one point codes from the hermitian curve in particular we employ zero dimensional schemes in the plane to characterize the minimum weight codewords of their dual codes providing explicit formulas for their number we discuss also some natural improvements of the duals of hermitian one point codes by means of geometric arguments finally some cohomological tools are developed to characterize the small weight codewords of such codes

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

lower bounds for an integral involving fractional laplacians and the generalized navier stokes equations in besov spaces when estimating solutions of dissipative partial differential equations in l p related spaces we often need lower bounds for an integral involving the dissipative term if the dissipative term is given by the usual laplacian lower bounds can be derived through integration by parts and embedding inequalities however when the laplacian is replaced by the fractional laplacian the approach of integration by parts no longer applies in this paper we obtain lower bounds for the integral involving by combining pointwise inequalities for with bernstein s inequalities for fractional derivatives as an application of these lower bounds we establish the existence and uniqueness of solutions to the generalized navier stokes equations in besov spaces the generalized navier stokes equations are the equations resulting from replacing in the navier stokes equations by

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

code automorphisms and permutation decoding of certain reed solomon binary images we consider primitive reed solomon rs codes over the field f2m of length n 2m 1 building on lacan s results for the case of binary extension fields we show that the binary images of certain two parity symbol rs n n 2 3 code have a code automorphism subgroup related to the general linear group gl m 2 for these codes we obtain a code automorphism subgroup of order m gl m 2 an explicit algorithm is given to compute a code automorphism if it exists that sends a particular choice of m binary positions into binary positions that correspond to a single symbol of the rs code if one such automorphism exists for a particular choice of m binary symbol positions we show that there are at least m of them computationally efficient permutation decoders are designed for the two parity symbol rs n n 2 3 codes simulation results are shown for the additive white gaussian noise awgn channel for the finite fields f23 and f24 we go on to derive subgroups of code automorphisms belonging to binary images of certain rs codes that have three parity symbols a table of code automorphism subgroup orders computed using the groups algorithms and programming gap software is tabulated for the fields f23 f24 and f25

global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity abstract we prove the global well posedness for the 2 d boussinesq system with the temperature dependent viscosity and thermal diffusivity

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

minimum rank skew hermitian solutions to the matrix approximation problem in the spectral norm in this paper we discuss the following two minimum rank matrix approximation problems in the spectral norm i for given a a h c m i m b c m i n determining x s 1 such that rank x min y s 1 rank y s 1 y y h c n i n a b y b h 2 min ii for given a a h c m i m b c m i n determining x s 2 such that rank x min y s 2 rank y s 2 y y h c n i n a b y b h 2 min by applying the norm preserving dilation theorem the hermitian type skew hermitian type generalized singular value decomposition hgsvd shgsvd we characterize the expressions of the minimum rank and derive a general form of minimum rank skew hermitian solutions to the matrix approximation problem

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

explicit kutta condition for unsteady two dimensional high order potential boundary element method an explicit unsteady pressure kutta condition is discribed that was directly and efficiently implemented in a time domain high order potential panel method so as to ensure the pressure equality on the upper and lower surfaces at the trailing edge of the airfoil at each time step

global well posedness for 2 d boussinesq system with the temperature dependent viscosity and supercritical dissipation abstract the present paper is dedicated to the global well posedness issue for the boussinesq system with the temperature dependent viscosity in r 2 motivated by the work abidi and zhang adv math 305 2017 1202 1249 on the critical dissipation d with 1 in the temperature equation of the system we consider the extension of their work to the supercritical dissipation for 2 3 1

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

a general preconditioner for linear complementarity problem with an m matrix abstract in this paper we first present a general preconditioner p for solving linear complementarity problem lcp associated with an m matrix a and a vector f and prove that the lcp a f is equivalent to the lcp p a p f then based on this general preconditioner p two preconditioned ssor methods for solving the linear complementarity problems are proposed we show that this general preconditioner p accelerates the convergence of two ssor methods under the assumption that p a is a z matrix in addition we also give a practically concrete choice for the preconditioner p satisfying aforementioned assumption numerical examples are used to illustrate the theoretical results obtained

recursive numerical recipes for the high efficient inversion of the confluent vandermonde matrices in the article we propose a recursive algorithm for calculating the inversion of the confluent vandermonde matrix besides contrary to other works in this field the article derives an explicit analytic formula for the calculation of the inverse of the confluent vandermonde matrix the results of this article do not require any symbolic calculations and therefore can be performed by a numerical algorithm implemented in any general purpose programming language moreover a ready to use c full implementation of the recursive algorithm is attached

global regularity for the 2d boussinesq equations with partial viscosity terms abstract in this paper we prove the global in time regularity for the 2d boussinesq system with either the zero diffusivity or the zero viscosity we also prove that as diffusivity viscosity tends to zero the solutions of the fully viscous equations converge strongly to those of zero diffusion viscosity equations our result for the zero diffusion system in particular solves the problem no 3 posed by moffatt in r l ricca ed kluwer academic publishers dordrecht the netherlands 2001 pp 3 10

numerical recipes for the high efficient inverse of the confluent vandermonde matrices abstract in this paper we derive a numerical recipe for the calculation of the inversion of the confluent vandermonde matrix the main result of this article does not require any symbolic calculations and therefore can be performed by a numerical algorithm implemented either in any high level like matlab or mathematica or low level programming language c c java pascal fortran etc the computational complexity of the presented algorithm is of an o n 2 class being by the linear term better than the ordinary gauss elimination method moreover a ready to use c full implementation of the algorithm is attached

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

lattice based threshold changeability for standard shamir secret sharing schemes we consider the problem of increasing the threshold parameter of a secret sharing scheme after the setup share distribution phase without further communication between the dealer and the shareholders previous solutions to this problem require one to start off with a nonstandard scheme designed specifically for this purpose or to have communication between shareholders in contrast we show how to increase the threshold parameter of the standard shamir secret sharing scheme without communication between the shareholders our technique can thus be applied to existing shamir schemes even if they were set up without consideration to future threshold increases our method is a new positive cryptographic application for lattice reduction algorithms inspired by recent work on lattice based list decoding of reed solomon codes with noise bounded in the lee norm we use fundamental results from the theory of lattices geometry of numbers to prove quantitative statements about the information theoretic security of our construction these lattice based security proof techniques may be of independent interest

regularity results for two dimensional flows of multiphase viscous fluids global regularity results for weak solutions of the navier stokes equations for two dimensional multiphase incompressible fluids are proved under suitable conditions on the viscosity without assuming positive lower bounds on the initial density as an application we deduce regularity properties for the integral curves of the corresponding velocity field finally we prove regularity results in the small for strong solutions

on the global well posedness for boussinesq system abstract in this paper we give a global well posedness result for the two dimensional boussinesq system with partial viscosity when the initial data v 0 b 1 1 r 2 l 2 r 2 and 0 b 2 1 0 r 2

effect of airfoil trailing edge thickness on the numerical solution of panel methods based on the dirichlet boundary condition the practical limit of airfoil thickness ratio for which acceptable engineering results are obtainable with the dirichlet boundary condition based numerical methods is investigated this is done by studying the effect of thickness on the calculated pressure distribution near the trailing edge and by comparing the aerodynamic coefficients with available exact solutions the first objective of this study owing to the wide use of such computational methods is to demonstrate the numerical symptoms that occur when the body or wing thickness approaches zero and to increase the awareness of potential users of these methods additionally an effort is made to obtain the practical limits of the trailing edge thickness where such problems will appear in the flow solution and to propose some possible cures for very thin airfoils or those with cusped trailing edges