decoding binary 2 d cyclic codes by the 2 d berlekamp massey algorithm a method of decoding two dimensional 2 d cyclic codes by applying the 2 d berlekamp massey algorithm is proposed to explain this decoding method the author introduces a subclass of 2 d cyclic codes which are called 2 d bch codes due to their similarity with bch codes it is shown that there are some short 2 d cyclic codes with a better cost parameter value the merit of the approach is verified by showing several simple examples of 2 d cyclic codes

a generalized euclidean algorithm for multisequence shift register synthesis the problem of finding a linear feedback shift register of shortest length capable of generating prescribed multiple sequences is considered a generalized euclidean algorithm which is based on a generalized polynomial division algorithm is presented a necessary and sufficient condition for the uniqueness of the solution is given when the solution is not unique the set of all possible solutions is also derived it is shown that the algorithm can be applied to the decoding of many cyclic codes for which multiple syndrome sequences are available when it is applied to the case of a single sequence the algorithm reduces to that introduced by y sugiyama et al inf control vol 27 p 87 9 feb 1975 in the decoding of bch codes

a generalized forney formula for algebraic geometric codes this correspondence contains a straightforward generalization of decoding of bch codes to the decoding of algebraic geometric codes couched in terms of varieties ideals and grobner bases this consists of 1 a berlekamp massey type lattice shifting row reduction algorithm with majority voting similar to algorithms in the current literature 2 a realization that it produces a minimal grobner basis b for the error locator ideal i v relative to a particular weighted total degree monomial ordering 3 a factoring of that basis into several minimal plex bases that facilitates finding the variety v of error positions and 4 a direct generalization of forney s formula to calculate error magnitudes using functions spl sigma p which are by products of this factoring

δ sequences and evaluation codes defined by plane valuations at infinity we introduce the concept of sequence a sequence generates a well ordered semigroup s in z or r we show how to construct and compute parameters for the dual code of any evaluation code associated with a weight function defined by from the polynomial ring in two indeterminates to a semigroup s as above we prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces called plane valuations at infinity we also give algorithms to construct an unlimited number of sequences of the different existing types and so this paper provides the tools to know and use a new large set of 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

efficient forney functions for decoding ag codes using a forney formula to solve for the error magnitudes in decoding algebraic geometric ag codes requires producing functions spl sigma sub p which are 0 at all but one point p of the variety of the error locator ideal the best such function is produced here in a reasonably efficient way from a lex grobner basis this lex basis is in turn produced efficiently from a weighted grevlex basis by using the fglm algorithm these two steps essentially complete the efficient decoding scheme based on a forney formula started in the author s previous work see ibid vol 42 p 1263 8 1996

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

on determining the independent point set for doubly periodic arrays and encoding two dimensional cyclic codes and their duals for the purpose of encoding two dimensional cyclic tdc codes an effective algorithm for finding the independent point set of an arbitrary module of doubly periodic dp arrays is proposed in addition a method for determining the characteristic ideal of a given set of dp arrays is exhibited with the aid of these methods it is possible to specify the structure of the generator and check ideals of tdc codes by applying this algorithm to nonsemisimple binary tdc codes with small areas several optimal linear codes have been found and their previously unknown tdc structures have been exhibited

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

rational approximation of 2 d linear discrete systems in this paper we present an efficient procedure to obtain a rational model for a 2 d linear shift invariant discrete system using first and second order data from it this procedure is a modification of the nonlinear least squares approximation and it generalizes the pade approximants and the spectral estimation modeling procedures the parameters of the approximating filter are obtained by solving a system of linear equations by means of an efficient recursive algorithm which is developed using the relation of the approximation problem with the theory of orthogonal polynomials on the unit bidisk we discuss some of the algebraic properties of the solution and apply them to define cases for which the bibo stability of the approximating filters is ensured the proposed procedure finds applications in the design and stabilization of 2 d recursive digital filters and in the autoregressive moving average arma modeling of stationary random fields

efficient inversion of toeplitz block toeplitz matrix an iterative algorithm for the inversion of a toeplitz block toeplitz matrix consisting of m m blocks of size p p is described the algorithm presented exploits the structure of the toeplitz block toeplitz matrix and outperforms akaike s algorithm by a factor of max 2 p m 2 the use of this algorithm for an iterative solution of a toeplitz block toeplitz set of linear equations is also presented

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

sub quadratic decoding of one point hermitian codes we present the first two sub quadratic complexity decoding algorithms for one point hermitian codes the first is based on a fast realization of the guruswami sudan algorithm using state of the art algorithms from computer algebra for polynomial ring matrix minimization the second is a power decoding algorithm an extension of classical key equation decoding which gives a probabilistic decoding algorithm up to the sudan radius we show how the resulting key equations can be solved by the matrix minimization algorithms from computer algebra yielding similar asymptotic complexities

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

achieving the designed error capacity in decoding algebraic geometric codes a decoding algorithm for codes arising from algebraic curves explicitly constructable by goppa s construction is presented any configuration up to the greatest integer less than or equal to d 1 2 errors is corrected by the algorithm whenever d or 6g where d is the designed minimum distance of the code and g is the genus of the curve the algorithm s complexity is at most o d sup 2 n where n denotes the length of the code application to hermitian codes and connections with well known algorithms are explained

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

permutation decoding of abelian codes a permutation decoding procedure for abelian codes is introduced by using the groebner bases theory this method is valid for decoding all the binary abelian codes some examples are given to show how powerful this method can be

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

fast decoding of algebraic geometric codes up to the designed minimum distance we present a decoding algorithm for algebraic geometric codes from regular plane curves in particular the hermitian curve which corrects all error patterns of weight less than d 2 with low complexity the algorithm is based on the majority scheme of feng and rao 1993 and uses a modified version of sakata s 1988 generalization of the berlekamp massey algorithm

fast decoding of multipoint codes from algebraic curves multipoint codes are a broad class of algebraic geometry codes derived from algebraic functions which have multiple poles and or zeros on an algebraic curve thus they are more general than one point codes which are an important class of algebraic geometry codes in the sense that they can be decoded efficiently using the berlekamp massey sakata algorithm we present a fast method for decoding multipoint codes from a plane curve particularly a hermitian curve our method with some adaptation can be applied to decode multipoint codes from a general algebraic curve embedded in the n dimensional affine space fqn over a finite field fq so that those algebraic geometry codes can be decoded efficiently if the dimension n of the affine space including the defining curve is small

uniform l1 error bounds for the semidiscrete solution of a volterra equation with completely monotonic convolution kernel abstract we study the second order continuous time galerkin approximation of the solution ut 0t t s au s ds 0 u 0 v t 0 where a is an elliptic partial differential operator and t is completely monotonic and locally integrable but not constant error estimates are derived in the norm of lt1 0 lx2

subquadratic time encodable codes beating the gilbert varshamov bound we construct explicit algebraic geometry codes built from the garcia stichtenoth function field tower beating the gilbert varshamov bound for alphabet sizes at least 192 messages are identified with functions in certain riemann roch spaces associated with divisors supported on multiple places encoding amounts to evaluating these functions at degree one places by exploiting algebraic structures particular to the garcia stichtenoth tower we devise an intricate deterministic omega 2 runtime exponent encoding and 1 omega 2 expected runtime exponent randomized unique and list decoding algorithms here omega is the matrix multiplication exponent if omega 2 as widely believed the encoding and decoding runtimes are respectively nearly linear and nearly quadratic prior to this work encoding time of code families beating the gilbert varshamov bound were quadratic or worse

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

minimum distance decoding of general algebraic geometry codes via lists algebraic geometry codes are defined by divisors d and g on a curve over a finite field f often g is supported by a single f rational point and the resulting code is called a one point code recently there has been interest in allowing the divisor g to be more general as this can result in superior codes in particular one may obtain a code with better parameters by allowing g to be supported by m distinct f rational points where m 1 in this paper we demonstrate that a multipoint algebraic geometry code c may be embedded in a one point code c exploiting this fact we obtain a minimum distance decoding algorithm for the multipoint code c this is accomplished via list decoding in the one point code c

partial permutation decoding for abelian codes in our earlier work we introduced a technique to construct an information set for every semisimple abelian code over an arbitrary field solely in terms of its defining set in this paper we apply the geometrical properties of those information sets to obtain sufficient conditions for a error correcting abelian code to have a b pd set for every b t these conditions are simply given in terms of the structure of the defining set of the code

error locator ideals for algebraic geometric codes the error locations for an algebraic geometric code c d mp are exactly the common zeros that is a projective variety v i of a set ideal i of error locator functions the paper gives a one dimensional berlekamp massey version of the feng rao 1993 algorithm for decoding algebraic geometric codes c d mp this produces a generating set for i as an ideal of size at most spl rho the smallest positive pole order at p of any function in l mp relative to any error of weight at most e

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

impulse response arrays of discrete space systems over a finite field the main thrust of this paper is directed towards an analysis of the structure and properties of the impulse response array of two dimensional 2 d discrete space systems characterizable by rational functions having coefficients in a field z q it is shown that such an array exhibits a row column type of periodicity expressions for the period are derived arrays with a maximum of three levels in the auto correlation function are identified and explicit expressions for these levels are given

lattice parameter autoregressive modeling of two dimensional fields part i the quarter plane case quarter plane autoregressive lattice parameter modeling of two dimensional fields is developed as a natural extension of one dimensional lattice parameter theory starting with a given data field four prediction error fields are generated their linear combination is used to define and calculate 2 d lattice parameter reflection factors for successive lattice parameter model stages in addition to developing the basic theory the presentation includes the relationships between the quarter plane two dimensional transfer function and the lattice parameters the lattice parameter synthesis model for generating a two dimensional field from a random field input and lattice parameter stability conditions the theory has been confirmed by computer simulations the extension of this lattice parameter theory for the asymmetric half plane case is to be presented in a subsequent paper

symmetric pseudo random matrices we consider the problem of generating symmetric pseudo random sign 1 matrices based on the similarity of their spectra to wigner s semicircular law using binary m sequences golomb sequences of lengths n 2 m 1 we give a simple explicit construction of circulant n times n sign matrices and show that their spectra converge to the semicircular law when n grows the kolmogorov complexity of the proposed matrices equals to that of golomb sequences and is at most 2 mathrm log 2 n bits

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

a systolic array implementation of the feng rao algorithm an efficient implementation of a parallel version of the feng rao algorithm on a one dimensional systolic array is presented in this paper by adopting an extended syndrome matrix syndromes of the same order lying on a slant diagonal in the extended syndrome matrix are scheduled to be examined by a series of cells simultaneously and therefore a high degree of concurrency of the feng rao algorithm can be achieved the time complexity of the proposed architecture is m g 1 by using a series of t g 1 2 1 nonhomogeneous but regular effective processors called pe cells and g trivial processors called d cells where t is designed as the half of the feng rao bound each d cell contains only delay units while each pe cell contains one finite field inverter and except the first one one or more finite field multipliers cell functions of each pe cell are basically the same and the overall control circuit of the proposed array is quite simple the proposed architecture requires in total t g 1 1 finite field inverters and t g 1 2 t g 1 2 1 2 finite field multipliers for a practical design this hardware complexity is acceptable

fast decoding of codes from algebraic plane curves improvement to an earlier decoding algorithm for codes from algebraic geometry is presented for codes from an arbitrary regular plane curve the authors correct up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance of the code and m is the degree of the curve the complexity of finding the error locator is o n sup 7 3 where n is the length of the code for codes from hermitian curves the complexity of finding the error values given the error locator is o n sup 2 and the same complexity can be obtained in the general case if only d 2 m sup 2 2 errors are corrected

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

stability of the difference type methods for linear volterra equations in hilbert spaces we study the stability of backward difference timestepping method for linear volterra equations of scalar type in a hilbert space framework the results and methods extend and simulate numerically those introduced by pruss for integrability with respect to continuous solutions

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

the numerical analysis on a volterra equation with asymptotically periodic solution we consider order one operational quadrature methods on a certain integro differential equation of volterra type on 0 with piecewise linear convolution kernels the forms of discretization solution are patterned after a continuous one of hannsgen 1979 2 an l 1 remainder stability and an error bound are derived

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

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

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

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

on the number of correctable errors for some ag codes an algorithm that for codes from a regular plane curve corrects up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance and m is the degree of the curve was presented in an earlier work see ibid vol 35 p 811 21 1989 it is now shown that this bound is the best possible for the algorithm considered

on the minimal partial realization of 2d discrete linear shift invariant systems the partial realization problem for 2d discrete linear shift invariant systems is discussed the authors attempt to establish a reasonable definition of minimality and propose a method for finding the degree and the system function of a minimal realization of a 2d linear system characterized by a given 2d impulse response array the method used is based on the 2d berlekamp massey algorithm which has a close connection with the 2d hankel matrix derived from the 2d array while it compares in efficiency with other methods for identification and approximation of 2d linear systems it gives a novel approach to the problem in the sense that one does not need any knowledge or assumptions concerning the system degree

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

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

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

determination of error values for algebraic geometry codes and the forney formula in the decoding of one point algebraic geometry codes with one defining equation in two variables we generalize the forney formula for the bose chaudhuri hocqungham bch codes

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

the inverses of block hankel and block toeplitz matrices a set of new formulae for the inverse of a block hankel or block toeplitz matrix is given the formulae are expressed in terms of certain matrix pade forms which approximate a matrix power series associated with the block hankel matrix by using frobenius type identities between certain matrix pade forms the inversion formulae are shown to generalize the formulae of gohberg heinig and in the scalar case the formulae of gohberg semencul and gohberg krupnik the new formulae have the significant advantage of requiring only that the block hankel matrix itself be nonsingular the other formulae require in addition that certain submatrices be nonsingular since effective algorithms for computing the required matrix pade forms are available the formulae are practical indeed some of the algorithms allow for the efficient calculation of the inverse not only of the given block hankel matrix but also of any nonsingular block principal minor

strict optimal rational approximants of multisequences the synthesis problem of multisequences of elements of a field f is studied a new way of approximation by rational functions to multisequences is introduced different from the usual approach by optimal rational approximant profile in the proposed strict optimal rational approximant of a multisequence the components of the optimal rational approximant are not required to have the same denominator an algorithm to obtain a strict optimal rational approximant profile is established the relation between optimal rational approximant and strict optimal rational approximant is studied

nonlinear operations on a class of modulation spaces abstract we discuss when the nonlinear operation f f f maps the modulation space m s p q r n 1 p q to the same space again it is known that m s p q r n is a multiplication algebra when s n n q hence it is true for this space if f is entire we claim that it is still true for non analytic f when q 4 3

the art of signaling fifty years of coding theory in 1948 shannon developed fundamental limits on the efficiency of communication over noisy channels the coding theorem asserts that there are block codes with code rates arbitrarily close to channel capacity and probabilities of error arbitrarily close to zero fifty years later codes for the gaussian channel have been discovered that come close to these fundamental limits there is now a substantial algebraic theory of error correcting codes with as many connections to mathematics as to engineering practice and the last 20 years have seen the construction of algebraic geometry codes that can be encoded and decoded in polynomial time and that beat the gilbert varshamov bound given the size of coding theory as a subject this review is of necessity a personal perspective and the focus is reliable communication and not source coding or cryptography the emphasis is on connecting coding theories for hamming and euclidean space and on future challenges specifically in data networking wireless communication and quantum information theory

convolution quadrature and discretized operational calculus ii operational quadrature rules are applied to problems in numerical integration and the numerical solution of integral equations singular integrals power and logarithmic singularities finite part integrals multiple timescale convolution volterra integral equations wiener hopf integral equations frequency domain conditions which determine the stability of such equations can be carried over to the discretization

soft decision list decoding of hermitian codes this paper proposes the first complete soft decision list decoding algorithm for hermitian codes based on the koetter vardy s reed solomon code decoding algorithm for hermitian codes interpolation processes trivariate polynomials which are defined over the pole basis of a hermitian curve in this paper the interpolated zero condition of a trivariate polynomial with respect to a multiplicity matrix m is redefined followed by a proof of the validity of the soft decision scheme this paper also introduces a new stopping criterion for the algorithm that tranforms the reliability matrix pi to the multiplicity matrix m geometric characterisation of the trivariate monomial decoding region is investigated resulting in an asymptotic optimal performance bound for the soft decision decoder by defining the weighted degree upper bound of the interpolated polynomial two complexity reducing modifications are introduced for the soft decision scheme elimination of unnecessary interpolated polynomials and pre calculation of the coefficients that relate the pole basis monomials to the zero basis functions of a hermitian curve our simulation results and analyses show that soft decision list decoding of hermitian code can outperform koetter vardy decoding of reed solomon code which is defined in a larger finite field but with less decoding complexity

unimodular fourier multipliers for modulation spaces abstract we investigate the boundedness of unimodular fourier multipliers on modulation spaces surprisingly the multipliers with general symbol e i where 0 2 are bounded on all modulation spaces but in general fail to be bounded on the usual l p spaces as a consequence the phase space concentration of the solutions to the free schrodinger and wave equations are preserved as a byproduct we also obtain boundedness results on modulation spaces for singular multipliers sin for 0

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

finding a basis for the characteristic ideal of an n dimensional linear recurring sequence an n dimensional linear recurring sequence sigma of elements from a field f for n or 1 is considered an algorithm chi base which determines a basis for the ideal characteristic polynomials of sigma under certain reasonable conditions is presented this analysis applied to doubly periodic arrays in particular

information sets from defining sets in abelian codes we describe a technique to construct a set of check positions and hence an information set for every abelian code solely in terms of its defining set this generalizes that given by imai in in the case of binary tdc codes

symbolic hamburger noether expressions of plane curves and applications to ag codes in this paper we consider some practical applications of the symbolic hamburger noether expressions for plane curves which are introduced as a symbolic version of the so called hamburger noether expansions more precisely we give and develop in symbolic terms algorithms to compute the resolution tree of a plane curve and the adjunction divisor in particular rational parametrizations for the branches of such a curve special adjoints with assigned conditions connected with different problems like the so called brill noether algorithm and the weierstrass semigroup at p together with functions for each value in this semigroup provided p is a rational branch of a singular plane model for the curve some other computational problems related to algebraic curves over perfect fields can be treated symbolically by means of such expressions but we deal just with those connected with the effective construction and decoding of algebraic geometry codes

pseudo wigner matrices we consider the problem of generating pseudo random matrices based on the similarity of their spectra to wigner s semicircular law we introduce the notion of an r independent pseudo wigner matrix ensemble and prove the closeness of the spectra of its matrices to the semicircular density in the kolmogorov distance we give an explicit construction of a family of n times n pseudo wigner ensembles using dual bch codes and show that the kolmogorov complexity of the obtained matrices is of the order of mathrm log n bits for a fixed designed kolmogorov distance precision we compare our construction with the quasi random graphs introduced by chung et al and demonstrate that the pseudo wigner matrices pass stronger randomness tests than the adjacency matrices of these graphs lifted by the mapping 0 rightarrow 1 and 1 rightarrow 1 do finally we provide numerical simulations verifying our theoretical results

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

codes on the klein quartic ideals and decoding corresp a sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field gf 2 3 in the sequence there is for instance a code of length 21 and dimension 10 with minimal distance 9 and a code of length 21 and dimension 16 with minimal distance 3 the codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by goppa the curve used in the present work is the klein quartic this curve has the maximal number of rational points over gf 2 3 allowed by serre s improvement of the hasse weil bound which together with the low genus accounts for the good code parameters the klein quartic has the frobenius group g of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes in fact the codes are given alternative descriptions as left ideals in the group algebra gf 2 3 g this description allows for easy decoding for instance in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 decoding is obtained by multiplication with an idempotent in the group algebra

finding recursions for multidimensional arrays we produce a concise description of sakata s algorithm for finding recursion relations that are valid for an n dimensional array the new analysis extends to limit points which are excluded by previous authors we show that under a natural hypothesis the algorithm becomes stationary below each limit point and in particular constructs a full set of recursion relations for a recursive array at some finite point the method used to justify the algorithm stresses the duality between the fundamental extension operation of sakata and the s polynomial operation of buchberger

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

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

functions operating on modulation spaces and nonlinear dispersive equations abstract the aim of this paper is two fold we show that if a complex function f on c operates in the modulation spaces m p 1 r n by composition then f is real analytic on r 2 c this answers negatively the open question posed in ruzhansky et al 2012 21 regarding the general power type nonlinearity of the form u u we also characterise the functions that operate in the modulation space m 1 1 r n the local well posedness of the nls nlw and nlkg equations for the real entire nonlinearities are also studied in some weighted modulation spaces m s p q r n

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

the hardness of decoding linear codes with preprocessing the problem of maximum likelihood decoding of linear block codes is known to be hard the fact that the problem remains hard even if the code is known in advance and can be preprocessed for as long as desired in order to device a decoding algorithm is shown the hardness is based on the fact that existence of a polynomial time algorithm implies that the polynomial hierarchy collapses thus some linear block codes probably do not have an efficient decoder the proof is based on results in complexity theory that relate uniform and nonuniform complexity classes

linear feedback shift registers and the minimal realization problem the first author is partially supported by grants mineco mtm2017 83624 p mtm2017 90682 redt and upv ehu giu16 42 the second author is partially supported by grants mineco mtm2017 83624 p and mtm2017 90682 redt

error locating pairs for cyclic codes a general decoding method for linear codes is investigated for cyclic codes the decoding consists of solving two systems of linear equations all but four binary cyclic codes of length less than 63 can so be decoded up to their actual distance a new family of codes is given for which the decoding needs only o n sup 2 operations

on the number of correctable errors of the feng rao decoding algorithm for ag codes in this correspondence we give geometric codes on one of the curves of garcia stichtenoth 1995 with the property that the true distances are bigger than feng rao distances moreover an example for which the feng rao algorithm cannot correct more than d sub fr 1 2 1 d sub true 1 2 errors is given

decoding of hermitian codes the key equation and efficient error evaluation this paper presents a generalization of the key equation to hermitian codes the syndrome is interpreted as a power series and the product of this power series with a locator polynomial gives the error evaluator polynomial the computation of the evaluator polynomial may be done iteratively using a modified version of a previously published computationally efficient algorithm for computing locator polynomials

decoding cyclic and bch codes up to actual minimum distance using nonrecurrent syndrome dependence relations the decoding capabilities of algebraic algorithms mainly the berlekamp massey algorithm the euclidean algorithm and the authors 1989 generalizations of these algorithms are basically constrained by the minimum distance bounds of the codes the authors introduce a more general procedure which breaks away from this restriction and which can determine the error locations from nonrecurrent dependence relations among the syndromes it can decode many cyclic and bch codes up to their actual minimum distance and is seen to be a generalization of the procedure introduced by w w peterson and e j weldon 1972

uniform l 1 behavior for time discretization of a volterra equation with completely monotonic kernel ii convergence a previous work on the time discretization for the solution of a volterra equation with completely monotonic convolution kernel is extended the considered time discretization method comes from part i x da ima j numer anal 22 2002 pp 133 151 where the backward euler is combined with order one convolution quadrature approximating the integral in the present paper the convergence properties of the discretization in time are given in the l t 1 0 infty h bigcap l t infty 0 infty h norm

generalized reed solomon codes from algebraic geometry a few years ago tsfasman em et al using results from algebraic geometry showed that there is a sequence of codes which are generalizations of goppa codes and which exceed the gilbert varshamov bound we show that a similar sequence of codes in fact the duals of the previous codes can be found by generalizing the construction of reed solomon codes our approach has the advantage that it uses less complicated concepts from algebraic geometry

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

decoding geometric goppa codes using an extra place decoding geometric goppa codes can be reduced to solving the key congruence of a received word in an affine ring if the codelength is smaller than the number of rational points on the curve then this method can correct up to 1 2 d l 2 s errors where d is the designed minimum distance of the code and s is the clifford defect the affine ring with respect to a place p is the set of all rational functions which have no poles except at p and it is somehow similar to a polynomial ring for a special kind of geometric goppa code namely c sub omega d mp the decoding algorithm is reduced to solving the key equation in the affine ring which can be carried out by the subresultant sequence in the affine ring with complexity o n sup 3 where n is the length of codewords

simplified understanding and efficient decoding of a class of algebraic geometric codes an efficient decoding algorithm for algebraic geometric codes is presented for codes from a large class of irreducible plane curves including hermitian curves it can correct up to d 1 2 errors where d is the designed minimum distance with it we also obtain a proof of d sub min spl ges d without directly using the riemann roch theorem the algorithm consists of gaussian elimination on a specially arranged syndrome matrix followed by a novel majority voting scheme a fast implementation incorporating block hankel matrix techniques is obtained whose worst case running time is o mn sup 2 where m is the degree of the curve applications of our techniques to decoding other algebraic geometric codes to decoding bch codes to actual minimum distance and to two dimensional shift register synthesis are also presented