decoding cyclic codes up to a new bound on the minimum distance a new lower bound on the minimum distance of q ary cyclic codes is proposed this bound improves upon the bose chaudhuri hocquenghem bound and for some codes upon the hartmann tzeng bound several boston bounds are special cases of our bound for some classes of codes the bound on the minimum distance is refined furthermore a quadratic time decoding algorithm up to this new bound is developed the determination of the error locations is based on the euclidean algorithm and a modified chien search the error evaluation is done by solving a generalization of forney s formula

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

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

the algebraic decoding of the 41 21 9 quadratic residue code a new algebraic approach for decoding the quadratic residue qr codes in particular the 41 21 9 qr code is presented the key ideas behind this decoding technique are a systematic application of the sylvester resultant method to the newton identities associated with the syndromes to find the error locator polynomial and next a method for determining error locations by solving certain quadratic cubic and quartic equations over gf 2 sup m in a new way which uses zech s logarithms for the arithmetic the logarithms developed for zech s logarithms save a substantial amount of computer memory by storing only a table of zech s logarithms these algorithms are suitable for implementation in a programmable microprocessor or special purpose vlsi chip it is expected that the algebraic methods developed can apply generally to other codes such as the bch and reed solomon codes

isospectral deformations i riemannian structures on two step nilspaces on montre comment construire des exemples de deformations isospectrales de metriques de riemann sur des varietes sur lesquelles des recouvrements de certains groupes de lie nilpotents agissent

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

two approximation techniques for functional differential equations abstract some authors proposed to approximate the solutions of delay differential equations by ordinary differential equations to eliminate in part the difficulties presented by time delays this technique termed the chain method and its modified version are discussed in this paper for general nonlinear time dependent delay differential equations the modified version is based on some heuristic approximation in biological modelling and the difference in properties of the original and the new modified approximation techniques is demonstrated on a delay differential equation of an epidemic model

an algebraic procedure for decoding beyond e sub bch we present a new way to find all possible error patterns up to a given weight greater than the designed error correcting capability e sub bch possible error positions are localized by indicators our method can be seen as an extension of the step by step decoder introduced by massey 1965 for bch codes we consider the decoding of binary cyclic codes

sign changes of the error term in weyl s law for heisenberg manifolds let r t be the error term in weyl s law for the 2l 1 dimensional heisenberg manifold hl gl in this paper several results on the sign changes and odd moments of r t are proved in particular it is proved that for some sufficiently large constant c r t changes sign in the interval t t c t for all large t moreover for a small constant c1 there exist infinitely many subintervals in t 2t of length c1 t log 5 t such that r t c1tl 1 4 holds on each of these subintervals

on double byte error correcting codes this correspondence shows that there is a flaw in the results presented by g l feng et al see ibid vol 44 no 3 p 1152 63 may 1998 a large family of the codes constructed are not double byte error correcting codes as originally claimed

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

new list decoding algorithms for reed solomon and bch codes in this paper we devise a rational curve fitting algorithm and apply it to the list decoding of reed solomon and bose chaudhuri hocquenghen bch codes the resulting list decoding algorithms exhibit the following significant properties

a new lower bound for the minimum distance of a cyclic code a new lower bound for the minimum distance of a linear code is derived when applied to cyclic codes both the bose chaudhuri hocquenghem bch bound and the hartmann tzeng ht bound are obtained as corollaries examples for which the new bound is superior to these two bounds as well as to the carlitz uchiyama bound are given

algebraic decoding of the 32 16 8 quadratic residue code an algebraic decoding algorithm for the 1 2 rate 32 16 8 quadratic residue qr code is found the key idea of this algorithm is to find the error locator polynomial by a systematic use of the newton identities associated with the code syndromes the techniques developed extend the algebraic decoding algorithm found recently for the 32 16 8 qr code it is expected that the algebraic approach developed here and by m elia 1987 applies also to longer qr codes and other bch type codes that are not fully decoded by the standard bch decoding algorithm

decoding the 47 24 11 quadratic residue code using bit error probability estimates a new algorithm is developed to facilitate faster decoding of the 47 24 11 quadratic residue qr code this decoder based on the idea first developed by reed in a 1959 mit lincoln laboratory report uses real channel data to estimate the individual bit error probabilities in a received word the algorithm then sequentially inverts the bits with the highest probability of error until one of the errors is canceled the remaining errors are then corrected by the use of algebraic decoding techniques this new algorithm called the reliability search algorithm is a complete decoder that significantly reduces the decoding complexity in terms of cpu time while maintaining the same bit error rate ber performance in fact this algorithm is an appropriate modification to the algorithm developed by chase

on the minimum distance of cyclic codes the main result is a new lower bound for the minimum distance of cyclic codes that includes earlier bounds i e bch bound ht bound roos bound this bound is related to a second method for bounding the minimum distance of a cyclic code which we call shifting this method can be even stronger than the first one for all binary cyclic codes of length with two exceptions we show that our methods yield the true minimum distance the two exceptions at the end of our list are a code and its even weight subcode we treat several examples of cyclic codes of length geq 63

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

spectra of sol manifolds arithmetic and quantum monodromy the spectral problem of three dimensional manifolds m 3 a admitting sol geometry in thurston s sense is investigated topologically m 3 a are torus bundles over a circle with a unimodular hyperbolic gluing map a the eigenfunctions of the corresponding laplace beltrami operators are described in terms of modified mathieu functions it is shown that the multiplicities of the eigenvalues are the same for generic values of the parameters in the metric and are directly related to the number of representations of an integer by a given indefinite binary quadratic form as a result the spectral statistics is shown to disagree with the berry tabor conjecture the topological nature of the monodromy for both classical and quantum systems on sol manifolds is demonstrated

the inaudible geometry of nilmanifolds we show that isospectral deformations of compact riemannian two step nilmanifolds can be systematically detected by simple changes in the behavior of their geodesics in spite of the fact that the length spectrum which measures the lengths of all closed geodesics remains constant

on squares of cyclic codes the square c 2 of a linear error correcting code c is the linear code spanned by the component wise products of every pair of non necessarily distinct words in c squares of codes have gained attention for several applications mainly in the area of cryptography and typically in those applications one is concerned about some of the parameters dimension and minimum distance of both c 2 and c in this paper motivated mostly by the study of this problem in the case of linear codes defined over the binary field squares of cyclic codes are considered general results on the minimum distance of the squares of cyclic codes are obtained and constructions of cyclic codes c with a relatively large dimension of c and minimum distance of the square c 2 are discussed in some cases the constructions lead to codes c such that both c and c 2 simultaneously have the largest possible minimum distances for their length and dimensions

the remainder in weyl s law for heisenberg manifolds we prove that the error term r in weyl s law is o 5 6 for certain three dimensional heisenberg manifolds we also show that the l 2 norm of the weyl error term integrated over the moduli space of left invariant heisenberg metrics is 3 4 we conjecture that r o 3 4 i s a sharp deterministic upper bound for heisenberg three manifolds

on the generalized hamming weights of cyclic codes we prove several results on the generalized hamming weights ghw s of linear codes particularly for cyclic codes based on these and previously known results we give some efficient algorithms for computing ghw hierarchy of cyclic codes we give complete weight hierarchy for each of the binary cyclic codes of odd lengths spl les 31 a table of second and third ghw s of binary cyclic codes of odd lengths spl les 57 is also presented we have also computed the second ghw of all binary cyclic codes of length 63 and the third ghw of one code from each dimension

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

algebraic decoding beyond e sub bch of some binary cyclic codes when e e sub bch for a number of binary cyclic codes with e e sub bch algebraic algorithms are given to find the error locator polynomial thus for these codes more errors can be corrected algebraically than by the berlekamp massey algorithm in some cases all error patterns of weight up to e can be decoded in other cases only error patterns of weight up to e with e sub bch

algebraic decoding of cyclic codes without error locator polynomials the algebraic decoding of a p ary cyclic code consists of four steps 1 computation of the known syndromes using the received word 2 computation of the unknown syndromes from the known syndromes 3 computation of the error positions by a use of the berlekamp massey bm algorithm and chien s search and 4 computation of the error values by solving a linear system this paper addresses two problems of determining the error positions and computing the unknown syndromes to solve the first problem a new matrix together with gaussian elimination instead of the bm algorithm and chien s search is proposed in this new simplified decoder finding an error locator polynomial is completely avoided a main advantage of the presented decoding method is when the bose chaudhuri hocquenghem bound is unequal to the minimum distance of the code some cyclic codes which do not have 2t consecutive known syndromes can be decoded up to their actual minimum distance by using the presented matrix once to solve the second problem two algorithms for different square matrices reported recently by lee et al are also provided to calculate the value of an unknown syndrome finally an algebraic decoding of the 47 23 15 ternary cyclic code is given

the method of lines for the solutions of some nonlinear partial differential equations abstract in this paper we deal with the method of lines in the case of certain first order nonlinear partial differential equations as well as equations of the parabolic type a new aspect compared to earlier investigations is that the system we consider may contain partial and ordinary differential equations alike and in addition the r h s of the equations may involve some nonlinear operators the domain of definition of the solutions is bounded by time dependent functions our general results are demonstrated on a simple system which describes a biological model

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

algebraic decoding of the ternary 13 7 5 quadratic residue code an algebraic decoding algorithm for the ternary 13 7 5 quadratic residue code is presented this seems to be the first attempt to provide an algebraic decoding algorithm for a quadratic residue code over a nonbinary field

a small scale density of states formula let m g be a c compact riemann manifold with classical hamiltonian h c t m assume that the corresponding quantization p1 op h is quantum completely integrable we establish an microlocal weyl law on short spectral intervals of size 2 e e 0 for various families of operators p1u u i containing p1 both in the mean and pointwise a e for u i the microlocalization refers to a small tubular neighbourhood of a non degenerate stable periodic bicharacteristic t m 0

thee invariant and the spectrum of the laplacian for compact nilmanifolds covered by heisenberg groups on presente une famille naturelle et explicite des varietes referencees qui representent des elements d ordre arbitrairement grand dans les groupes d homotopie stables des spheres

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

extension of the bch decoding algorithm to decode binary cyclic codes up to their maximum error correction capacities the bch algorithm is extended to correct more errors than indicated by the bch bound in the first step of the decoding procedure a number of errors corresponding to a particular case of the hartmann tzeng bound are corrected in the second step full error correction is the goal a measure for the worst case number of field elements of an extension field gf 2 sup m that must be tested for this purpose is given for binary cyclic linear unequal error protection codes as well as for conventional binary cyclic codes

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

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

decoding the ternary golay code a decoding algorithm is presented for the ternary 11 6 5 golay code this algorithm is an algebraic decoding and relies on solving the newton identities associated with the syndromes of the code however to complete the decoding some explicit values of zech s logarithms need to be established

an explicit extrapolated box scheme for the gurtin maccamy equation abstract an explicit extrapolated box method is considered for a nonlinear partial integrodifferential equation the gurtin maccamy equation subject to a nonlocal boundary condition this problem describes the evolution in time of the age structure of a population the consistency stability and convergence properties of the method are studied and its second order accuracy is demonstrated both analytically and numerically

some mathematical aspects of modelling cell population dynamics abstract the purpose of this paper is to discuss different types of models of cell population dynamics based on differential equations and to point out the connection between them from the starting point of the mckendrick von foerster type model we derive lotka volterra type integral equations and some delaying differential equations we establish a connection between the original partial differential equation model and the takahashi ordinary differential equations model based on compartmental analysis by means of a numerical procedure named the method of lines

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

decoding the 47 24 11 quadratic residue code the techniques needed to decode the 47 24 11 quadratic residue qr code differ from the schemes developed for cyclic codes by finding certain nonlinear relations between the known and unknown syndromes for this special code two methods are developed to decode up to the true minimum distance of the 47 24 11 qr code these algorithms can be utilized to decode effectively the 1 2 rate 48 24 12 qr code for correcting five errors and detecting six errors

circular symmetry and the trace formula let m be a compact n dimensional manifold without boundary and let p be a positive first order elliptic self adjoint classical pseudodifferential operator on lz m the hilbert space of half densities on m the spectrum of p is discrete and accumulates only at we denote the eigenvalues with multiplicities by 2o 21 2 recall that the singularities of the inverse fourier transform of the spectral measure