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

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

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

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

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

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

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

variations and estimators for self similarity parameters via malliavin calculus using multiple stochastic integrals and the malliavin calculus we analyze the asymptotic behavior of quadratic variations for a specific non gaussian self similar process the rosenblatt process we apply our results to the design of strongly consistent statistical estimators for the self similarity parameter h although in the case of the rosenblatt process our estimator has non gaussian asymptotics for all h 1 2 we show the remarkable fact that the process s data at time 1 can be used to construct a distinct compensated estimator with gaussian asymptotics for h in 1 2 2 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

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

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

tail asymptotics of the brownian signature the signature of a path gamma is a sequence whose n th term is the order n iterated integrals of gamma it arises from solving multidimensional linear differential equations driven by gamma we are interested in relating the path properties of gamma with its signature if gamma is c 1 then an elegant formula of hambly and lyons relates the length of gamma to the tail asymptotics of the signature we show an analogous formula for the multidimensional brownian motion with the quadratic variation playing a similar role to the length in the proof we study the hyperbolic development of brownian motion and also obtain a new subadditive estimate for the asymptotic of signature which may be of independent interest as a corollary we strengthen the existing uniqueness results for the signatures of brownian motion

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

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 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