algebraic decoding of cyclic codes using partial syndrome matrices cyclic codes have been widely used in many applications of communication systems and data storage systems this paper proposes a new procedure for decoding cyclic codes up to actual minimum distance the decoding procedure consists of two steps 1 computation of known syndromes and 2 computation of error positions and error values simultaneously to do so a matrix whose all entries are syndromes is called syndrome matrix a matrix whose entries are either syndromes or the elements of a finite field is said to be partial syndrome matrix in this paper two novel methods are presented to determine error positions and error values simultaneously and directly the first method uses a new partial syndrome matrix along with gaussian elimination the partial syndrome matrices for binary respectively ternary cyclic codes of lengths from 69 to 99 respectively 16 to 37 are tabulated for some cyclic codes the partial syndrome matrices contain unknown syndromes the second method constructs a matrix from a system of equations which is generated by the determinants of different partial syndrome matrices and makes use of gaussian elimination to determine its row rank many more cyclic codes beyond the bose chaudhuri hocquenghem bound can be decoded with these methods

minimum weights and weight enumerators of bbz_ 4 linear quadratic residue codes a fast method to compute the minimum lee weight and the symmetrized weight enumerator of extended quadratic residue codes xqr codes over the ring z4 is developed our approach is based on the classical brouwer zimmermann algorithm and additionally takes advantage of the large group of automorphisms and the self duality of the z4 linear xqr codes as well as the projection to the binary xqr codes as a result the hitherto unknown minimum lee distances of all z4 linear xqr codes of lengths between 72 and 104 and the minimum euclidean distances for the lengths 72 80 and 104 are computed it turns out that the binary gray image of the z4 linear xqr codes of lengths 80 and 104 has higher minimum distance than any known linear binary code of equal length and cardinality furthermore the z4 linear xqr code of length 80 is a new example of an extremal z4 linear typeii code additionally we give the symmetrized weight enumerator of the z4 linear xqr codes of lengths 72 and 80 and we correct the weight enumerators of the z4 linear xqr code of length 48 given by pless and qian and bonnecaze et al

algebraic decoding of 71 36 11 79 40 15 and 97 49 15 quadratic residue codes recently a new algebraic decoding algorithm for quadratic residue qr codes was proposed by truong et al using that decoding scheme we now develop three decoders for the qr codes with parameters 71 36 11 79 40 15 and 97 49 15 which have not been decoded before to confirm our results an exhaustive computer simulation has been executed successfully

weight enumerators of extremal singly even 60 30 12 codes conway and sloane 1990 have listed the possible weight enumerators for extremal self dual codes up to length 72 in this correspondence we construct extremal singly even self dual 60 30 12 codes whose weight enumerator does not appear in this list in addition we present the possible weight enumerators for extremal self dual codes of length 60

self dual codes with an automorphism of order 11 using a method for constructing self dual codes having an automorphism of odd prime order we classify up to equivalence all binary self dual codes with an automorphism of order 11 with 6 cycles and minimum distance 12 this classification gives new 72 36 12 codes with weight enumerator that was previously not obtained as well as many 66 33 12 68 34 12 and 70 35 12 codes with new values of the parameters in their respective weight enumerators

a new scheme to determine the weight distributions of binary extended quadratic residue codes this letter proposes a novel scheme which consists of a weight counting algorithm the combinatorial designs of the assmus mattson theorem and the weight polynomial of gleason s theorem to determine the weight distributions of binary extended quadratic residue codes as a consequence the weight distributions of binary 138 69 22 and 168 84 24 extended quadratic residue codes are given

existence of some extremal self dual codes two new singly even extremal self dual codes are constructed a 52 26 10 code and a 54 27 10 code

existence and nonexistence of positive solutions of p kolmogorov equations perturbed by a hardy potential abstract in this article we establish the phenomenon of existence and nonexistence of positive weak solutions of parabolic quasi linear equations perturbed by a singular hardy potential on the whole euclidean space depending on the controllability of the given singular potential to control the singular potential we use a weighted hardy inequality with an optimal constant which was recently discovered in hauer and rhandi 2013 our results in this paper extend the ones in goldstein et al 2012 concerning a linear kolmogorov operator significantly in several ways firstly by establishing existence of positive global solutions of singular parabolic equations involving nonlinear operators of p laplace type with a nonlinear convection term for 1 p and secondly by establishing nonexistence locally in time of positive weak solutions of such equations without using any growth conditions

doubly even self dual code of length 96 we prove that 23 11 and 7 do not divide the order of the automorphism group of a binary 96 48 20 doubly even self dual code we construct 25 new inequivalent binary 96 48 16 doubly even self dual codes via an automorphism of order 23

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

on solving wave equations on fixed bounded intervals involving robin boundary conditions with time dependent coefficients abstract in this paper it is shown how characteristic coordinates or equivalently how the well known formula of d alembert can be used to solve initial boundary value problems for wave equations on fixed bounded intervals involving robin type of boundary conditions with time dependent coefficients a robin boundary condition is a condition that specifies a linear combination of the dependent variable and its first order space derivative on a boundary of the interval analytical methods such as the method of separation of variables sov or the laplace transform method are not applicable to those types of problems the obtained analytical results by applying the proposed method are in complete agreement with those obtained by using the numerical finite difference method for problems with time independent coefficients in the robin boundary condition s the results of the proposed method also completely agree with those as for instance obtained by the method of separation of variables or by the finite difference method

an improved decoding algorithm of the 71 36 11 quadratic residue code without determining unknown syndromes in this paper a new algebraic method to decode the 71 36 11 qr code up to five errors is proposed it completely avoids computing the unknown syndromes and uses the previous scheme of decoding this qr code up to three errors but corrects four and five errors with a new different method in the four error case the new algorithm directly determines the coefficients of the error locator polynomial by eliminating unknown syndromes in newton identities subsequently the shift search algorithm can be utilized to decode the fifth error and the concept of bit reliability is also introduced to accelerate the decoding process in other words a weight five error pattern can be decoded in terms of the four error case by inverting an incorrect bit of the received word in ascending order of reliability particularly a threshold parameter gamma can be preset to limit the number of inverting bits one by one and a corresponding upper bound of the probability that decoding fails is derived finally simulation and analysis show that the proposed new decoding algorithm for the abovementioned qr code not only significantly reduces the decoding complexity in terms of cpu time but also saves a lot of memory while maintaining the same error rate performance additionally the introduction of gamma achieves a better tradeoff between the decoding performance and the computational complexity

reflection and damping properties for semi infinite string equations with non classical boundary conditions abstract in this paper initial boundary value problems for a linear wave string equation are considered the main objective is to study boundary reflection and damping properties of waves in semi infinite strings this problem is of considerable practical interest in the context of vibration suppression at boundaries of elastic structures solutions of wave equations will be constructed for two different classes of boundary conditions in the first class a massless system consisting of a spring and damper will be considered at the boundary in the second class an additional mass will be added to the system at the boundary the d alembert method will be used to construct explicit solutions of the boundary value problem for the one dimensional wave equation on the semi infinite domain it will also be shown how waves are damped and reflected at these boundaries and how much energy is dissipated at the boundary

sharp upper bound on the blow up rate for the critical nonlinear schrodinger equation we consider the critical nonlinear schrodinger equation iu t delta u u 4 n with initial condition u 0 x u0

quasi cyclic codes over z sub 4 and some new binary codes previously linear codes over z sub 4 and quasi cyclic qc codes over fields have been shown to yield useful results in coding theory combining these two ideas we study z sub 4 qc codes and obtain new binary codes using the usual gray map among the new codes the lift of the famous golay code to z sub 4 produces a new binary code a 92 2 sup 24 28 code which is the best among all binary codes linear or nonlinear moreover we characterize cyclic codes corresponding to free modules in terms of their generator polynomials

the weight distributions of some binary quadratic residue codes the weight distributions of binary quadratic residue codes c can be computed from the weight distribution of a subset of c containing one fourth resp one eighth of the codewords in c when the length of the code is congruent to 1 resp 1 modulo 8 an algorithm to determine the weight distributions of binary cyclic codes is given as a consequence the weight distributions of 73 37 13 89 45 17 and 97 49 15 quadratic residue codes are determined precisely

some self dual codes invariant under the hall janko group abstract we construct self dual codes of length 100 invariant under the hall janko group j 2 we also construct a 2 100 22 168 design with the automorphism group j 2 2 which is the extension of j 2 by its outer automorphism

the hardy inequality and the asymptotic behaviour of the heat equation with an inverse square potential we study the well posedness and describe the asymptotic behavior of solutions of the heat equation with inverse square potentials for the cauchy dirichlet problem in a bounded domain and also for the cauchy problem in rn in the case of the bounded domain we use an improved form of the so called hardy poincare inequality and prove the exponential stabilization towards a solution in separated variables in rn we first establish a new weighted version of the hardy poincare inequality and then show the stabilization towards a radially symmetric solution in self similar variables with a polynomial decay rate this work complements and explains well known work by baras and goldstein on the existence of global solutions and blow up for these equations in the present article the sign restriction on the data and solutions is removed the functional framework for well posedness is described and the asymptotic rates calculated examples of non uniqueness are also given

on the existence of a certain 64 32 12 extremal code j h conway and n j a sloane see ibid vol 36 no 6 p 1319 33 nov 1990 give weight enumerators of several self dual codes with the highest possible minimal distance whose existence was not known a generator matrix for one of these a type i 64 32 12 code is given proving its existence the method of construction is described

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

cyclic codes and self dual codes over f sub 2 uf sub 2 we introduce linear cyclic codes over the ring f sub 2 uf sub 2 0 1 u u u 1 where u sup 2 0 and study them by analogy with the z sub 4 case we give the structure of these codes on this new alphabet self dual codes of odd length exist as in the case of z sub 4 codes unlike the z sub 4 case here free codes are not interesting some nonfree codes give rise to optimal binary linear codes and extremal self dual codes through a linear gray map

extremal self dual codes with an automorphism of order 2 a method to design binary self dual codes with an automorphism of order two without fixed points is presented extremal self dual codes with lengths 40 42 44 54 58 68 are constructed many of them have weight enumerators for which extremal codes were previously not known to exist

fast algorithm for computing the roots of error locator polynomials up to degree 11 in reed solomon decoders the central problem in the implementation of a reed solomon code is finding the roots of the error locator polynomial in 1967 berlekamp et al found an algorithm for finding the roots of an affine polynomial in gf 2 sup m that can be used to solve this problem in this paper it is shown that this berlekamp rumsey solomon 1967 algorithm together with the chien 1964 search method makes possible a fast decoding algorithm in the standard basis representation that is naturally suitable in a software implementation finally simulation results for this fast algorithm are given

new extremal binary 44 22 8 codes one of the problems in coding theory is constructing self dual codes whose weight enumerators are not yet known to exist we use the concept of neighbors and construct extremal binary self dual 44 22 8 codes whose weight enumerators are not yet known to exist

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen

continuation of blowup solutions of nonlinear heat equations in several space dimensions the possible continuation of solutions of the nonlinear heat equation in r n r ut u m u p with m 0 p 1 after the blowup time is studied and the different continuation modes are discussed in terms of the exponents m and p thus for m p 2 we find a phenomenon of nontrivial continuation where the regionfx u x t 1g is bounded and propagates with finite speed this we call incomplete blowup for n 3 and p m n 2 n 2 we find solutions that blow up at finite t t and then become bounded again for t t otherwise we find that blowup is complete for a wide class of initial data in the analysis of the behavior for large p a list of critical exponents appears whose role is described we also discuss a number of related problems and equations we apply the same technique of analysis to the problem of continuation after the onset of extinction for example for the equation ut u m u p m 0

on nonexistence of baras goldstein type for higher order parabolic equations with singular potentials the celebrated result by baras and goldstein 1984 established that the heat equation with singular inverse square potential in a smooth bounded domain n n 3 such that 0 e u t u c x 2 u in 0 t u 0 in the supercritical range c c hardy 1 n 2 2 2 does not have a solution for any nontrivial l 1 initial data uo x 0 in or for a positive measure namely it was proved that a regular approximation of a possible solution by a sequence u n x t of classical solutions of uniformly parabolic equations with bounded truncated potentials given by v x c x 2 v n x min c x 2 n n 1 diverges and as n oo u n x t in x 0 t in the present paper we reveal the connection of this very singular evolution with a spectrum of some limiting operator the proposed approach allows us to consider more general higher order operators for which hardy s inequalities were known since rellich 1954 and initial data that are not necessarily positive in particular it is established that under some natural hypothesis the divergence result is valid for any 2mth order parabolic equation with singular potential u t m u c x 2m u in 0 t where c c h m m 1 with zero dirichlet conditions on and for a wide class of initial data in particular typically the divergence holds for any data satisfying u 0 x is continuous at x 0 and uo 0 0 similar nonexistence i e divergence as e 0 results are also derived for time dependent potentials e 2m q x e t e 2m and nonlinear reaction terms u p e 2m x 2m with p 1 applications to other linear and semilinear schrodinger and wave pdes are discussed

sharp constants in the hardy rellich inequalities optimal constants are found in hardy rellich inequalities containing derivatives of arbitrary not necessarily integer order l some new inequalities of this type are also obtained

on wave reflections and energetics for a semi infinite traveling string with a nonclassical boundary support abstract the free linear transverse vibrations of a semi infinite axially moving string are considered in order to study the reflection of an incident wave and the damping properties of different types of boundary conditions such as free fixed spring dashpot and mass spring dashpot to obtain the response to the initial conditions the method of d alembert is applied furthermore analytical expressions for the time rate of change of the total mechanical energy of the system are derived the obtained results give insight into the most efficient way of placing a boundary support depending on the direction of the transport velocity moreover for nonclassical boundary conditions the dynamics of the string is described by the relative values of the system parameters

construction of an extremal self dual code of length 62 we construct an extremal singly even self dual 62 31 12 code the existence of such a self dual code was not previously known combined with known results the above code also determines all lengths for which there are self dual codes with minimum weight 12

on the shape of the general error locator polynomial for cyclic codes general error locator polynomials were introduced in 2005 as an alternative decoding for cyclic codes we now present a conjecture on their sparsity which would imply polynomial time decoding for all cyclic codes a general result on the explicit form of the general error locator polynomial for all cyclic codes is given along with several results for specific code families providing evidence to our conjecture from these a theoretical justification of the sparsity of general error locator polynomials is obtained for all binary cyclic codes with t leq 2 and n as well as for t 3 and n except for some cases where the conjectured sparsity is proved by a computer check moreover we summarize all related results previously published and we show how they provide further evidence to our conjecture finally we discuss the link between our conjecture and the complexity of bounded distance decoding of the cyclic codes

shadow codes and weight enumerators the technique of using shadow codes to build larger self dual codes is extended to codes over arbitrary fields it is shown how to build the codes and how to determine the new weight enumerator as well for codes over fields equipped with a square root of 1 and not of characteristic 2 a self dual code of length n 2 can be built from a self dual code of length n for codes over a field without a square root of 1 and not of characteristic 2 a self dual code of length n 4 is built from a self dual code of length n and for codes over fields of characteristic 2 the length of the new self dual code depends on the presence of the all one vector in the subcode chosen in certain cases using the subcode of vectors orthogonal to the all one vector the new weight enumerator can be calculated directly from the original weight enumerator specific examples of the technique are illustrated for codes over f sub 3 f sub 4 and f sub 5

on the classification of extremal 36 18 8 binary self dual codes in this correspondence we give a new recursive method to classify extremal self dual codes as an application we classify all the 41 extremal binary 36 18 8 self dual codes

algebraic decoding of 103 52 19 and 113 57 15 quadratic residue codes in this paper two algebraic decoders for the 103 52 19 and 113 57 15 quadratic residue codes which have lengths greater than 100 are presented the results have been verified by software simulation that programs in c language have been executed to check possible error patterns of both quadratic residue codes

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

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

disjoint linear codes from algebraic function fields in this correspondence we study disjoint linear codes and give constructions of families of disjoint linear codes based on algebraic function fields it turns out that for some parameters our constructions improve on a result of johansson and pasalic

a performance comparison of the binary quadratic residue codes with the 1 2 rate convolutional codes the 1 2 rate binary quadratic residue qr codes using binary phase shift keyed bpsk modulation and hard decoding are presented as an efficient system for reliable communication performance results of error correction are obtained both theoretically and by means of computer calculations for a number of binary qr codes these results are compared with the commonly used 1 2 rate convolutional codes with constraint lengths from 3 to 7 for the hard decision case the binary qr codes of different lengths are shown to be equivalent in error correction performance to some 1 2 rate convolutional codes each of which has a constraint length k that corresponds to the error control rate d n and the minimum distance d of the qr codes

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

existence of certain extremal self dual codes two new singly even extremal self dual codes are given these are a 42 21 8 code and a 58 29 10 code

on decoding of the 89 45 17 quadratic residue code in this paper three decoding methods of the 89 45 17 binary quadratic residue qr code to be presented are hard soft and linear programming decoding algorithms firstly a new hybrid algebraic decoding algorithm for the 89 45 17 qr code is proposed it uses the laplace formula to obtain the primary unknown syndromes as done in lin et al s algorithm when the number of errors v is less than or equal to 5 whereas gaussian elimination is adopted to compute the unknown syndromes when v 6 secondly an appropriate modification to the algorithm developed by chase is also given in this paper therefore combining the proposed algebraic decoding algorithm with the modified chase ii algorithm called a new soft decision decoding algorithm becomes a complete soft decoding of qr codes thirdly in order to further improve the error correcting performance of the code linear programming lp is utilized to decode the 89 45 17 qr code simulation results show that the proposed algebraic decoding algorithm reduces the decoding time when compared with lin et al s hard decoding algorithm and thus significantly reduces the decoding complexity of soft decoding while maintaining the same bit error rate ber performance moreover the lp based decoding improves the error rate performance almost without increasing the decoding complexity when compared with the new soft decision decoding algorithm it provides a coding gain of 0 2 db at ber 2 x 10 6

a method for constructing self dual codes with an automorphism of order 2 in this paper we investigate binary self dual codes with an automorphism of order 2 with c cycles and f fixed points a method for constructing such codes using self orthogonal codes of length c and self dual codes of length f is presented we apply this method to construct extremal self dual codes of lengths 40 42 44 52 54 and 58 some of them have weight enumerators for which self dual codes were previously not known to exist we prove that there do not exist self dual 50 25 10 and 96 48 20 codes with an automorphism of order 2 with f fixed points for f 0 in their automorphism groups

a reflected wave superposition method for vibration and energy of a travelling string this paper considers the analytical free time domain response and energy in an axially translating and laterally vibrating string the domain of the string is either a constant or variable length dependent upon the general initial conditions the translating tensioned strings possess either fixed fixed or fixed free boundaries an alternative analytical solution using a reflected wave superposition method is presented for a finite translating string firstly the cycles of vibration for both constant and variable length strings are provided which for the latter are dependent upon the variable string length each cycle is divided into three time intervals according to the magnitude and the direction of the translating string velocity applying d alembert s method combined with the reflection properties expressions for the reflected waves at the two boundaries are obtained subsequently superposition of all of the incident and reflected waves provides results for the free vibration of the string over the three time intervals the variation in the total mechanical energy of the string system is also shown the accuracy and efficiency of the proposed method are confirmed numerically by comparison to simulations produced using a newmark beta method solution and an existing state space function representation of the string dynamics

cubic self dual binary codes we study binary self dual codes with a fixed point free automorphism of order three all binary codes of that type can be obtained by a cubic construction that generalizes turyn s we regard such cubic codes of length 3 spl lscr as codes of length spl lscr over the ring f sub 2 spl times f sub 4 classical notions of type ii codes shadow codes and weight enumerators are adapted to that ring two infinite families of cubic codes are introduced new extremal binary codes in lengths spl les 66 are constructed by a randomized algorithm necessary conditions for the existence of a cubic 72 36 16 type ii code are derived

some new extremal codes of lengths 76 and 78 we find new extremal 76 38 14 and 78 39 14 binary self dual codes the code of length 76 is the first known self dual code with these parameters it is shadow extremal and the shortest known self dual code with minimal distance 14 it has an automorphism of order 19 without fixed coordinates the code of length 78 has a new weight enumerator and an automorphism of order 19 with two fixed coordinates

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

there is no self dual bbz _ 4 linear code whose gray image has the parameters 72 2 36 16 it is shown that there is no self dual bbz4 linear code whose gray image has the parameters 72 236 16

algebraic decoding of the 89 45 17 quadratic residue code recently an algebraic decoding algorithm suggested by truong 2005 for some quadratic residue codes with irreducible generating polynomials has been designed that uses the inverse free berlekamp massey bm algorithm to determine the error locator polynomial in this paper based on the ideas of the algorithm mentioned above an algebraic decoder for the 89 45 17 binary quadratic residue code the last one not decoded yet of length less than 100 is proposed it was also verified theoretically for all error patterns within the error correcting capacity of the code moreover the verification method developed in this paper can be extended for all cyclic codes without checking all error patterns by computer simulations

new binary singly even self dual codes in this paper we construct new binary singly even self dual codes with larger minimum weights than the previously known singly even self dual codes for several lengths several known construction methods are used to construct the new self dual codes

on decoding of the 73 37 13 quadratic residue code in this paper a method to search the set of syndromes indices needed in computing the unknown syndromes for the 73 37 13 quadratic residue qr code is proposed according to the resulting index sets one computes the unknown syndromes and thus finds the corresponding error locator polynomial by using an inverse free berlekamp massey bm algorithm based on the modified chase ii algorithm the performance of soft decision decoding for the 73 37 13 qr code is given this result is new moreover the error rate performance of linear programming lp decoding for the 73 37 13 qr code is also investigated and lp based decoding is shown to be significantly superior in performance to the algebraic soft decision decoding while requiring almost the same computational complexity in fact the algebraic hard decision and soft decision decoding of the 89 45 17 qr code outperforms that of the 73 37 13 qr code because the former has a larger minimal distance however experimental results indicate that the 73 37 13 qr code outperforms the 89 45 17 qr code with much fewer arithmetic operations when using the lp based decoding algorithms the pseudocodewords analysis partially explains this seemingly strange phenomenon

a method for constructing inequivalent self dual codes with applications to length 56 necessary and sufficient conditions are presented for two doubly even codes with an automorphism of odd prime order to be equivalent under certain assumptions for the codes as an application we obtain all extremal doubly even codes of length 56 with an automorphism of order 13 sixteen such codes are inequivalent

the heat equation with a singular potential of concern is the singular problem du dt au c x 2 iz x tz x 0 u0 x and its generalizations here c 0 x g rn t 0 and and t 0 are nonnegative and not both identically zero there is a dimension dependent constant c n such that the problem has no solution for c ct n for c s ct a necessary and sufficient conditions are found for and u0 so that a nonnegative solution exists

construction of self dual codes construction methods for self dual codes are given by using these methods some new extremal self dual 66 33 12 and 68 34 12 codes are obtained

on a class of self dual codes derived from quadratic residues in 1976 moore investigated a class of self dual double circulant codes based on primes of the form q 3 mod 8 this paper shows that the corresponding length 168 code contains a weight 28 codeword and thus is not extremal this answers a long standing question regarding this code the next five codes in this class are also shown not to be extremal

classification of extremal and s extremal binary self dual codes of length 38 in this paper we classify all extremal and s extremal binary self dual codes of length 38 there are exactly 2744 extremal self dual codes two s extremal codes and 1730 s extremal codes we obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self dual codes of length 36 and from a generalization of this recursive algorithm for the shadow the classification of extremal codes permits to achieve the classification of all extremal codes with

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

a generalized treatment of the energetics of translating continua part i strings and second order tensioned pipes the energetics of translating one dimensional uniform strings and highly tensioned pipes with vanishing bending stiffness and flowing fluid are analyzed for fixed free and damped boundary conditions the interaction between the translating continua and the boundary supports causes energy transfer at a fixed boundary the transverse component of tension does work and the coriolis forces at a free end cause energy flux into the second order continuum under a symmetric boundary configuration the total energy of free oscillation varies periodically at the fundamental natural frequency asymmetric boundary supports in the pipe fluid system lead to damped or self excited motions at a viscously damped boundary the condition for maximal energy dissipation the destabilizing effect of dissipation and the stabilizing effect of negative damping are examined analytically using travelling wave solutions the energies transferred at the different boundary supports are quantified by energy reflection coefficients which are determined completely by the boundary conditions numerical simulations verify the analytically predicted energy variations

a construction of quantum stabilizer codes based on syndrome assignment by classical parity check matrices in this paper a new but simple construction of stabilizer codes and related entanglement assisted quantum error correcting codes is proposed based on syndrome assignment by classical parity check matrices this method turns the construction of quantum stabilizer codes to the construction of classical parity check matrices satisfying a specific commutative condition the designed minimum distance 2t 1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity check matrix with classical minimum distance 4t m where the parameter m 0 m 2t depends on a property of the parity check matrix as m decreases there is an increasing set of additional correctable error operators beyond the designed error correcting capability t the asymptotic coding efficiency is at least comparable to that of css codes a class of quantum reed muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum reed muller codes previously developed in the literature quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters

cyclic codes over z sub 4 locator polynomials and newton s identities certain nonlinear binary codes contain more codewords than any comparable linear code presently known these include the kerdock 1972 and preparata 1968 codes that can be very simply constructed as binary images under the gray map of linear codes over z sub 4 that are defined by means of parity checks involving galois rings this paper describes how fourier transforms on galois rings and elementary symmetric functions can be used to derive lower bounds on the minimum distance of such codes these methods and techniques from algebraic geometry are applied to find the exact minimum distance of a family of z sub 4 linear codes with length 2 sup m m odd and size 2 2 sup m 1 5m 2 the gray image of the code of length 32 is the best 64 2 sup 37 code that is presently known this paper also determines the exact minimum lee distance of the linear codes over z sub 4 that are obtained from the extended binary two and three error correcting bch codes by hensel lifting the gray image of the hensel lift of the three error correcting bch code of length 32 is the best 64 2 sup 32 code that is presently known this code also determines an extremal 32 dimensional even unimodular lattice

algebraic decoding of a class of binary cyclic codes via lagrange interpolation formula in this paper three algebraic decoding algorithms are proposed for the binary quadratic residue qr codes generated by irreducible polynomials the polynomial relations among the syndromes and the coefficients of the error locator polynomials have been computed with lagrange interpolation formula lif unlike some previous qr decoders which may take several iterations to decode a corrupted word the iteration number of the first two algorithms is at most one the processes in the first algorithm are the calculation of consecutive syndromes inverse free berlekamp massey algorithm ifbma and the chien search one of orsini sala s results on the structure of general error locator polynomials is generalized and applied to derive the second respectively third algorithm that consists of the determination of general error locator polynomial respectively classical error locator polynomials and the chien search finally the 17 9 5 23 12 7 and 41 21 9 qr decoders are illustrated and their complexity analyses are given

on a degenerate heat equation with a singular potential using a new method we generalize the blow up and existence result from p baras and j a goldstein 1984 trans amer math soc 284 121 139 to heat equations on the heisenberg group in doing so we need to overcome the difficulty that the equation in this case is both degenerate and of variable coefficients comparing with the euclidean case an interesting new result is that solutions can blow up even when the singularity of the potential is weaker than the inverse square of the distance function

a geometrical version of hardy s inequality we prove a version of hardy s type inequality in a domain omega subset of r n which involves the distance to the boundary and the volume of omega in particular we obtain a result which gives a positive answer to a question asked by h brezis and m marcus

a systematic construction of self dual codes a new coding construction scheme of block codes using short base codes and permutations that enables the construction of binary self dual codes is presented in cadic et al 2001 and carlach et al 1999 2000 the scheme leads to doubly even resp singly even self dual codes provided the base code is a doubly even self dual code and the number of permutations is even resp odd we study the particular case where the base code is the 8 4 4 extended hamming in this special case we construct a new 88 44 16 extremal doubly even self dual code and we give a new unified construction of the five 32 16 8 extremal doubly even self dual codes

a characterization of a 56 28 extremal self dual code corresp a 54 27 code is characterized in terms of a 27 times 27 matrix made up of circulant 3 times 3 matrices the minimum distance of this code is found to be 11 and its weight distribution is computed the 54 27 d 11 code is extended to a 56 28 d 12 external self dual code a 54 27 d 10 self dual code is also described in terms of the incidence matrix of a 27 13 6 group difference set defined on gf 3 3

encoding and decoding for the minimization of message symbol error rates in linear block codes given any fixed linear block code the error rates for the message symbols depend both on the encoding function and on the decoding map this research shows how to optimize the choice of a generator matrix and decoding map simultaneously to minimize the error rates for all message symbols the model used assumes that the distribution of messages is flat and that the distribution of error vectors defining the channel is independent of the message transmitted in addition it is shown that with proper choice of coset leaders standard array decoding is optimal in this circumstance the results generalize previously known results on unequal error protection and are sufficiently general to apply when a code is used for error detection only

the existence of certain extremal 54 27 10 self dual codes some new extremal binary 54 27 10 self dual codes are constructed using automorphisms of order 7

some new extremal self dual codes with lengths 44 50 54 and 58 we construct extremal self dual codes with lengths 44 50 54 and 58 they have weight enumerators for which extremal codes were previously not known to exist two methods are used for constructing the codes using self dual codes of same or smaller length to obtain the codes we use a combinatorial optimization search

the covering radius of extremal self dual code d11 and its application d11 constructed by conway and sloane is an extremal self dual 56 28 12 code with weight enumerator 1 8190y sup 12 622314y sup 16 11699688y sup 20 we find that the covering radius of d11 is 10 this shows that the inequality r c sub 56 spl les 10 cannot be improved any more where r c sub 56 is the covering radius of any extremal doubly even 56 28 12 code c sub 56 and we use it to get ten new 58 29 10 codes

automorphisms of codes with applications to extremal doubly even codes of length 48 general results on automorphisms of self dual binary codes are given these results are applied to the study of extremal self dual doubly even binary codes of length 48 the main theorem proved is that an extremal self dual doubly even code of length 48 with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code interesting constructions of the binary extended golay code as well as a conjecture about a possible connection between an extremal self dual doubly even code of length 72 and an extremal quaternary code of length 24 arc yielded by techniques used in the proof

a new extremal self dual code of length 64 in this correspondence we construct an extremal singly even self dual 64 32 12 code with a weight enumerator which was not previously known to exist

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

some restrictions on weight enumerators of singly even self dual codes in this correspondence we give some restrictions on weight enumerators of singly even self dual n n 2 d codes whose shadows have minimum weight d 2 as a consequence we determine the weight enumerators for which there is an extremal singly even self dual 40 20 8 code and an optimal singly even self dual 50 25 10 code

linear parabolic equations with strong singular potentials using an extension of a recent method of cabre and martel 1999 we extend the blow up and existence result in the paper of baras and goldstein 1984 to parabolic equations with variable leading coefficients under almost optimal conditions on the singular potentials this problem has been left open in baras and goldstein these potentials lie at a borderline case where standard theories such as the strong maximum principle and boundedness of weak solutions fail even in the special case when the leading operator is the laplacian we extend a recent result in cabre and martel from bounded smooth domains to unbounded nonsmooth domains