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

nonlinear scattering theory at low energy abstract we study the scattering theory of a conservative nonlinear one parameter group of operators on a hilbert space x relative to a group of linear unitary operators under certain hypotheses the scattering operator carries a neighborhood of 0 in x into x the theory is designed to apply to the semilinear schrodinger and klein gordon equations

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

the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

decoding the golay code with venn diagrams a decoding algorithm based on venn diagrams for decoding the 23 12 7 golay code is presented the decoding algorithm is based on the design properties of the parity sets of the code as for other decoding algorithms for the golay code decoding can be easily done by hand

on the decoding of the 24 12 8 golay code an improved syndrome shift register decoding algorithm called the syndrome weight decoding algorithm is proposed for decoding three possible errors and detecting four errors in the 24 12 8 golay code this method can also be extended to decode two other short codes such as the 15 5 7 cyclic code and the 31 16 7 quadratic residue qr code the proposed decoding algorithm makes use of the properties of cyclic codes the weight of syndrome and the syndrome decoder with a reduced size lookup table rslt in order to reduce the number of syndromes and their corresponding coset leaders this approach results in a significant reduction in the memory requirement for the lookup table thereby yielding a faster decoding algorithm simulation results show that the decoding speed of the proposed algorithm is approximately 3 6 times faster than that of the algebraic decoding algorithm

blow up of solutions to semilinear wave equations with variable coefficients and boundary this paper is devoted to studying initial boundary value problems for semilinear wave equations and derivative semilinear wave equations with variable coefficients on exterior domain with subcritical exponents in n space dimensions we will establish blow up results for the initial boundary value problems it is proved that there can be no global solutions no matter how small the initial data are and also we give the life span estimate of solutions for the problems

two step runge kutta methods of order over five with reduced function evaluations two step runge kutta formulae of order five with reduced function evaluations are discussed and both of the order conditions for two step runge kutta formulae of order five and six are offered numerical experiments show these new formulae are feasible and effective

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

weighted strichartz estimates and global existence for semilinear wave equations in this paper we prove a sharp global existence theorem in all dimensions for nonlinear wave equations with power type nonlinearities the proof is based on a weighted strichartz estimate involving powers of the lorentz distance

finite time blow up for critical wave equations in high dimensions we prove that solutions to the critical wave equation 1 1 with dimension n 4 can not be global if the initial values are positive somewhere and nonnegative this completes the solution to the famous strauss conjecture about semilinear wave equations of the form u t2u u p 0 the rest of the cases the lower dimensional case n 3 and the sub or super critical cases were settled many years earlier by the work of several authors

on the existence and nonexistence of global solutions for certain semilinear exterior problems with nontrivial robin boundary conditions abstract we consider three types of semilinear equations elliptic parabolic and hyperbolic posed in the n dimensional exterior domain r n d where n 2 and d is the closed unit ball in r n a nontrivial robin boundary condition is imposed on the boundary of d using a test function approach with judicious choices of the test functions we show that the considered problems share a common critical behavior we discuss separately the cases n 2 and n 3 moreover in the case n 3 the dependence of the critical exponent on initial data is discussed to the best our knowledge the study of the critical behavior in an exterior domain with a nontrivial robin boundary condition has never been studied in the literature

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

decoding the golay codes we introduce exceptionally simple decoding algorithms for the two extended golay codes the algorithms are based on recent methods of conway and curtis of finding the unique blocks containing five points in either the 5 8 24 steiner system or the 5 6 12 steiner system these decoding methods are simple enough to enable decoding extended golay codes by hand both of the methods involve relations between the extended golay codes and other self dual codes proofs are given explaining these relationships and why the decoding methods work the decoding algorithms are explained and illustrated with many examples 3 chap 12 has facts about the mathieu group and some details about decoding the golay codes

a blow up result for a nonlinear damped wave equation in exterior domain the critical case abstract we consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain we prove a blow up result which generalizes the result of non existence of global solutions of ogawa and takeda 2009 we also show that the critical exponent belongs to the blow up case

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

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

order conditions for integrators and mappings with applications abstract we regard the elementary weight function as a mapping from input to output so that the mappings allow us to derive the order conditions for a given method more easily and concisely this paper presents the order conditions of one step runge kutta methods runge kutta type methods with derivatives a class of two step runge kutta methods and three step runge kutta methods based on the elementary weight mappings the composition formula of two mappings as well as the mappings i d r e defined on the set t of all rooted trees from an input to an output it is pointed out that the order conditions can be easily achieved for linear multistep methods and second derivative linear multistep methods of order p and it suffices to consider only bushy trees of order up to p together with the empty tree 0 for a linear multistep method or a second derivative linear multistep method of order p

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

nonexistence of global solutions to semilinear wave equations in high dimensions on considere le probleme de cauchy pour l equation d onde semi lineaire 2 t 2 u x t u x t p p 1 u x o f x u t x o g x avec f g c o r n

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

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

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

new blow up phenomena for hyperbolic inequalities with combined nonlinearities abstract we study for the first time the large time behavior of solutions to certain hyperbolic inequalities with nonlinearities involving the function and its gradient namely we consider the problem u t t u u p u q f t x in 0 r n where p q 1 and f 0 f 0 we obtain general criteria for the nonexistence of global solutions to this problem next we discuss some special cases of the inhomogeneous term f in particular when n 3 and f depends only on the variable space we obtain a discontinuous fujita critical exponent p n q namely p n q 1 2 n 2 if q 1 1 n 1 and p n q if q 1 1 n 1 next we extend our study to the exterior problem u t t u u p u q in 0 d c under the boundary condition u n f t x in 0 d where d is the unit open ball in r n d c r n d and n is the outward relative to d c unit normal of d

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

two dimensional exterior mixed problem for semilinear damped wave equations abstract we consider two dimensional exterior mixed problems for a semilinear damped wave equation with a power type nonlinearity u p for compactly supported initial data which have a small energy we shall derive global in time existence results in the case when the power of the nonlinearity satisfies 2 p this generalizes a previous result of j differential equations 200 2004 53 68 which dealt with a radially symmetric solution

critical exponents of fujita type for inhomogeneous parabolic equations and systems abstract we consider the large time behavior of sign changing solutions of inhomogeneous parabolic equations and systems for example for u t u u p w x in r n 0 t we show the following if n 3 and r n w x dx 0 and 1 p n n 2 then all solutions blow up in finite time while if p n n 2 there are both global and nonglobal solutions we show by example that global solutions exist for all p 1 and w satisfying r n w x dx n 1 2 and r n w x dx 0 no solution can exist for all time extensions of the above result to various geometries and some other problems are indicated

new blow up results for nonlinear boundary value problems in exterior domains abstract we consider exterior boundary value problems with nontrivial boundary conditions including three types of semilinear equations heat equation wave equation and damped wave equation using a test function approach with a choice of a suitable family of test functions we obtain the critical exponents in the sense of fujita for the considered problems our obtained results close some open questions from the literature

critical criteria of fujita type for a system of inhomogeneous wave inequalities in exterior domains abstract we consider blow up results for a system of inhomogeneous wave inequalities in exterior domains we will handle three type boundary conditions dirichlet type neumann type and mixed boundary conditions we use a unified approach to show the optimal criteria of fujita type for each case our study yields naturally optimal nonexistence results for the corresponding stationary wave system and equation we provide many new results and close some open questions

the recursions of order conditions for some runge kutta like methods abstract it is well known that hairer and wanner reformulated butcher s approach to taylor series and introduced the name of b series in this paper the recursions of order conditions using b series are presented for some runge kutta like methods

on the minimum distance of some quadratic residue codes corresp we find the minimum distances of the binary 113 57 and ternary 37 19 61 31 71 36 and 73 37 quadratic residue codes and the corresponding extended codes these distances are 15 10 11 17 and 17 respectively for the nonextended codes and are increased by one for the respective extended codes we also characterize the minimum weight codewords for the 113 57 binary code and its extended counterpart

finding roots of polynomials over finite fields we propose an improved algorithm for finding roots of polynomials over finite fields this makes possible significant speedup of the decoding process of bose chaudhuri hocquenghem reed solomon and some other error correcting codes

non existence of weak solutions to nonlinear damped wave equations in exterior domains abstract we consider the initial boundary value problem of the nonlinear damped wave equation in an exterior domain t 2 u u t u u p t 0 x u 0 x u 0 x t u 0 x u 1 x x u 0 t 0 x when 1 p 1 2 n and the initial data u 0 u 1 h 0 1 l 2 having compact support we prove the non existence of non negative global solutions of the above problem we employ the kaplan fujita h fujita on the blowing up of solutions of the cauchy problem for u t u u 1 j sci univ tokyo sec i 13 1966 109 124 s kaplan on the growth of solutions quasi linear parabolic equations comm pure appl math 16 1963 305 330 method to avoid the difficulty of the reflection from the boundary

a class of runge kutta formulae of order three and four with reduced evaluations of function traditionally an explicit runge kutta method of order three requires three evaluations of function and an explicit runge kutta method of order four requires four evaluations of function in this paper a class of runge kutta processes of order three and order four with reduced evaluations of function are proposed the numerical experiments shows that these new runge kutta processes are more attractive than the traditional ones

on fujita critical exponent for a nonlinear ultraparabolic equation in an exterior domain abstract in this paper we consider the nonlinear ultraparabolic equation t u s u u u p t 0 s 0 x d c u t s x f x t 0 s 0 x d u t 0 x u 1 t x t 0 x d c u 0 s x u 2 s x s 0 x d c where d b 0 1 is the closed unit ball in r n n 2 d c is its complement p 1 u i 0 i 1 2 and d f x d s x 0 we derive the critical exponent for the considered problem in the sense of fujita we discuss separately the cases n 2 and n 3 to the best of our knowledge this is the first work dealing with the blow up of solutions to multi time equations in an exterior domain

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