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

attractors for second order lattice dynamical systems abstract we consider the existence and the approximation of the global attractor for second order damped lattice dynamical systems

synchronization in lattices of coupled oscillators abstract we consider coupled nonlinear oscillators with external periodic forces and the dirichlet boundary conditions we prove that synchronization occurs provided that the coupling is dissipative and the coupling coefficients are sufficiently large the synchronization here is of an obvious type the size of an attractor is comparable to the difference of the subsystems

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

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

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

pattern formation in the brusselator system in the paper we deal with a reaction diffusion system well known as the brusselator model and some improved results for the steady states of this model are presented we first give an a priori estimates positive upper and lower bounds of positive steady states then we obtain the non existence and existence of positive non constant steady states as the parameters and b are varied which means some certain conditions under which the pattern formation occurs or not

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

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

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

fractal dimension for fractal structures a hausdorff approach revisited abstract in this paper we use fractal structures to study a new approach to the hausdorff dimension from both continuous and discrete points of view we show that it is possible to generalize the hausdorff dimension in the context of euclidean spaces equipped with their natural fractal structure to do this we provide three definitions of fractal dimension for a fractal structure and study their relationships and mathematical properties one of these definitions is in terms of finite coverings by elements of the fractal structure we prove that this dimension is equal to the hausdorff dimension for compact subsets of euclidean spaces this may be the key for the creation of new algorithms to calculate the hausdorff dimension of these kinds of space

stationary patterns caused by cross diffusion for a three species prey predator model we study a system arising from a three component prey predator model with prey dependent and ratio dependent functional responses where cross diffusion is included in such a way that predator chases the prey and the prey runs away from the predator we prove that the cross diffusion can generate the stationary patterns nonconstant positive steady states

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

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

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

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

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

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

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

spatial pattern in an ecosystem of phytoplankton nutrient from remote sensing abstract in this paper we consider an ecosystem of phytoplankton nutrient which comes from pelagic zone of the ocean by using remote sensing we are concerned with the pattern formation of the model we first give the asymptotic behavior of positive solutions of parabolic dynamical system then we discuss the stability of constant positive solution and the boundedness of stationary solutions finally the non existence and existence results of the pattern are investigated by considering some parameters the result shows that under certain range of these parameters the pattern of the model occurs

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

non constant positive steady states of the sel kov model abstract this paper deals with the reaction diffusion system known as the sel kov model with the homogeneous neumann boundary condition this model has been applied to various problems in chemistry and biology we first give a priori estimates positive upper and lower bounds of positive steady states and then study the non existence bifurcation and global existence of non constant positive steady states as the parameters and are varied

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

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

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

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

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

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

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

stationary patterns of strongly coupled prey predator models abstract we study some elliptic systems arising from 3 component predator prey models where cross diffusions are included in such a way that predator chases the prey and the prey runs away from the predator we establish the existence and non existence of non constant positive solutions our results show that the cross diffusions can create the stationary patterns

exponential attractor and its fractal dimension for a second order lattice dynamical system abstract we construct an exponential attractor for a second order lattice dynamical system with nonlinear damping arising from spatial discretization of wave equations in r k and we obtain fractal dimension of the exponential attractor and its finite dimensional approximation

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

coexistence states for a diffusive one prey and two predators model with b d functional response abstract in this paper we study a diffusive one prey and two predators system with beddington deangelis functional response the sufficient and necessary conditions for the existence of coexistence states are obtained by means of the fixed point index theory in addition the stability and uniqueness of coexistence states are investigated finally we give the sufficient conditions for extinction and permanence of the time dependent system

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

positive steady state solutions of the noyes field model for belousov zhabotinskii reaction abstract in the paper we investigate the noyes field model for belousov zhabotinskii reaction and study positive steady state solutions of this model with the homogeneous neumann boundary condition we obtain the existence and non existence of non constant positive steady state solutions

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

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

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

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

the analysis of a finite element method for the three species lotka volterra competition diffusion with dirichlet boundary conditions a galerkin finite element method is developed for the two dimensional three dimensional nonlinear time dependent three species lotka volterra competition diffusion equations on a bounded domain the existence and uniqueness of the solution to the numerical formulation are proved an error estimate for the numerical solution is obtained numerical computations are carried out to examine the expected orders of accuracy in the error estimates

coupled map lattices some topological and ergodic properties abstract lattice dynamical systems ldss form the class of extended systems that is the intermediate one between partial differential equations pdes and cellular automata the most popular class of ldss is formed by coupled map lattices cmls while being introduced rather recently ldss allowed already to clarify and to define exactly some notions that form the basis of the modern phenomenological theory of spatio temporal dynamics and to obtain some new and even rigorous results on space time chaos intermittency and pattern formation we discuss the rigorous results that were obtained in this area but eventually give also some applications

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

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

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

instability of stationary solutions for a lotka volterra competition model with diffusion abstract we consider a lotka volterra competition model with diffusion onrwhich describes the dynamics of the population of two competing species and study the stability of positive stationary solutions of the model relative to the spacexof bounded uniformly continuous functions with the supremum norm in consideration of earlier results we shall arrive at the study of the distribution of the real eigenvalues of the linearized operator around the stationary solution and prove the existence of only one positive eigenvalue whose eigenfunction exponentially decays to 0 as this suggests that the instability of the stationary solution is not due to the spacex

the n dimensional key equation and a decoding application the author introduce the n dimensional key equation which exhibits the error locator polynomial of an n dimensional cyclic code as a product of n univariate polynomials and the error evaluator polynomial as an n variable polynomial they then reinterpret these polynomials in the context of linear recurring sequences in particular they reduce the decoding problem to successive application of the berlekamp massey algorithm with this new method they are able to decode up to half their minimum distance many codes in a table of 2 d cyclic codes due to jensen 1985

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

stationary patterns created by cross diffusion for the competitor competitor mutualist model abstract in this paper we consider a competitor competitor mutualist model with cross diffusion we prove some existence and non existence results concerning non constant positive steady states patterns in particular we demonstrate that the cross diffusion can create patterns when the corresponding model without cross diffusion fails

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

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

stationary patterns for a prey predator model with prey dependent and ratio dependent functional responses and diffusion abstract we study a reaction diffusion system arising from a three species prey predator model with prey dependent and ratio dependent functional responses the main aim of this paper is to study the global existence and bifurcation of non constant positive steady states in particular we will show that even though the unique positive constant steady state is stable for the ode dynamics non constant positive steady states exist for the reaction diffusion system this demonstrates that stationary patterns arise as a result of diffusion

pattern formation of a coupled two cell brusselator model abstract in this paper we study the stationary problems for the coupled two cell brusselator model as follows d 1 u 1 b 1 u b u 2 v c w u in d 2 v u u 2 v in d 1 w 1 b 1 w b w 2 z c u w in d 2 z w w 2 z in u v w z 0 on we first study the stability of the unique positive constant solution u v w z 1 1 1 1 then we give a priori estimate positive upper and lower bounds to the positive solution at last we obtain the non existence and existence of positive non constant solutions as parameters d 1 d 2 and b varied

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

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

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

existence and instability of neumann layer solutions for a 3 component lotka volterra model with diffusion abstract we consider a 3 component lotka volterra model with diffusion which describes the dynamics of the population of two competing prey and one predator and we discuss the existence of positive stationary solutions and their stability property to do this the singular perturbation method and the associated singular limit spectral analysis are employed

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

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

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

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