on the inequivalence of generalized preparata codes if m is odd and sigma in aut gf 2 m is such that x rightarrow x sigma 2 1 is 1 1 there is a 2 m 1 1 2 m l 2m 2 nonlinear binary code p sigma having minimum distance 5 all the codes p sigma have the same distance and weight enumerators as the usual preparata codes which rise as p sigma when x sigma x 2 it is shown that p sigma and p tau are equivalent if and only if tau sigma pm 1 and aut p sigma is determined

complete distances of all negacyclic codes of length 2 s over bbz _ 2 a various kinds of distances of all negacyclic codes of length 2s over zopf2 a are completely determined using our structure theorems of negacyclic codes of length 2 s over zopf2 a we first calculate the hamming distances of all such negacyclic codes which particularly lead to the hamming weight distributions and hamming weight enumerators of several codes these hamming distances are then used to obtain their homogeneous lee and euclidean distances our techniques are extendable to the more general class of constacyclic codes namely the lambda constacyclic codes of length 2s over zopf2 a where lambda is any unit of zopf2 a with the form 4k 1 we establish the hamming homogeneous lee and euclidean distances of all such constacyclic codes

cyclic and negacyclic codes over finite chain rings the structures of cyclic and negacyclic codes of length n and their duals over a finite chain ring r are established when n is not divisible by the characteristic of the residue field r some cases where n is divisible by the characteristic of the residue field r are also considered namely the structure of negacyclic codes of length 2 sup t over spl zopf sub 2 sup m and that of their duals are derived

nested cartesian grid method in incompressible viscous fluid flow in this work the local grid refinement procedure is focused by using a nested cartesian grid formulation the method is developed for simulating unsteady viscous incompressible flows with complex immersed boundaries a finite volume formulation based on globally second order accurate central difference schemes is adopted here in conjunction with a two step fractional step procedure the key aspects that needed to be considered in developing such a nested grid solver are proper imposition of interface conditions on the nested block boundaries and accurate discretization of the governing equations in cells that are with block interface as a control surface the interpolation procedure adopted in the study allows systematic development of a discretization scheme that preserves global second order spatial accuracy of the underlying solver and as a result high efficiency accuracy nested grid discretization method is developed herein the proposed nested grid method has been widely tested through effective simulation of four different classes of unsteady incompressible viscous flows thereby demonstrating its performance in the solution of various complex flow structure interactions the numerical examples include a lid driven cavity flow and pearson vortex problems flow past a circular cylinder symmetrically installed in a channel flow past an elliptic cylinder at an angle of attack and flow past two tandem circular cylinders of unequal diameters for the numerical simulations of flows past bluff bodies an immersed boundary ib method has been implemented in which the solid object is represented by a distributed body force in the navier stokes equations the main advantages of the implemented immersed boundary method are that the simulations could be performed on a regular cartesian grid and applied to multiple nested block cartesian structured grids without any difficulty through the numerical experiments the strength of the solver in effectively accurately simulating various complex flows past different forms of immersed boundaries is extensively demonstrated in which the nested cartesian grid method was suitably combined together with the fractional step algorithm to speed up the solution procedure

constacyclic codes of length p s over fpm ufpm abstract for any prime p all constacyclic codes of length p s over the ring r f p m u f p m are considered the units of the ring r are of the forms and u where and are nonzero elements of f p m which provides p m p m 1 such constacyclic codes first the structure and hamming distances of all constacyclic codes of length p s over the finite field f p m are obtained they are used as a tool to establish the structure and hamming distances of all u constacyclic codes of length p s over r we then classify all cyclic codes of length p s over r and obtain the number of codewords in each of those cyclic codes finally a one to one correspondence between cyclic and constacyclic codes of length p s over r is constructed via ring isomorphism which carries over the results regarding cyclic codes corresponding to constacyclic codes of length p s over r

negacyclic codes over z sub 4 of even length this paper generalizes the results from wolfmann see ibid vol 45 p 2527 2532 nov 1999 and vol 47 p 1773 1779 july 2001 classifying all negacyclic codes over z sub 4 of even length using a transform approach it is then shown which linear binary cyclic codes are images of negacyclic codes under the gray map in the process the concatenated structure of both negacyclic codes and binary repeated root cyclic codes is given

a transform approach to permutation groups of cyclic codes over galois rings berger and charpin see ibid vol 42 p 2194 2209 1996 and des codes cuyptogr vol 18 no 1 3 p 29 53 1999 devised a theoretical method of calculating the permutation group of a primitive cyclic code over a finite field using permutation polynomials and a transform description of such codes we extend this method to cyclic and extended cyclic codes over the galois ring gr p sup a m developing a generalization of the mattson solomon polynomial in particular we classify all affine invariant codes of length 2 sup m over z sub 4 thus generalizing the corresponding result of kasami lin and peterson 1967 and giving an alternative proof to abdukhalikov we give a large class of codes over z sub 4 with large permutation groups which include generalizations of bose chaudhuri hocquenghem bch and reed muller rm codes

log periodic corrections to the cole cole expression in dielectric relaxation abstract a model of the self similar process of relaxation is given and a method of derivation of the kinetic equations for the total polarization based on the ideas of fractional kinetics is suggested the derived kinetic equations contain integro differential operators having non integer order they lead to the cole cole expression for the complex dielectric permittivity it is shown rigorously that the power law exponent in the cole cole expression coincides with the dimension of the mixed space temporal fractal ensemble if the discrete scale invariance for the temporal space structure of the dielectric medium considered becomes important then the expression for the complex dielectric permittivity contains log periodic corrections oscillations and hence it generalizes the conventional cole cole expression the corrections obtained in this model suggest another way of interpretation and analysis of dielectric spectra for different complex materials

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

on repeated root cyclic codes a parity check matrix for a q ary repeated root cyclic code is derived using the hasse derivative then the minimum distance of a q ary repeated root cyclic code is expressed in terms of the minimum distance of a certain simple root cyclic code with the help of this result several binary repeated root cyclic codes of lengths up to n 62 are shown to contain the largest known number of codewords for their given length and minimum distance the relative minimum distance d sub min n of q ary repeated root cyclic codes of rate r or r is proven to tend to zero as the largest multiplicity of a root of the generator g x increases to infinity it is further shown that repeated root cycle codes cannot be asymptotically better than simple root cyclic codes

a conservative finite volume method for incompressible navierstokes equations on locally refined nested cartesian grids a second order accurate finite volume method is developed for the solution of incompressible navierstokes equations on locally refined nested cartesian grids numerical accuracy and stability on locally refined nested cartesian grids are achieved using a finite volume discretization of the incompressible navierstokes equations based on higher order conservation principles i e in addition to mass and momentum conservation kinetic energy conservation in the inviscid limit is used to guide the selection of the discrete operators and solution algorithms hanging nodes at the interface are virtually slanted to improve the pressurevelocity projection while the other parts of the grid maintain an orthogonal cartesian grid topology the present method is straight forward to implement and shows superior conservation of mass momentum and kinetic energy compared to the conventional methods employing interpolation at the interface between coarse and fine grids

repeated root cyclic codes in the theory of cyclic codes it is common practice to require that n q 1 where n is the word length and f sub q is the alphabet it is shown that the even weight subcodes of the shortened binary hamming codes form a sequence of repeated root cyclic codes that are optimal in nearly all other cases one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime this statement is based on an analysis for lengths up to 100 a theorem shows why this was to be expected but it also leads to low complexity decoding methods this is an advantage especially for the codes that are not much worse than corresponding codes of odd length it is demonstrated that a binary cyclic code of length 2n n odd can be obtained from two cyclic codes of length n by the well known mod u mod u v mod construction this leads to an infinite sequence of optimal cyclic codes with distance 4 furthermore it is shown that low complexity decoding methods can be used for these codes the structure theorem generalizes to other characteristics and to other lengths some comparisons of the methods using earlier examples are given

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

binary images of cyclic codes over z sub 4 we determine all linear cyclic codes over z sub 4 of odd length whose gray images are linear codes or equivalently whose nechaev gray 1989 image are linear cyclic codes or are linear cyclic codes

anomalous diffusion with under and overshooting subordination a competition between the very large jumps in physical and operational times in this paper we present an approach to anomalous diffusion based on subordination of stochastic processes application of such a methodology to analysis of the diffusion processes helps better understanding of physical mechanisms underlying the nonexponential relaxation phenomena in the subordination framework we analyze a coupling between the very large jumps in physical and two different operational times modeled by under and overshooting subordinators respectively we show that the resulting diffusion processes display features by means of which all experimentally observed two power law dielectric relaxation patterns can be explained we also derive the corresponding fractional equations governing the spatiotemporal evolution of the diffusion front of an excitation mode undergoing diffusion in the system under consideration the commonly known type of subdiffusion corresponding to the mittag leffler or cole cole relaxation appears as a special case of the studied anomalous diffusion processes

negacyclic and cyclic codes over z sub 4 the negashift spl nu of z sub 4 sup n is defined as the permutation of z sub 4 sup n such that spl nu a sub 0 a sub 1 spl middot spl middot spl middot a sub i spl middot spl middot spl middot a sub n 1 a sub n 1 a sub 0 spl middot spl middot spl middot a sub i spl middot spl middot spl middot a sub n 2 and a negacyclic code of length n over z sub 4 is defined as a subset c of z sub 4 sup n such that spl nu c c we prove that the gray image of a linear negacyclic code over z sub 4 of length n is a binary distance invariant not necessary linear cyclic code we also prove that if n is odd then every binary code which is the gray image of a linear cyclic code over z sub 4 of length n is equivalent to a not necessary linear cyclic code and this equivalence is explicitely described this last result explains and generalizes the existence already known of versions of kerdock preparata and others codes as doubly extended cyclic codes furthermore we introduce a family of binary linear cyclic codes which are gray images of z sub 4 linear negacyclic codes

constacyclic codes of length 2 s over galois extension rings of bbf _ 2 u bbf _2 we study all constacyclic codes of length 2s over gr rfr m the galois extension ring of dimension m of the ring rfr f2 uf2 the units of the ring gr rfr m are of the forms alpha and alpha ubeta where alpha beta are nonzero elements of f2m which correspond to 2 m 2m 1 such constacyclic codes first the structure and hamming distances of 1 ugamma constacyclic codes are established we then classify all cyclic codes of length 2s over gr rfr m and obtain a formula for the number of those cyclic codes as well as the number of codewords in each code finally one to one correspondences between cyclic and alpha constacyclic codes as well as 1 ugamma constacyclic and alpha ubeta constacyclic codes are provided via ring isomorphisms that allow us to carry over the results about cyclic and 1 ugamma constacyclic accordingly to all constacyclic codes of length 2s over gr rfr m

the permutation group of affine invariant extended cyclic codes the extended cyclic codes of length p sup m p is a prime which are invariant under the affine group acting on f sub pm are called affine invariant codes following results of berger 1996 1994 we present the formal expression of the permutation group of these codes afterwards we give several tools in order to determine effectively the group of a given code or of some infinite class of codes we next prove by studying some examples that our tools are efficient in the end we give our main application the permutation group of primitive bch codes defined on any prime field

negacyclic codes of length 2 sup s over galois rings codes over the ring of integers modulo 4 have been studied by many researchers negacyclic codes such that the length n of the code is odd have been characterized over the alphabet zopf4 and furthermore have been generalized to the case of the alphabet being a finite commutative chain ring in this paper we investigate negacyclic codes of length 2s over galois rings the structure of negacyclic codes of length 2s over the galois rings gr 2a m as well as that of their duals are completely obtained the hamming distances of negacyclic codes over gr 2a m in general and over zopf2 a in particular are studied among other more general results the hamming distances of all negacyclic codes over zopf2 a of length 4 8 and 16 are given the weight distributions of such negacyclic codes are also discussed

langevin picture of levy walks and their extensions in this paper we derive langevin picture of levy walks applying recent advances in the theory of coupled continuous time random walks we find a limiting process of the properly scaled levy walk next we introduce extensions of levy walks in which jump sizes are some functions of waiting times we prove that under proper scaling conditions such generalized levy walks converge in distribution to the appropriate limiting processes we also derive the corresponding fractional diffusion equations and investigate behavior of the mean square displacements of the limiting processes showing that different coupling functions lead to various types of anomalous diffusion

an exponential number of generalized kerdock codes if n 1 is an odd composite integer then there are at least 2 1 2 n pairwise inequivalent binary error correcting codes of length 2 n size 2 2 n and minimum distance 2 n 1 2 1 2 n 1

multistability and symmetry breaking in the two dimensional flow around a square cylinder we use numerical methods to study two dimensional flow passing a square cylinder at low reynolds numbers and observe period 1 and 3 vortices behind the cylinder at the same reynolds number re when re increases from a small number to a critical value re c approximately equal to 320 the system could change from bistability which maintains the spatial symmetry to tristability which breaks the spatial symmetry our results suggest many interesting problems for further studies