levels of knotting of spatial handlebodies if h is a spatial handlebody i e a handlebody embedded in the 3 sphere a spine of h is a graph s3 such that h is a regular neighbourhood of usually h is said to be unknotted if it admits a planar spine this suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties building on this remark we define several levels of knotting of spatial handlebodies and we provide a complete description of the relationships between these levels focusing our attention on the case of genus 2 we also relate the knotting level of a spatial handlebody h to classical topological properties of its complement m s3 h such as its cut number more precisely we show that if h is not highly knotted then m admits special cut systems for m and we discuss the extent to which the converse implication holds along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies these obstructions are based on recent quandle coloring invariants for spatial handlebodies on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links on the analysis of appropriate coverings of handlebody complements and on the study of the classical alexander elementary ideals of their fundamental groups

upper bounds on generalized distances we derive new asymptotic bounds for generalized distances our approach extends the classical hamming plotkin and elias bounds the latter bound involves extending the definition of generalized distances to nonlinear codes

a novel adaptive finite element method for the ground state solution of bose einstein condensates abstract in this study we propose a novel adaptive finite element method for the ground state solution of bose einstein condensates bec different from the classical adaptive scheme applied to bec which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low dimensional correction space further the linear elliptic boundary value problem is solved by adaptive multigrid iterations since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces the solving efficiency can be improved evidently in addition the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically

vectorized multigrid poisson solver for the cdc cyber 205 the full multigrid fmg method is applied to the two dimensional poisson equation with dirichlet boundary conditions this has been chosen as a relatively simple test case for examining the efficiency of fully vectorizing of the multigrid method data structure and programming considerations and techniques are discussed accompanied by performance details

the weight hierarchies of some product codes bounds on the weight hierarchies of the product of two simplex codes two first order reed muller codes and the product of a simplex code and a first order reed muller code are determined the weight hierarchies of the product of two hamming codes and the product of a hamming code and an even weight code are also discussed

bounds on the weight hierarchies of linear codes of dimension 4 the weight hierarchy of a linear n k q code c over gf q is the sequence d sub 1 d sub 2 d sub k where d sub r is the smallest support of an r dimensional subcode of c the codes of dimension 4 are collected in classes for each class bounds and extremal codes are discussed

on superregular matrices and mdp convolutional codes superregular matrices are a type of lower triangular toeplitz matrix that arises in the context of constructing convolutional codes having a maximum distance profile these matrices are characterized by the property that the only submatrices having a zero determinant are those whose determinants are trivially zero due to the lower triangular structure in this paper we discuss how superregular matrices may be used to construct codes having a maximum distance profile we also present an upper bound on the minimum size a finite field must have in order that a superregular matrix of a given size can exist over that field this in turn gives an upper bound on the smallest field size over which an mdp n k convolutional code can exist

there are no unexpected tunnel number one knots of genus one author s scharlemann martin abstract we show that the only knots that are tunnel number one and genus one are those that are already known 2 bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by eudave munoz and by morimoto and sakuma this confirms a conjecture first made by goda and teragaito

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

the duals of mmd codes are proper for error detection a linear code when used for error detection on a symmetric channel is said to be proper if the corresponding undetected error probability increases monotonically in spl epsiv the symbol error probability of the channel such codes are generally considered to perform well in error detection a number of well known classes of linear codes are proper e g the perfect codes mds codes macdonald s codes mmd codes and some near mds codes the aim of this work is to show that also the duals of mmd codes are proper

the mmd codes are proper for error detection the undetected error probability of a linear code used to detect errors on a symmetric channel is a function of the symbol error probability spl epsi of the channel and involves the weight distribution of the code the code is proper if the undetected error probability increases monotonously in spl epsi proper codes are generally considered to perform well in error detection we show in this correspondence that maximum minimum distance mmd codes are proper

superregular matrices and applications to convolutional codes abstract the main results of this paper are twofold the first one is a matrix theoretical result we say that a matrix is superregular if all of its minors that are not trivially zero are nonzero given a a b a b superregular matrix over a field we show that if all of its rows are nonzero then any linear combination of its columns with nonzero coefficients has at least a b 1 nonzero entries secondly we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code namely the rate and the forney indices these results answer some open questions on distances and constructions of convolutional codes posted in the literature 6 9

a numerical study of the two dimensional navier stokes equations in vorticity velocity variables abstract solution methods for compact finite difference schemes are applied to the vorticity velocity form of the two dimensional unsteady navier stokes equations numerical experiments for stagnation point and driven cavity flows are described

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

driven cavity flows by efficient numerical techniques abstract efficient and reliable numerical techniques of high order accuracy are presented for solving problems of steady viscous incompressible flow in the plane and are used to obtain accurate solutions for the driven cavity a solution is obtained at reynolds number 10 000 on a 180 180 grid the numerical methods combine an efficient linear system solver an adaptive newton like method for nonlinear systems and a continuation procedure for following a branch of solutions over a range of reynolds numbers

a new class of superregular matrices and mdp convolutional codes abstract this paper deals with the problem of constructing superregular matrices that lead to mdp convolutional codes these matrices are a type of lower block triangular toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular we present a new class of matrices that are superregular over a sufficiently large finite field f such construction works for any given choice of characteristic of the field f and code parameters n k such that n k we also discuss the size of f needed so that the proposed matrices are superregular

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

the decoding error probability of linear codes over the erasure channel in this paper we study the decoding error probability of linear codes over the erasure channel under the list decoding the notion of the q ell incorrigible sets of linear codes is introduced to characterize its decoding error probability under the list decoding or the maximum likelihood decoding by calculating the q ell incorrigible set distributions the decoding error probability of a linear code over the erasure channel under the list decoding or the maximum likelihood decoding is expressed by its support weight distributions for the ensemble of all n k linear codes the average decoding error probability under the maximum likelihood decoding and the average unsuccessful decoding probability under unambiguous decoding are determined furthermore the error exponent of the average unsuccessful decoding probability under the unambiguous decoding is determined for the ensemble of all n nr linear codes

on generator matrices of mds codes corresp it is shown that the family of q ary generalized reed solomon codes is identical to the family of q ary linear codes generated by matrices of the form i a where i is the identity matrix and a is a generalized cauchy matrix using cauchy matrices a construction is shown of maximal triangular arrays over gf q which are constant along diagonals in a hankel matrix fashion and with the property that every square subarray is a nonsingular matrix by taking rectangular subarrays of the described triangles it is possible to construct generator matrices i a of maximum distance separable codes where a is a hankel matrix the parameters of the codes are n k d for 1 leq n leq q 1 1 leq k leq n and d n k 1

bounds on the minimum support weights the minimum support weight d sub r c of a linear code c over gf q is the minimal size of the support of an r dimensional subcode of c a number of bounds on d sub r c are derived generalizing the plotkin bound and the griesmer bound as well as giving two new existential bounds as the main result it is shown that there exist codes of any given rate r whose ratio d sub rd sub 1 is lower bounded by a number ranging from q sup r 1 q sup r q sup r 1 to r depending on r

the fast adaptive composite grid method fac algorithms for advanced computers fac is a fast systematic process for the numerical solution of partial differential equations on adapted grids this paper reports on numerical studies with fac in advancing computing environments including simulations with multiprocessor array architectures and actual tests on the hep we also analyze the parallel complexity of fac for general local grid applications

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

the weight hierarchy of product codes the weight of a code is the number of coordinate positions where no codeword is zero the rth minimum weight d sub r is the least weight of an r dimensional subcode wei and yang 1993 gave a conjecture about the minimum weights for some product codes in this article we find a relation between product codes and the segre embedding of a pair of projective systems and we use this to prove the conjecture

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

successive overrelaxation multigrid and preconditioned conjugate gradients algorithms for solving a diffusion problem on a vector computer the purpose of this paper is the treatment of three numerical algorithms successive overrelaxation sor multigrid mg and conjugate gradients preconditioned by a fast poisson solver cg for solving large but mildly behaved diffusion problems on a vector computer with memory to memory architecture the problem is a symmetric nonnegative definite matrix equation arising from a cell centered finite difference approximation of a 3 d diffusion equation with full neumann boundary conditions

error detection with a class of irreducible binary cyclic codes and their dual codes the irreducible binary cyclic codes of even dimension introduced by delsarte and goethals in 1970 constitute a parametric class with three parameters we show that these codes and their duals are either proper or not good for error detection

on the generalized hamming weights of convolutional codes motivated by applications in cryptology wei 1991 introduced the concept of a generalized hamming weight for a linear block code in this correspondence we define generalized hamming weights for the class of convolutional codes and we derive several of their basic properties by restricting to convolutional codes having a generator matrix g d with bounded kronecker indices we are able to derive upper and lower bounds on the weight hierarchy

on the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes it was shown earlier that for a punctured reed muller rm code or a primitive bch code which contains a punctured rm code of the same minimum distance as a large subcode the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions the problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized hamming weight hierarchy of a code and it is shown that for rm codes the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to v k wei 1991 the state complexity of the trellis diagram for the extended and permuted 64 24 bch code is discussed

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

on the probability of undetected error for the maximum distance separable codes in this paper we investigate the performance of maximum distance separable codes with symbols from gf q when they are used for pure error detection or for simultaneous error correction and detection over a q input and q output discret memoryless channel with symbol error probability e first we show that the probability of undetected error for an mds code used for pure error detection is upper bounded by q r and decreases monotonically as edecreases from q 1 q to 0 where r is the number of parity check symbols of the code then we show that the probability of undetected error for an mds code used for correcting t or fewer symbol errors is upper bounded by q r sum min i 0 max t min i max n q 1 i and decreases monotonically as e decreases from q 1 q to 0 these results show that the mds codes are effective for both pure error detection and simultaneous error correction and detection

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

weight hierarchy of a product code formulas to compute the first members of the weight hierarchy of a product code are given and proved basic patterns for the supports of subcodes in which the minimum cardinalities can be found are presented hence formulas for every term of the hierarchy can be given and the proof for each case will be a generalization of the proofs shown here

the weight hierarchies of some product codes we determine the weight hierarchies of the product of an n tuple space and an arbitrary code the product of an m dimensional even weight code and the 24 12 8 extended golay code and the product of an m dimensional even weight code and the 8 4 4 extended hamming code the conjecture d sub r d sub r is proven for all three cases

the intractability of computing the minimum distance of a code it is shown that the problem of computing the minimum distance of a binary linear code is np hard and the corresponding decision problem is np complete this result constitutes a proof of the conjecture of berlekamp mceliece and van tilborg 1978 extensions and applications of this result to other problems in coding theory are discussed

almost mds and near mds codes for error detection the error detection capability of almost mds amds and nearly mds nmds codes is studied necessary and sufficient conditions for the codes to be proper or good for error detection are given

on the weight hierarchy of codes coming from semigroups with two generators the weight hierarchy of one point algebraic geometry codes can be estimated by means of the generalized order bounds which are described in terms of a certain weierstrass semigroup the asymptotical behavior of such bounds for r 2 differs from that of the classical feng rao distance r 1 by the so called feng rao numbers this paper is addressed to compute the feng rao numbers for numerical semigroups of embedding dimension two with two generators obtaining a closed simple formula for the general case by using numerical semigroup techniques these involve the computation of the apery set with respect to an integer of the semigroups under consideration the formula obtained is applied to lower bounding the generalized hamming weights improving the bound given by kirfel and pellikaan in terms of the classical feng rao distance we also compare our bound with a modification of the griesmer bound improving this one in many cases

the weight hierarchies of q ary codes of dimension 4 the weight hierarchy of a linear n k q code c over gf q is the sequence d sub 1 d sub 2 d sub k where d sub spl tau is the smallest support of an spl tau dimensional subcode of c the possible weight hierarchies of n 4 q codes are studied in particular the possible weight hierarchies of n 4 3 codes are determined

self dual near mds codes from elliptic curves in recent years self dual mds codes have attracted a lot of attention due to theoretical interest and practical importance similar to self dual mds codes self dual near mds nmds for short codes have nice structures as well from both theoretical and practical points of view it is natural to study self dual nmds codes although there has been lots of work on nmds codes in literature little is known for self dual nmds codes it seems more challenging to construct self dual nmds codes than self dual mds codes the only work on construction of self dual nmds codes shows existence of q ary self dual nmds codes of length q 1 for odd prime power q or length up to 16 for some small primes q with q le 197 in this paper we make use of properties of elliptic curves to construct self dual nmds codes it turns out that as long as 2 q and n is even with 4 le n le q lfloor 2 sqrt q rfloor 2 one can construct a self dual nmds code of length n over mathbb f q furthermore for odd prime power q there exists a self dual nmds code of length n over mathbb f q if q ge 4 n 3 times n 3 2

code construction for the noiseless binary switching multiple access channel the noiseless coding problem is considered for a discrete memoryless multiple access channel that is a counterpart to the well known binary adder channel upper and lower bounds on the number of code words in a uniquely decodable code pair are given from which the zero error capacity region of the channel is derived this region coincides with the classical capacity region of this channel the proof uses the notion of second order distance of a code for several values of n and k good code pairs of block length n are constructed with the first code being n k linear some of these are found to be optimal some convolutional codes are investigated that yield additional good rate pairs

minimum support weights of binary codes some relations between the minimum support weights are discussed in particular the possible weight hierarchies of codes of dimension 4 are determined

on multigrid for overlapping grids the solution of elliptic partial differential equations on composite overlapping grids using multigrid is discussed an approach is described that provides a fast and memory efficient scheme for the solution of boundary value problems in complex geometries the key aspects of the new scheme are an automatic coarse grid generation algorithm an adaptive smoothing technique for adjusting residuals on different component grids and the use of local smoothing near interpolation boundaries other important features include optimizations for cartesian component grids the use of over relaxed red black smoothers and the generation of coarse grid operators through galerkin averaging numerical results in two and three dimensions show that very good multigrid convergence rates can be obtained for dirichlet neumann and mixed boundary conditions a comparison to krylov based solvers shows that the multigrid solver can be much faster and require significantly less memory

a new upper bound on the block error probability after decoding over the erasure channel motivated by cryptographic applications we derive a new upper bound on the block error probability after decoding over the erasure channel the bound works for all linear codes and is in terms of the generalized hamming weights it turns out to be quite useful for reed muller codes for which all the generalized hamming weights are known whereas the full weight distribution is only partially known for these codes the error probability is related to the cryptographic notion of algebraic immunity we use our bound to show that the algebraic immunity of a random balanced m variable boolean function is of order m 2 1 o 1 with probability tending to 1 as m goes to infinity

matrix characterization of linear codes with arbitrary hamming weight hierarchy the support of an n k linear code c over a finite field f q is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position the rth generalized hamming weight d r c 1 leq r leq k of c is defined as the minimum of the cardinalities of the supports of all n r subcodes of c the sequence d 1 c d 2 c d k c is called the hamming weight hierarchy hwh of c the hwh d r c n k r r 1 2 k characterizes maximum distance separable mds codes therefore the matrix characterization of mds codes is also the characterization of codes with the hwh dr c n k r r 1 2 k a linear code c with systematic check matrix i p where i is the n k x n k identity matrix and p is a n k x k matrix is mds iff every square submatrix of p is nonsingular in this paper we extend this characterization to linear codes with arbitrary hwh using this result we characterize near mds codes near near mds n 2 mds codes and a mu mds codes the mds rank of c is the smallest integer n such that d n 1 n k n 1 and the defect vector of c with mds rank n is defined as the ordered set mu1 c mu2 c mu3 c mu n c mu n 1 c where mu i c n k i di c we call c a dually defective code if the defect vector of the code and its dual are the same we also discuss matrix characterization of dually defective codes further the codes meeting the generalized greismer bound are characterized in terms of their generator matrix the hwh of dually defective codes meeting the generalized greismer bound are also reported