a lower bound on the undetected error probability and strictly optimal codes error detection is a simple technique used in various communication and memory systems to enhance reliability we study the probability that a q ary linear or nonlinear block code of length n and size m fails to detect an error a lower bound on this undetected error probability is derived in terms of q n and m the new bound improves upon other bounds mentioned in the literature even those that hold only for linear codes block codes whose undetected error probability equals the new lower bound are investigated we call these codes strictly optimal codes and give a combinatorial characterization of them we also present necessary and sufficient conditions for their existence in particular we find all values of n and m for which strictly optimal binary codes exist and determine the structure of all of them for example we construct strictly optimal binary coded decimal codes of length four and five and we show that these are the only possible lengths of such codes

optimal codes for error detection the probability of undetected error for codes over gf 2 sup m having a generator matrix over gf 2 is studied the optimal codes of dimensions four or less are determined for higher dimensions some properties of optimal codes are determined

an overlapping domain decomposition preconditioner for a class of discontinuous galerkin approximations of advection diffusion problems we consider a scalar advection diffusion problem and a recently proposed discontinuous galerkin approximation which employs discontinuous finite element spaces and suitable bilinear forms containing interface terms that ensure consistency for the corresponding sparse non symmetric linear system we propose and study an additive two level overlapping schwarz preconditioner consisting of a coarse problem on a coarse triangulation and local solvers associated to suitable problems defined on a family of subdomains this is a generalization of the corresponding overlapping method for approximations on continuous finite element spaces related to the lack of continuity of our approximation spaces some interesting new features arise in our generalization which have no analog in the conforming case we prove an upper bound for the number of iterations obtained by using this preconditioner with gmres which is independent of the number of degrees of freedom of the original problem and the number of subdomains the performance of the method is illustrated by several numerical experiments for different test problems using linear finite elements in two dimensions

new upper bounds on error exponents we derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels this is an improvement on the best earlier known bounds by shannon gallager berlekamp 1967 and mceliece omura 1977 for the probability of undetected error the new bounds are better than the bounds by levenshtein 1978 1989 and the bound by abdel ghaffar see ibid vol 43 p 1489 502 1997 moreover we further extend the range of rates where the undetected error exponent is known to be exact the new bounds are based on an analysis of possible distance distributions of the codes along with some inequalities relating the distance distributions to the error probabilities

preconditioning methods for local discontinuous galerkin discretizations a multilevel interior penalty method is used as an efficient preconditioner for the schur complement of the local discontinuous galerkin ldg discretization of a poisson problem the method is then used in a block triangular preconditioner of the ldg saddle point system the block preconditioner is of the same efficiency as the schur complement version finally the block preconditioner is extended to the discretization of the stokes problem by the ldg method again the preconditioned saddle point problem can be solved in about as many steps as the schur complement the influence of several parameters on the performance of these methods is investigated

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

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

a bddc algorithm for a class of staggered discontinuous galerkin methods a bddc balancing domain decomposition by constraints algorithm is developed and analyzed for a staggered discontinuous galerkin dg finite element approximation of second order scalar elliptic problems on a quite irregular subdomain partition an optimal condition number bound is proved for two dimensional problems in addition a sub optimal but scalable condition number bound is obtained for three dimensional problems these bounds are shown to be independent of coefficient jumps in the subdomain partition numerical results are also included to show the performance of the algorithm

deflation accelerated preconditioning of the poisson neumann schur problem on long domains with a high order discontinuous element based collocation method a combination of block jacobi and deflation preconditioning is used to solve a high order discontinuous element based collocation discretization of the schur complement of the poisson neumann system as arises in the operator splitting of the incompressible navier stokes equations the preconditioners and deflation vectors are chosen to mitigate the effects of ill conditioning due to highly elongated domains typical of simulations of strongly non hydrostatic environmental flows and to achieve generalized minimum residual method gmres convergence independent of the size of the number of elements in the long direction the ill posedness of the poisson neumann system manifests as an inconsistency of the schur complement problem but it is shown that this can be accounted for with appropriate projections out of the null space of the schur complement matrix without affecting the accuracy of the solution the block jacobi preconditioner is shown to yield gmres convergence independent of the polynomial order and only weakly dependent on the number of elements within a subdomain in the decomposition the combined deflation and block jacobi preconditioning is compared with two level non overlapping block jacobi preconditioning of the schur problem and while both methods achieve convergence independent of the grid size deflation is shown to require half as many gmres iterations and 25 less wall clock time for a variety of grid sizes and domain aspect ratios the deflation methods shown to be effective for the two dimensional poisson neumann problem are extensible to the three dimensional problem assuming a fourier discretization in the third dimension a fourier discretization results in a two dimensional helmholtz problem for each fourier component that is solved using deflated block jacobi preconditioning on its schur complement here again deflation is shown to be superior to two level non overlapping block jacobi preconditioning requiring about half as many gmres iterations and 15 less time

piecewise solenoidal vector fields and the stokes problem nonconforming finite element approximations to solutions of the stokes equations are constructed optimal rates of convergence are proved for the velocity and pressure approximations for the pressure approximation c 0 piecewise polynomial functions are used the class of vector fields used to approximate the velocity field have piecewise polynomial components discontinuous across interelement boundaries on each triangle these vector fields satisfy the incompressibility condition pointwise it is shown that these piecewise solenoidal vector fields possess optimal approximation properties to smooth solenoidal vector fields on domains with curved boundaries

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

smoothed aggregation multigrid for the discontinuous galerkin method the aim of this paper is to investigate theoretically as well as experimentally an algebraic multilevel algorithm for the solution of the linear systems that arise from the discontinuous galerkin method the smoothed aggregation multigrid introduced by vanek for the conforming finite element method is applied to low order discretizations of convection diffusion equations for the elliptic model problem the algorithm is shown to be quasi optimal adjustments for the case of nonvanishing advection such as directionally implicit smoothing and a suitable splitting of the operator are discussed several numerical experiments are presented for two dimensional problems including a newton type linearization of the compressible navier stokes equations

scalable two level preconditioning and deflation based on a piecewise constant subspace for sip dg systems for diffusion problems this paper is focused on the preconditioned conjugate gradient cg method for linear systems resulting from symmetric interior penalty discontinuous galerkin sipg discretizations for stationary diffusion problems in particular it concerns two level preconditioning strategies where the coarse space is based on piecewise constant dg basis functions in this paper we show that both the two level preconditioner and the corresponding bnn or adef2 deflation variant yield scalable convergence of the cg method independent of the mesh element diameter these theoretical results are illustrated by numerical experiments we consider a solver for the discontinuous galerkin method we use the symmetric interior penalty variant the deflated solver appears to be scalable our solver is independent of the jump in the coefficients

a multilevel discontinuous galerkin method a variable v cycle preconditioner for an interior penalty finite element discretization for elliptic problems is presented an analysis under a mild regularity assumption shows that the preconditioner is uniform the interior penalty method is then combined with a discontinuous galerkin scheme to arrive at a discretization scheme for an advection diffusion problem for which an error estimate is proved a multigrid algorithm for this method is presented and numerical experiments indicating its robustness with respect to diffusion coefficient are reported

two level additive schwarz methods for a discontinuous galerkin approximation of second order elliptic problems we present some two level nonoverlapping and overlapping additive schwarz methods for a discontinuous galerkin method for solving second order elliptic problems it is shown that the condition numbers of the preconditioned systems are of the order o frac h h for the nonoverlapping schwarz methods and of the order o frac h delta for the overlapping schwarz methods where h and h stand for the fine mesh size and the coarse mesh size respectively and delta denotes the size of the overlaps between subdomains numerical experiments are provided to gauge the efficiency of the methods and to validate the theory

on the generalized hamming weights of cyclic codes we prove several results on the generalized hamming weights ghw s of linear codes particularly for cyclic codes based on these and previously known results we give some efficient algorithms for computing ghw hierarchy of cyclic codes we give complete weight hierarchy for each of the binary cyclic codes of odd lengths spl les 31 a table of second and third ghw s of binary cyclic codes of odd lengths spl les 57 is also presented we have also computed the second ghw of all binary cyclic codes of length 63 and the third ghw of one code from each dimension

the calderon problem with partial data in two dimensions we prove for a two dimensional bounded domain that the cauchy data for the schrodinger equation measured on an arbitrary open subset of the boundary determines uniquely the potential this implies for the conductivity equation that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset we can determine uniquely the conductivity we use carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results

two level additive schwarz preconditioners for c 0 interior penalty methods we study two level additive schwarz preconditioners that can be used in the iterative solution of the discrete problems resulting from c0 interior penalty methods for fourth order elliptic boundary value problems we show that the condition number of the preconditioned system is bounded by c 1 h3 3 where h is the typical diameter of a subdomain measures the overlap among the subdomains and the positive constant c is independent of the mesh sizes and the number of subdomains

multilevel preconditioners for discontinuous galerkin approximations of elliptic problems with jump coefficients in this article we develop and analyze two level and multi level methods for the family of interior penalty ip discontinuous galerkin dg discretizations of second order elliptic problems with rough coecients exhibiting large jumps across interfaces in the domain these methods are based on a decomposition of the dg nite element space that inherently hinges on the diusion coecient of the problem our analysis of the proposed preconditioners is presented for both symmetric and non symmetric ip schemes and we establish both robustness with respect to the jump in the coecient and near optimality with respect to the mesh size following the analysis we present a sequence of detailed numerical results which verify the theory and illustrate the performance of the methods

interior penalty preconditioners for mixed finite element approximations of elliptic problems it is established that an interior penalty method applied to second order elliptic problems gives rise to a local operator which is spectrally equivalent to the corresponding nonlocal operator arising from the mixed finite element method this relation can be utilized in order to construct preconditioners for the discrete mixed system as an example a family of additive schwarz preconditioners for these systems is constructed numerical examples which confirm the theoretical results are also presented

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

a massively parallel nonoverlapping additive schwarz method for discontinuous galerkin discretization of elliptic problems a second order elliptic problem with discontinuous coefficient in 2 d or 3 d is considered the problem is discretized by a symmetric weighted interior penalty discontinuous galerkin finite element method with nonmatching simplicial elements and piecewise linear functions the resulting discrete problem is solved by a two level additive schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element in this way the method has a potential for a very high level of fine grained parallelism condition number estimate depending on the relative sizes of the underlying grids is provided the rate of convergence of the method is independent of the jumps of the coefficient if its variation is moderate inside coarse grid substructures or on local solvers subdomain boundaries numerical experiments are reported which confirm theoretical results

a nonconforming finite element method for the stationary navier stokes equations approximations to solutions of the inhomogeneous boundary value problem for the navier stokes equations are constructed via a nonstandard finite element method the velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element the pressure is approximated by c0 functions optimal rates of convergence results are obtained requiring only local quasi uniformity assumptions on the meshes

norm preconditioners for discontinuous galerkin hp finite element methods we consider a norm preconditioning approach for the solution of discontinuous galerkin finite element discretizations of second order partial differential equations with a nonnegative characteristic form our solution method is a norm preconditioned three term gmres routine we find that for symmetric positive definite diffusivity tensors the convergence of our solver is independent of discretization while for the semidefinite case both theory and experiment indicate dependence on both h and p numerical results are included to illustrate performance on several test cases

a unified analysis of balancing domain decomposition by constraints for discontinuous galerkin discretizations the bddc algorithm is extended to a large class of discontinuous galerkin dg discretizations of second order elliptic problems an estimate of c 1 log h h 2 is obtained for the condition number of the preconditioned system where c is a constant independent of h or h or large jumps in the coefficient of the problem numerical simulations are presented which confirm the theoretical results a key component for the development and analysis of the bddc algorithm is a novel perspective presenting the dg discretization as the sum of elementwise local bilinear forms the elementwise perspective allows for a simple unified analysis of a variety of dg methods and leads naturally to the appropriate choice for the subdomainwise local bilinear forms additionally this new perspective enables a connection to be drawn between the dg discretization and a related continuous finite element discretization to simplify the analysis of the bddc algorithm

analysis of schwarz methods for a hybridizable discontinuous galerkin discretization schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized for hybridizable discontinuous galerkin hdg methods this is a relatively new field of research because hdg methods impose continuity across elements using a robin condition while classical schwarz solvers use dirichlet transmission conditions robin conditions are used in optimized schwarz methods to get faster convergence compared to classical schwarz methods and this even without overlap when the robin parameter is well chosen we present in this paper a rigorous convergence analysis of schwarz methods for the concrete case of the hybridizable interior penalty iph method we show that the penalization parameter needed for convergence of iph leads to slow convergence of the classical additive schwarz method and we propose a modified solver which leads to much faster convergence our analysis is entirely at the discrete level and thus holds for arbitrary in

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

discontinuous hp finite element methods for advection diffusion reaction problems we consider the hp version of the discontinuous galerkin finite element method dgfem for second order partial differential equations with nonnegative characteristic form this class of equations includes second order elliptic and parabolic equations advection reaction equations as well as problems of mixed hyperbolic elliptic parabolic type our main concern is the error analysis of the method in the absence of streamline diffusion stabilization in the hyperbolic case an hp optimal error bound is derived here we consider only advection reaction problems which satisfy a certain standard positivity condition in the self adjoint elliptic case an error bound that is h optimal and p suboptimal by frac 1 2 a power of p is obtained these estimates are then combined to deduce an error bound in the general case for elementwise analytic solutions the method exhibits exponential rates of convergence under p refinement the theoretical results are illustrated by numerical experiments