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

an improvement of the van wee bound for binary linear covering codes the van wee bound gives a lower bound on the cardinality of block codes with a prescribed covering radius the author presents an improvement of this bound for binary linear codes and compares the results with other bounds mentioned in the literature almost all reported bounds can be obtained via the present bound as well moreover the present authors obtain a number of new lower bounds

a polyhedral study of the generalized vertex packing problem the traditional vertex packing problem defined on an undirected graph identifies the largest weighted independent set of nodes that is a set of nodes whose induced subgraph contains no edges in this paper we examine a generalized vertex packing problem gvp k in which k violations to the independent set restriction are permitted whereby k edges may exist within the subgraph induced by the chosen set of nodes a particular context in which such problems arise is in the national airspace planning model of sherali smith and trani 2000 where a set of flight plans need to be composed for various flights subject to conflict workload and equity considerations the gvp k structure arises in modeling the air traffic control sector workload restrictions which stipulate that for each sector and during each discretized time slot the number of aircraft conflicts that would need to be resolved should not exceed k for some k 1 we derive several classes of facetial valid inequalities for gvp k for certain specially structured subgraphs identifying polynomial sized convex hull representations for some of these cases related constraint generation routines are also developed and some computational results are provided to demonstrate the efficacy of utilizing the proposed valid inequalities in solving gvp k for different values of k

an upper bound on the covering radius as a function of the dual distance p delsarte 1973 developed a method that gives an upper bound on the cardinality of a code as a function of its minimum distance it is shown that using a modification of that method one gets an upper bound on the covering radius of a code as a function of its dual distance as an interesting special case the covering radius of the dual of a bch code is considered

modified bounds for covering codes the covering radius of binary codes is studied bounds on k n r the minimum cardinality of any binary code of length n and covering radius r are found modifications of the van wee lower bounds are proved for k n r the minimal number of codewords in any binary code of length n and covering radius r the first of the two van wee bounds is based on studying the hamming spheres of radius 1 centered at the points which have distance r to the code c the points covered by more than one codeword are divided into several classes and better estimates for some of these classes are obtained using a suitable averaging process the lower bound for k n r when r or 2 is improved the second van wee bound studies spheres of radius 2 centered at the points which have distance r 1 or r to the code c these points are divided essentially into two classes the points that are covered by only one codeword of c and the points that are covered by more than one codeword

continuous characterizations of the maximum clique problem given a graph g whose adjacency matrix is a the motzkin strauss formulation of the maximum clique problem is the quadratic program max xtax xte 1 x 0 it is well known that the global optimum value of this qp is 1 1 g where g is the clique number of g here we characterize the following properties pertaining to the above qp 1 first order optimality 2 second order optimality 3 local optimality 4 strict local optimality these characterizations reveal interesting underlying discrete structures and are polynomial time verifiable a parametrization of the motzkin strauss qp is then introduced and its properties are investigated finally an extension of the motzkin strauss formulation is provided for the weighted clique number of a graph

on the structure of linear codes with covering radius two and three we obtain new bounds on l m r the minimum length of a linear code with codimension m and covering radius r all bounds are derived in a uniform way we employ results from coding theory some earlier results on covering codes and combinatorial arguments we prove the following bounds l 6 2 13 l 7 2 19 l 8 2 spl ges 25 l 9 2 spl ges 34 l 2m l 2 spl ges 2 sup m 1 for m spl ges 3 l 14 2 spl ges 182 l 16 2 spl ges 363 l 18 2 spl ges 725 l 20 2 spl ges 1449 l 22 2 spl ges 2897 l 24 2 spl ges 5794 l 9 3 spl ges 17 l 10 3 spl ges 21 l 12 3 spl ges 31 l 13 3 spl ges 38

short codes with a given covering radius the covering radius r of a code is the maximum distance from any vector in the space containing the code to the nearest codeword the authors introduce a new function l m r called the length function which equals the smallest length of a binary code of codimension m and covering radius r they investigate basic properties of the length function projective geometries over larger fields are used to construct families of codes which improve significantly the upper bound for l m 2 obtained by amalgamation of hamming codes general methods are developed for ruling out the existence of codes of covering radius 2 with a given codimension and length resulting in lower bounds for l m 2 a table is presented which gives the best results now known for l m r with m

on the covering radius of codes the covering radius r of a code is the maximal distance of any vector from the code this work gives a number of new results concerning t n k the minimal covering radius of any binary code of length n and dimension k for example t n 4 and t n 5 are determined exactly and reasonably tight bounds on t n k are obtained for any k when n is large these results are found by using several new constructions for codes with small covering radius one construction the amalgamated direct sum involves a quantity called the norm of a code codes with norm leq 2 r 1 are called normal and may be combined efficiently the paper concludes with a table giving bounds on t n k for n leq 64

linear inequalities for covering codes ii triple covering inequalities the lower bounds on k n r minimum number of codewords of any binary code of length n and covering radius r are improved a new technique combining the hamming association scheme and the results of a classic problem of covering pairs by k tuples is introduced the new lower bounds are obtained by studying various linear inequalities satisfied by covering codes and such an inequality is derived the lower bounds for k n r are studied for some special values of n and r which serve as examples showing how the methods developed are applied to concrete cases

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

on clique relaxation models in network analysis abstract increasing interest in studying community structures or clusters in complex networks arising in various applications has led to a large and diverse body of literature introducing numerous graph theoretic models relaxing certain characteristics of the classical clique concept this paper analyzes the elementary clique defining properties implicitly exploited in the available clique relaxation models and proposes a taxonomic framework that not only allows to classify the existing models in a systematic fashion but also yields new clique relaxations of potential practical interest some basic structural properties of several of the considered models are identified that may facilitate the choice of methods for solving the corresponding optimization problems in addition bounds describing the cohesiveness properties of different clique relaxation structures are established and practical implications of choosing one model over another are discussed

improved sphere bounds on the covering radius of codes the sphere bound is a trivial lower bound on k n r the minimal cardinality of any binary code of length n and with covering radius r by simple arguments it is considerably improved to k n 1 or 2 sup n n for n even a table of lower and upper bounds on k n r for n

lower bounds on t n k from linear inequalities the linear inequality method for covering codes is used to improve the lower bounds of t n k the smallest covering radius of any n k binary linear code to make better use of the strength of this method the relation between the covering radius of a code and the minimum distance of its dual code is studied the authors obtained 65 improved lower bounds for t n k within the range of n

clique relaxations in social network analysis the maximum k plex problem this paper introduces and studies the maximum k plex problem which arises in social network analysis and has wider applicability in several important areas employing graph based data mining after establishing np completeness of the decision version of the problem on arbitrary graphs an integer programming formulation is presented followed by a polyhedral study to identify combinatorial valid inequalities and facets a branch and cut algorithm is implemented and tested on proposed benchmark instances an algorithmic approach is developed exploiting the graph theoretic properties of a k plex that is effective in solving the problem to optimality on very large sparse graphs such as the power law graphs frequently encountered in the applications of interest

inequalities for covering codes any code c with covering radius r must satisfy a set of linear inequalities that involve the lloyd polynomial l sub r x these generalize the sphere bound syndrome graphs associated with a linear code c are introduced to help keep track of low weight vectors in the same coset of c if there are too many such vectors c cannot exist illustrations show that t 17 10 3 and t 23 15 3 where t n k is the smallest covering radius of any n k code

solving quadratic programming by cutting planes we propose new cutting planes for strengthening the linear relaxations that appear in the solution of nonconvex quadratic problems with linear constraints by a famous result of motzkin and straus

np hardness of deciding convexity of quartic polynomials and related problems we show that unless p np there exists no polynomial time or even pseudo polynomial time algorithm that can decide whether a multivariate polynomial of degree four or higher even degree is globally convex this solves a problem that has been open since 1992 when n z shor asked for the complexity of deciding convexity for quartic polynomials we also prove that deciding strict convexity strong convexity quasiconvexity and pseudoconvexity of polynomials of even degree four or higher is strongly np hard by contrast we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time

the minimal covering radius t 15 6 of a six dimensional binary linear code of length 15 is equal to 4 the nonexistence of a binary linear 15 6 code with covering radius 3 is proved by geometrically based arguments the author also considers the wider application of the techniques

constructions for perfect mixed codes and other covering codes a construction for an infinite family of perfect mixed codes with covering radius 2 is presented these are the first known nontrivial perfect mixed codes with covering radius greater than 1 based on mixed codes constructions for binary covering codes that lead to a considerable improvement of upper bounds on the sizes of covering codes are presented these codes and some other codes can be obtained by the blockwise direct sum construction two infinite families of codes are of special interest they are quasi perfect nonlinear union of their disjoint translates covers the space and their density as covering codes is remarkably low

new lower bounds for covering codes some new lower bounds on mod c mod for a binary linear n k r code c with n 1 t r 1 r 0 2 odd or with n 1 t r 1 1 t 2 even are obtained these bounds improve the sphere covering bound considerably and give several new values and lower bounds for the function t n k the smallest covering radius of any n k code

the maximum principle for relaxed hereditary differential systems with function space end condition this paper contains a proof of the global pointwise maximum principle for relaxed hereditary differential systems with general function space end condition first a multiplier theorem establishes the existence of lagrange multipliers l 0 l where to l 0 in mathbb r and l is in the dual of the sobolev space w n infty r 0 then l can be identified with an element of w n infty r 0 provided that the optimal trajectory satisfies a certain regularity condition this yields two equivalent forms of the maximum principle using the results on regular reachability obtained in a companion paper the maximum principle is shown to be in a certain sense generically valid

a twofold spline approximation for finite horizon lqg control of hereditary systems in this paper an approximation scheme is developed for the solution of the linear quadratic gaussian lqg control on a finite time interval for hereditary systems with multiple noncommensurate delays and distributed delay the solution here proposed is achieved by means of two approximating subspaces the first one to approximate the riccati equation for control and the second one to approximate the filtering equations since the approximating subspaces have finite dimension the resulting equations can be implemented the convergence of the approximated control law to the optimal one is proved simulation results are reported on a wind tunnel model showing the high performance of the method

continuous cubic formulations for cluster detection problems in networks the celebrated motzkin straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex it was originally developed as a way of proving turan s theorem in graph theory but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization clique relaxations such as s defective clique and s plex emerged as attractive more practical alternatives to cliques in network based cluster detection models arising in numerous applications this paper establishes continuous cubic formulations for the maximum s defective clique problem and the maximum s plex problem by generalizing the motzkin straus formulation to the corresponding clique relaxations the formulations are used to extend turan s theorem and other known lower bounds on the clique number to the considered clique relaxations results of preliminary numerical experiments with the conopt solver demonstrate that the proposed formulations can be used to quickly compute high quality solutions for the maximum s defective clique problem and the maximum s plex problem the proposed formulations can also be used to generate interesting test instances for global optimization solvers