quasipotential and exit time for 2d stochastic navier stokes equations driven by space time white noise we are dealing with the navier stokes equation in a bounded regular domain mathcal o of mathbb r 2 perturbed by an additive gaussian noise partial w q delta partial t which is white in time and colored in space we assume that the correlation radius of the noise gets smaller and smaller as delta searrow 0 so that the noise converges to the white noise in space and time for every delta 0 we introduce the large deviation action functional s delta t and the corresponding quasi potential u delta and by using arguments from relaxation and gamma convergence we show that u delta converges to u u 0 in spite of the fact that the navier stokes equation has no meaning in the space of square integrable functions when perturbed by space time white noise moreover in the case of periodic boundary conditions the limiting functional u is explicitly computed finally we apply these results to estimate of the asymptotics of the expected exit time of the solution of the stochastic navier stokes equation from a basin of attraction of an asymptotically stable point for the unperturbed system

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

new good quasi cyclic ternary and quaternary linear codes let n k d q codes be linear codes of length n dimension k and minimum hamming distance d over gf q the following quasi cyclic codes are constructed in this paper 44 11 20 3 55 11 26 3 66 11 32 3 48 12 21 3 60 12 28 3 56 13 24 3 65 13 29 3 56 14 23 3 60 15 23 3 64 16 25 3 36 9 19 4 90 9 55 4 99 9 61 4 30 10 14 4 50 10 27 4 55 10 30 4 33 11 15 4 44 11 22 4 55 11 29 4 36 12 16 4 48 12 23 4 60 12 31 4 all of these codes have established or exceed the respective lower bounds on the minimum distance given by brouwer

on k t subnormal covering codes the concept of a k t subnormal covering code is defined it is discussed how an amalgamated direct sumlike construction can be used to combine such codes the existence of optimal q n m 1 codes c is discussed such that by puncturing the first coordinate of c one obtains a code with q 1 subnorm 2

on the covering radius of mds codes for a linear maximum distance separable mds code with redundancy r the covering radius is either r or r 1 however for r 3 few examples of q ary linear mds codes with radius r 1 are known including the reed solomon codes with length q 1 in this paper for redundancies r as large as sqrt 12 q infinite families of q ary mds codes with covering radius r 1 and length less than q 1 are constructed these codes are obtained from algebraic geometric codes arising from elliptic curves for most pairs r q with r le sqrt 12 q these are the shortest q ary mds codes with covering radius r 1

a note on bounds for q ary covering codes two strongly seminormal codes over z sub 5 are constructed to prove a conjecture of ostergard see ibid vol 37 no 3 p 660 4 1991 it is shown that a result of honkala see ibid vol 37 no 4 p 1203 6 1991 on k t subnormal codes holds also under weaker assumptions a lower bound and an upper bound on k sub q n r the minimal cardinality of a q ary code of length n with covering radius r are obtained these give improvements in seven upper bounds and twelve lower bounds by ostergard for k sub q n r for q 3 4 and 5

isothermic surfaces in sphere geometries as moutard nets we give an elaborated treatment of discrete isothermic surfaces and their analogues in different geometries projective mobius laguerre and lie we find the core of the theory to be a novel characterization of discrete isothermic nets as moutard nets the latter are characterized by the existence of representatives in the space of homogeneous coordinates satisfying the discrete moutard equation moutard nets admit also a projective geometric characterization as nets with planar faces with a five point property a vertex and its four diagonal neighbours span a three dimensional space restricting the projective theory to quadrics we obtain moutard nets in sphere geometries in particular moutard nets in mobius geometry are shown to coincide with discrete isothermic nets the five point property in this particular case states that a vertex and its four diagonal neighbours lie on a common sphere which is a novel characterization of discrete isothermic surfaces discrete laguerre isothermic surfaces are defined through the corresponding five plane property which requires that a plane and its four diagonal neighbours share a common touching sphere equivalently laguerre isothermic surfaces are characterized by having an isothermic gauss map s isothermic surfaces as an instance of moutard nets in lie geometry are also discussed

linear codes with covering radius 2 and other new covering codes infinite families of linear binary codes with covering radius r 2 and minimum distance d 3 and d 4 are given using the constructed codes with d 3 r 2 families of covering codes with r 2 are obtained the parameters of many constructed codes with r

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

improved upper bounds on binary identifying codes in binary hamming spaces we construct new 1 identifying codes from 2 fold 1 coverings that are 1 identifying we improve on previously known upper bounds for the cardinalities of 1 identifying codes of many lengths when n 10 we construct t identifying codes using the direct sum of t 1 identifying codes this solves partly an open problem posed by blass honkala and litsyn in 2001 we also prove a general result concerning the direct sum of a t identifying code with the whole space of any dimension

lower bounds for binary codes of covering radius one let k n l denote the minimal cardinality of a binary code of length n and covering radius one blass and litsyn proved a lower bound k n l for in the case n equiv 5 mod6 we give a simplification of the proof which yields a slightly better result

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

further results on k t subnormal covering codes the concept of k t subnormal covering codes is discussed generalizing some of the earlier results in a similar way k t normal covering codes are defined using the results including some new constructions upper bounds for covering codes are improved it is shown how simulated annealing can be used to find acceptable partitions for codes

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

constructions families and tables of binary linear covering codes presents constructions and infinite families of binary linear covering codes with covering radii r 2 3 4 using these codes the authors obtain a table of constructive upper bounds on the length function l r r for r spl les 64 and r 2 3 4 where l r r is the smallest length of a binary linear code with given codimension r and covering radius r they obtain also upper bounds on l r r for r 21 28 r 5 parameters of the constructed codes are better than parameters of previously known codes

new lower bounds for binary covering codes g d chen et al ibid vol it 32 p 680 94 1986 presented two new lower bounds for k n r where k n r denotes the minimum cardinality of a binary code of length n and covering radius r the author shows that a slight modification gives further improvements and some examples are given to confirm the argument codes that have a certain partitioning property are considered

constructions of nonlinear covering codes constructions of nonlinear covering codes are given using any nonlinear starting code of covering radius r spl ges 2 these constructions form an infinite family of codes with the same covering radius a nonlinear code is treated as a union of cosets of a linear code new infinite families of nonlinear covering codes are obtained concepts of r l objects r l partitions and r l length are described for nonlinear codes

large deviations for the invariant measure of a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the invariant measure of the non gaussian stochastic partial differential equation spde t v e v e f x v e e x v e ddot w tx here is a strongly elliptic second order operator with constant coefficients h dh xx h and the space variablex takes values on the unit circles 1 the functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic pde and in particular we require that 0 m m wherem andm are some finite positive constants the perturbationw is a brownian sheet it is well known that under some simple assumptions the solutionv 2 is ac k s 1 valued markov process for each 0 1 2 wherec s 1 is the banach space of real valued continuous functions ons 1 which are holder continuous of exponent we prove under some further natural assumptions onf and which imply that the zero element ofc s 1 is a globally exponentially stable critical point of the unperturbed equation t 0 0 f x 0 that e has a unique stationary distributionv k on c s 1 c k s 1 when the perturbation parameter e is small enough some further calculations show that as e tends to zero v k tends tov k 0 the point mass centered on the zero element ofc s 1 the main goal of this paper is to show that in factv k is governed by a large deviations principle ldp our starting point in establishing the ldp forv k is the ldp for the process e which has been shown in an earlier paper our methods of deriving the ldp forv k based on the ldp for e are slightly non standard compared to the corresponding proofs for finite dimensional stochastic differential equations since the state spacec s 1 is inherently infinite dimensional

disproof of a conjecture on the existence of balanced optimal covering codes the minimum number of codewords in a binary code with length n and covering radius r is denoted by k n r and corresponding codes are called optimal a code with m words is said to be balanced in a given coordinate if the number of 0 s and 1 s in this coordinate are at least spl lfloor m 2 spl rfloor a code is balanced if it is balanced in all coordinates it has been conjectured that among optimal covering codes with given parameters there is at least one balanced code by using a computational method for classifying covering codes it is shown that there is no balanced code attaining k 9 1 62

isothermic submanifolds of symmetric r spaces we extend the classical theory of isothermic surfaces in conformal 3 space due to bour christoffel darboux bianchi and others to the more general context of submanifolds of symmetric r spaces with essentially no loss of integrable structure

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

quasi perfect codes with small distance the main purpose of this paper is to give bounds on the length of the shortest and longest binary quasi perfect codes with a given hamming distance covering radius and redundancy we consider codes with hamming distance 4 and 5 and covering radius 2 and 3 respectively we discuss the blockwise direct sum bds construction which has an important role in finding these bounds

invariant sets under linear operator and covering codes over modules let a be a finite commutative ring with identity a subset h of the a module an is called an r short covering of an if every vector of this module can be written as a sum of a scalar multiple of an element in h and an a linear combination with at most r canonical vectors let c a n r denote the smallest cardinality of an r short covering of an in order to explore the symmetries of such coverings we investigate a characterization of invariant sets under certain linear operator over a finite ring a based on its factorization into local rings as a consequence new classes of upper bounds on c a n r are obtained extending previous results moreover an optimal class on c a n r is derived from a construction of mds codes over certain rings

binary linear quasi perfect codes are normal whether quasi perfect codes are normal is addressed let c be a code of length n dimension k covering radius r and minimal distance d it is proved that c is normal if d or 2r 1 hence all quasi perfect codes are normal consequently any n k r binary linear code with minimal distance d or 2r 1 is normal

quasi perfect linear codes with minimum distance 4 some new infinite families of short quasi perfect linear codes are described such codes provide improvements on the currently known upper bounds on the minimal length of a quasi perfect n n m 4 q code when either 1 q 16 m ges 5 m odd or 2 q 2i 7 les i les 15 m ges 4 or 3 q 22lscr lscr ges 8 m ges 5 m odd as quasi perfect n n m 4 q codes and complete n caps in projective spaces pg m 1 q are equivalent objects new upper bounds on the size of the smallest complete cap in pg m 1 q are obtained

further results on the covering radius of codes a number of upper and lower bounds are obtained for k n r the minimal number of codewords in any binary code of length n and covering radius r several new constructions are used to derive the upper bounds including an amalgamated direct sum construction for nonlinear codes this construction works best when applied to normal codes and we give some new and stronger conditions which imply that a linear code is normal an upper bound is given for the density of a covering code over any alphabet and it is shown that k n 2 r 1 leq k n r holds for sufficiently large n

more binary covering codes are normal it is shown that every optimal binary code with covering radius r 1 is normal this parity proves a conjecture of cohen lobstein and sloane 1986 it is also proved that codes with minimal distance 2r or 2r 1 are normal a generalization of frankl s construction 1987 of abnormal codes is given

several new lower bounds on the size of codes with covering radius one we derive several new lower bounds on the size of binary codes with covering radius one in particular we prove k

upper bounds for q ary covering codes new methods for constructing q ary covering codes are presented the author introduces the concepts of p seminormal and strongly p seminormal codes and shows how seminormal codes and punctured hamming codes can be combined to construct new covering codes using these methods upper bounds for ternary covering codes are improved the new bounds are k sub 3

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

some results on the norm of codes two upper bounds for the norm n c of a binary linear code c with minimal weight d and covering radius r are given the second of these bounds implies that c is normal if r 3

asymptotic compactness and absorbing sets for 2d stochastic navier stokes equations on some unbounded domains we introduce a notion of an asymptotically compact ac random dynamical system rds we prove that for an ac rds the limit set b of any bounded set b is nonempty compact strictly invariant and attracts the set b we establish that the 2d navier stokes equations nses in a domain satisfying the poincare inequality perturbed by an additive irregular noise generate an ac rds in the energy space h as a consequence we deduce existence of an invariant measure for such nses our study generalizes on the one hand the earlier results by flandoli crauel 1994 and schmalfuss 1992 obtained in the case of bounded domains and regular noise and on the other hand the results by rosa 1998 for the deterministic nses

simple factor dressing and the lopez ros deformation of minimal surfaces in euclidean 3 space the aim of this paper is to investigate a new link between integrable systems and minimal surface theory the dressing operation uses the associated family of flat connections of a harmonic map to construct new harmonic maps since a minimal surface in 3 space is a willmore surface its conformal gauss map is harmonic and a dressing on the conformal gauss map can be defined we study the induced transformation on minimal surfaces in the simplest case the simple factor dressing and show that the well known lopez ros deformation of minimal surfaces is a special case of this transformation we express the simple factor dressing and the lopez ros deformation explicitly in terms of the minimal surface and its conjugate surface in particular we can control periods and end behaviour of the simple factor dressing this allows to construct new examples of doubly periodic minimal surfaces arising as simple factor dressings of scherk s first surface

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

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

generalized dpw method and an application to isometric immersions of space forms let g be a complex lie group and g denote the group of maps from the unit circle mathbb s 1 into g of a suitable class a differentiable map f from a manifold m into g is said to be of connection order a b if the fourier expansion in the loop parameter of the mathbb s 1 family of maurer cartan forms for f namely f lambda 1 rm d f lambda is of the form sum i a b alpha i lambda i most integrable systems in geometry are associated to such a map roughly speaking the dpw method used a birkhoff type splitting to reduce a harmonic map into a symmetric space which can be represented by a certain order 1 1 map into a pair of simpler maps of order 1 1 and 1 1 respectively conversely one could construct such a harmonic map from any pair of 1 1 and 1 1 maps this allowed a weierstrass type description of harmonic maps into symmetric spaces we extend this method to show that for a large class of loop groups a connection order a b map for a 0 b splits uniquely into a pair of a 1 and 1 b maps as an application we show that constant non zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds reducing the problem at least locally to the flat case to extend the dpw method sufficiently to handle this problem requires a more general iwasawa type splitting of the loop group which we prove always holds at least locally

stochastic nonlinear beam equations an extensible beam equation with a stochastic force of a white noise type is studied lyapunov functions techniques being used to prove existence of global mild solutions and asymptotic stability of the zero solution

packing and covering properties of rank metric codes this paper investigates packing and covering properties of codes with the rank metric first we investigate packing properties of rank metric codes then we study sphere covering properties of rank metric codes derive bounds on their parameters and investigate their asymptotic covering properties

on the size of optimal binary codes of length 9 and covering radius 1 the minimum number of codewords in a binary code with length n and covering radius r is denoted by k n r the values of k n 1 are known up to length 8 and the corresponding optimal codes have been classified it is known that 57 spl les k 9 1 spl les 62 in the current work the lower bound is improved to settle k 9 1 62 in the approach which is computer aided possible distributions of codewords in subspaces are refined until each subspace is of dimension zero consists of only one word repeatedly a linear programming problem is solved considering only inequivalent distributions a connection between this approach and weighted coverings is also presented the computations give new results for such coverings as a by product

frequency computation and bounded queries there have been several papers over the last ten years that consider the number of queries needed to compute a function as a measure of its complexity the following function has been studied extensively in that light faa x1 xa a x1 a xa we are interested in the complexity in terms of the number of queries of approximating faa let b a and let f be any function such that faa x1 xa and f x1 xa agree on at least b bits for a general set a we have matching upper and lower bounds on f that depend on coding theory these are applied to get exact bounds for the case where a is semirecursive a is superterse and assuming p np a sat we obtain exact bounds when a is the halting problem using different methods

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

unidirectional covering codes a code c spl sube z sup n sub 2 where z sub 2 0 1 has unidirectional covering radius r if r is the smallest integer so that any word in z sup n sub 2 can be obtained from at least one codeword c spl isin c by replacing either 1s by 0s in at most r coordinates or 0s by 1s in at most r coordinates the minimum cardinality of such a code is denoted by e n r upper bounds on this function are here obtained by constructing codes using tabu search lower bounds on the other hand are mainly obtained by integer programming and exhaustive search best known bounds on e n r for n spl les 13 and r spl les 6 are tabulated

new binary covering codes obtained by simulated annealing new binary covering codes of radius 1 obtained by simulated annealing are presented these constructions establish that k 9 1 spl les 62 and k 12 1 spl les 380 the article is concerned with finding upper bounds on k n r the minimum cardinality of any binary code of length n and with covering radius r

some new lower bounds for binary and ternary covering codes a modified lower bound for binary codes with covering radius one is derived let k sub q n r denote the minimum cardinality of a q ary code with covering radius r the new bound shows that k sub 2 11 1 or 177 and that k sub 2 17 1 or 7391 improvements of the best lower bounds known the authors also generalize a known lower bound for binary codes to the case of arbitrary q for q 3 this simple bound improves the best lower bounds known in several cases an updated version of a table for k sub 3 n r is included

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

some new upper bounds on the covering radius of binary linear codes a griesmer like upper bound on the covering radius r is given to the author s knowledge this is the only upper bound which explicitly depends on all three parameters n k and d an upper bound on r for cyclic codes is then given which depends on the generator polynomial of the cyclic code and which in many cases leads to an improvement of the previous bound an upper bound on the irreducible generator polynomial cyclic codes is also given new interpretations and applications of the so called norse bounds and necessary and sufficient conditions to attain one of these bounds are provided generalizations of most of the results for codes over gf q are outlined

asymptotic behavior of stochastic discrete wave equations with nonlinear noise and damping in this article we study the asymptotic behavior for a class of discrete wave equations with nonlinear noise and damping defined on a k dimensional integer set the well posedness of the system is established when the nonlinear drift function and the nonlinear diffusion term are only locally lipschitz continuous the mean random dynamical system associated with the non autonomous system is shown to possess a unique tempered weak pullback mean random attractor in l2 f l2 l2 the existence of invariant measures for the autonomous system is also derived by using the krylov bogolyubov method the difficulty in proving the tightness of a family of distribution laws of the solutions is overcome by using the idea of uniform estimates on the tails of solutions

the multicovering radii of codes the covering radius of a code is the least r such that the set of balls of radius r around codewords covers the entire ambient space we introduce a generalization of the notion of covering radius the m covering radius of a code is the least radius such that the set of balls of that radius covers all m tuples of elements in the ambient space we investigate basic properties of m covering radii we investigate whether codes exist with given m covering radii not always we derive bounds on the size of the smallest code with a given m covering radius based on generalizations of the sphere bound and the method of counting excesses

on the optimal linear receiving filter for digital transmission over nonlinear channels a recent paper by e biglieri et al see ibid vol 35 p 620 5 may 1989 shows that the matched filter tapped delay line structure is optimum not only for linear pulse modulated signals and linear channel distortion but also for nonlinear finite alphabet pulse modulation and some nonlinear channel distortion this has important practical applications therefore its connection with other work reported in the literature is presented

on a new class of codes for identifying vertices in graphs we investigate a new class of codes for the optimal covering of vertices in an undirected graph g such that any vertex in g can be uniquely identified by examining the vertices that cover it we define a ball of radius t centered on a vertex spl upsi to be the set of vertices in g that are at distance at most t from spl upsi the vertex spl upsi is then said to cover itself and every other vertex in the ball with center spl upsi our formal problem statement is as follows given an undirected graph g and an integer t spl ges 1 find a minimal set c of vertices such that every vertex in g belongs to a unique set of balls of radius t centered at the vertices in c the set of vertices thus obtained constitutes a code for vertex identification we first develop topology independent bounds on the size of c we then develop methods for constructing c for several specific topologies such as binary cubes nonbinary cubes and trees we also describe the identification of sets of vertices using covering codes that uniquely identify single vertices we develop methods for constructing optimal topologies that yield identifying codes with a minimum number of codewords finally we describe an application of the theory developed in this paper to fault diagnosis of multiprocessor systems

curved flats and isothermic surfaces we show how pairs of isothermic surfaces are given by curved flats in a pseudo riemannian symmetric space and vice versa calapso s fourth order partial differential equation is derived and using a solution of this equation a mobius invariant frame for an isothermic surface is built

covering radius survey and recent results all known results on covering radius are presented as well as some new results there are a number of upper and lower bounds including asymptotic results a few exact determinations of covering radius some extensive relations with other aspects of coding theory through the reed muller codes and new results on the least covering radius of any linear n k code there is also a recent result on the complexity of computing the covering radius

an improved sphere covering bound for the codes with n 3r 2 let c be a binary code not necessarily linear with covering radius r and length n 3r 2 the sphere covering bound on the cardinality of c is improved considerably provided c has minimal distance d 2 some new results on the function t n k the smallest covering radius of any binary linear code with length n and dimension k t 38 6 or 13 t 47 7 or 16 t 59 8 or 20 are given

explicit codes with low covering radius a series of explicit low rate binary linear codes that have relatively low covering radius and can be rapidly decoded is exhibited these codes can be derived from higher dimensional analogs of the gale berlekamp switching game conjectures of independent interest involving hadamard matrices are given that could yield semiexplicit covering codes of very low density

a new binary code of length 10 and covering radius 1 a mixed code of covering radius 1 that has 60 codewords is constructed this code is then used to show that k 10 1

constructions and families of covering codes and saturated sets of points in projective geometry in davydov 1990 constructions of linear binary covering codes were considered in the present paper constructions and techniques of the earlier paper are developed and modified for q ary linear nonbinary covering codes q spl ges 3 and new constructions are proposed the described constructions design an infinite family of codes with covering radius r based on a starting code of the same covering radius for arbitrary r spl ges 2 q spl ges 3 new infinite families of nonbinary covering codes with good parameters are obtained with the help of an iterative process when constructed codes are the starting codes for the following steps the table of upper bounds on the length function for codes with q 3 r 2 3 and codimension up to 24 is given the author proposes to use saturated sets of points in projective geometries over finite fields as parity check matrices of starting codes new saturated sets are obtained

bounds for abnormal binary codes with covering radius one the normality of binary codes is studied the minimum cardinality of a binary code of length n with covering radius r is denoted by k n r it is assumed that c is an n m r code that is a binary code of length n with m codewords and covering radius r it is shown that if c is an n m 1 code then it is easy to find a normal n m 1 code by changing c in a suitable way and that all the optimal n m 1 codes i e those for which m k n 1 are normal and their every coordinate is acceptable it is shown that if c is an abnormal n m code then n or 9 and an abnormal 9118 1 code which is the smallest abnormal code known at present is constructed lower bounds on the minimum cardinality of a binary abnormal code of length n with covering radius 1 are derived and it is shown that if an n m 1 code is abnormal then m or 96

on the covering radius of subcodes of a code let c be a binary linear code with covering radius r and c sub 0 a subcode of c of codimension i an upper bound is obtained for the covering radius of c sub 0 in terms of r and i when c sub 0 0 the bound becomes the sphere covering bound for r

lower bounds on the covering radius of the non binary and binary irreducible goppa codes new lower bounds on the covering radius for different subclasses of irreducible goppa codes are obtained

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed

large deviations principle for the invariant measures of the 2d stochastic navier stokes equations on a torus abstract we prove here the validity of a large deviation principle for the family of invariant measures associated to a two dimensional navier stokes equation on a torus perturbed by a smooth additive noise

on normal and subnormal q ary codes the authors extend to the q ary case the notions of a normal code a subnormal code and the amalgamated direct sum construction in order to investigate problems related to the covering radius of codes for example the authors prove that every nonbinary nontrivial perfect code is absubnormal they also include some linear programming lower bounds on ternary codes with covering radius 2 or 3

lower bounds for q ary covering codes the authors prove combinatorial lower bounds for k sub q n r the minimal cardinality of any q ary code of length n and covering radius r tables of lower bounds for k sub q n r are presented for q 3 4 5

a new construction for covering codes a novel method for constructing normal binary nonlinear covering codes is presented the construction improves several upper bounds on k n r the minimum cardinality of a binary code of length n and covering radius r

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation