binary 18 11 2 codes do not exist nor do 64 53 2 codes a binary linear block code c with block length n and dimension n is commonly denoted by n k or if its minimum distance is d by n k d the code s covering radius r c can be defined as the smallest number r such that any binary column vector of length n k can be written as a sum of r or fewer columns of a parity check matrix of c an n k code with covering radius r is denoted by n k r r a brualdi et al 1989 showed that l m r is defined to be the smallest n such that an n n m r code exists l m 2 is known for m or 6 while it is shown by brualdi et al that 17 or l

the heat equation with rough boundary conditions and holomorphic functional calculus abstract in this paper we consider the laplace operator with dirichlet boundary conditions on a smooth domain we prove that it has a bounded h calculus on weighted l p spaces for power weights which fall outside the classical class of a p weights furthermore we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity in particular we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

boundedness of riesz transforms for elliptic operators on abstract wiener spaces let e h be an abstract wiener space and let dv vd where d denotes the malliavin derivative and v is a closed and densely defined operator from h into another hilbert space h given a bounded operator b on h coercive on the range r v we consider the operators a v bv in h and a vv b in h as well as the realisations of the operators l dv bdv and l dvdv b in lp e and lp e h respectively where 1 p our main result asserts that the following four assertions are equivalent 1 d l d dv with lf p dvf p for f d l 2 l admits a bounded h functional calculus on r dv 3 d a d v with ah vh for h d a 4 a admits a bounded h functional calculus on r v moreover if these conditions are satisfied then d l d dv2 d da the equivalence 1 4 is a non symmetric generalisation of the classical meyer inequalities of malliavin calculus where h h v i b 12i a one sided version of 1 4 giving lp boundedness of the riesz transform dv l in terms of a square function estimate is also obtained as an application let a generate an analytic c0 contraction semigroup on a hilbert space h and let l be the lp realisation of the generator of its second quantisation our results imply that two sided bounds for the riesz transform of l are equivalent with the kato square root property for a the boundedness of the riesz transform is used to obtain an lp domain characterisation for the operator l

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

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

difference norms for vector valued bessel potential spaces with an application to pointwise multipliers abstract in this paper we prove a randomized difference norm characterization for bessel potential spaces with values in umd banach spaces the main ingredients are r boundedness results for fourier multiplier operators which are of independent interest as an application we characterize the pointwise multiplier property of the indicator function of the half space on these spaces all results are proved in the setting of weighted spaces

comment on the paper interval criteria for oscillation of second order nonlinear neutral differential equations comput math appl 46 5 6 2003 903 918

covering radii of ternary linear codes of small dimensions and codimensions the covering radii and the least covering radii of all two and three dimensional codes over gf 3 are given in the present work the exact values of the least covering radii of some ternary linear codes of codimensions up to 8 are also determined and a table of values of t sub 3 n k for n spl les 27 is presented

stochastic evolution equations in umd banach spaces we discuss existence uniqueness and space time holder regular ity for solutions of the parabolic stochastic evolution equation du t au t f t u t dt b t u t dwh t t 2 0 t0 u 0 u0 where a generates an analytic c0 semigroup on a umd banach space e and wh is a cylindrical brownian motion with values in a hilbert space h we prove that if the mappings f 0 t e e and b 0 t e l h e satisfy suitable lipschitz conditions and u0 is f0 measurable and bounded then this problem has a unique mild solution which has trajectories in c 0 t d a provided 0 and 0 satisfy 1 2 vari ous extensions of this result are given and the results are applied to parabolic stochastic partial dierential equations

locally optimal nonshortening linear covering codes and minimal saturating sets in projective spaces a concept of locally optimal lo linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields an lo code is nonshortening in the sense that one cannot remove any column from a parity check matrix without increasing the code covering radius several q sup m concatenating constructions of lo covering codes are described taking a starting lo code as a seed such constructions produce an infinite family of lo codes with the same covering radius the infinite families of lo codes are designed using minimal saturating sets as starting codes new upper bounds on the length function are given new extremal and classification problems for linear covering codes are formulated and investigated in particular the spectrum of possible lengths of lo codes including the greatest possible length the complete computer classification of the minimal saturating sets in small geometries and of the corresponding lo codes is obtained

schauder estimates for stochastic transport diffusion equations with levy processes abstract in this study we consider a transport diffusion equation with levy noise and holder continuous coefficients by using the heat kernel estimates we derive the schauder estimates for the mild solutions moreover when the transport term vanishes and p 2 we show that the holder index in the space variable is optimal

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

a bmo estimate for stochastic singular integral operators and its application to spdes abstract in this paper we investigate kernel conditions on k t s x so that the stochastic singular integral operator 0 t k t s g s x d w s has a bounded mean oscillation as an application we prove that for the solution u of the stochastic heat equation 0 1 d u t x a i j t u x i x j d t g t k x d w t k u 0 0 t t the q th order bmo quasi norm of the derivatives of u is controlled by g l

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

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

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

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

pathwise holder convergence of the implicit linear euler scheme for semi linear spdes with multiplicative noise in this article we prove pathwise holder convergence with optimal rates of the implicit euler scheme for the abstract stochastic cauchy problem begin aligned left begin aligned du t au t dt f t u t dt g t u t dw h t quad t in 0 t u 0 x 0 end aligned right end aligned here a is the generator of an analytic c 0 semigroup on a umd banach space x w h is a cylindrical brownian motion in a hilbert space h and the functions f 0 t times x rightarrow x theta f and g 0 t times x rightarrow fancyscript l h x theta g satisfy appropriate local lipschitz conditions the results are applied to a class of second order parabolic spdes driven by multiplicative space time white noise

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

constructions and families of nonbinary linear codes with covering radius 2 new constructions of linear nonbinary codes with covering radius r 2 are proposed they are in part modifications of earlier constructions by the author and in part are new using a starting code with r 2 as a seed these constructions yield an infinite family of codes with the same covering radius new infinite families of codes with r 2 are obtained for all alphabets of size q spl ges 4 and all codimensions r spl ges 3 with the help of the constructions described the parameters obtained are better than those of known codes new estimates for some partition parameters in earlier known constructions are used to design new code families complete caps and other saturated sets of points in projective geometry are applied as starting codes a table of new upper bounds on the length function for q 4 5 7 r 2 and r spl les 24 is included

traces and embeddings of anisotropic function spaces in this paper we characterize the trace spaces of a class of weighted function spaces of intersection type with mixed regularities to a large extent we can overcome the difficulty of mixed scales by employing a microscopic improvement in sobolev and mixed derivative embeddings with fixed integrability we apply the general results to prove maximal l p l q regularity for the linearized fully inhomogeneous two phase stefan problem with gibbs thomson correction

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

stochastic maximal lp regularity in this article we prove a maximal lp regularity result for stochastic convolutions which extends krylov s basic mixed lp lq inequality for the laplace operator on d to large classes of elliptic operators both on d and on bounded domains in d with various boundary conditions our method of proof is based on mcintosh s h functional calculus r boundedness techniques and sharp lp lq square function estimates for stochastic integrals in lq spaces under an additional invertibility assumption on a a maximal space time lp regularity result is obtained as well

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

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

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

binary and ternary linear quasi perfect codes with small dimensions the aim of this work is a systematic investigation of the possible parameters of quasi perfect qp binary and ternary linear codes of small dimensions and preparing a complete classification of all such codes first we give a list of infinite families of qp codes which includes all binary ternary and quaternary codes known to us we continue further with a list of sporadic examples of binary and ternary qp codes later we present the results of our investigation where binary qp codes of dimensions up to 14 and ternary qp codes of dimensions up to 13 are classified

on the regularity of the stochastic heat equation on polygonal domains in r2 abstract we establish existence uniqueness and higher order weighted l p sobolev regularity for the stochastic heat equation with zero dirichlet boundary condition on angular domains and on polygonal domains in r 2 we use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable in this way we can capture the influence of both main sources for singularities the incompatibility between noise and boundary condition on the one hand and the singularities of the boundary on the other hand the range of admissible powers of the distance to the vertexes is described in terms of the maximal interior angle and is sharp

conical stochastic maximal l p regularity for 1 leqslant p infty let a mathrm div a cdot nabla be a second order divergence form elliptic operator on mathbb r n with bounded measurable real valued coefficients and let w be a cylindrical brownian motion in a hilbert space h our main result implies that the stochastic convolution process begin aligned u t int 0 t e t s a g s dw s quad t geqslant 0 end aligned satisfies for all 1 leqslant p infty a conical maximal l p regularity estimate begin aligned mathbb e vert nabla u vert t 2 p 2 mathbb r times mathbb r n p leqslant c p p mathbb e vert g vert t 2 p 2 mathbb r times mathbb r n h p end aligned here t 2 p 2 mathbb r times mathbb r n and t 2 p 2 mathbb r times mathbb r n h are the parabolic tent spaces of real valued and h valued functions respectively this contrasts with krylov s maximal l p regularity estimate begin aligned mathbb e vert nabla u vert l p mathbb r l 2 mathbb r n mathbb r n p leqslant c p mathbb e vert g vert l p mathbb r l 2 mathbb r n h p end aligned which is known to hold only for 2 leqslant p infty even when a delta and h mathbb r the proof is based on an l 2 estimate and extrapolation arguments which use the fact that a satisfies suitable off diagonal bounds our results are applied to obtain conical stochastic maximal l p regularity for a class of nonlinear spdes with rough initial data

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

erratum to perturbation and interpolation theorems for the h calculus with applications to differential operators we prove comparison theorems for the h calculus that allow to transfer the property of having a bounded h calculus from one sectorial operator to another the basic technical ingredient are suitable square function estimates these comparison results provide a new approach to perturbation theorems for the h calculus in a variety of situations suitable for applications our square function estimates also give rise to a new interpolation method the rademacher interpolation we show that a bounded h calculus is characterized by interpolation of the domains of fractional powers with respect to rademacher interpolation this leads to comparison and perturbation results for operators defined in interpolation scales such as the l p scale we apply the results to give new proofs on the h calculus for elliptic differential operators including schrodinger operators and perturbed boundary conditions as new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded h calculus in certain sobolev spaces and that the stokes operator on bounded domains with c 1 1 has a bounded h calculus in the helmholtz scale l p p 1

stochastic integration in umd banach spaces in this paper we construct a theory of stochastic integration of processes with values in l h e where h is a separable hilbert space and e is a umd banach space i e a space in which martingale differences are unconditional the integrator is an h cylindrical brownian motion our approach is based on a two sided l p decoupling inequality for umd spaces due to garling which is combined with the theory of stochastic integration of l h e valued functions introduced recently by two of the authors we obtain various characterizations of the stochastic integral and prove versions of the ito isometry the burkholder davis gundy inequalities and the representation theorem for brownian martingales

linear codes with covering radius r 2 3 and codimension tr let n n r sub q r denote a linear code over f sub q with length n codimension r and covering radius r we use a modification of constructions of 2q 1 2q 3 sub q 2 and 3q 1 3q 5 sub q 3 codes q spl ges 5 to produce infinite families of good codes with covering radius 2 and 3 and codimension tr

well posedness and optimal regularity of stochastic evolution equations with multiplicative noises abstract in this paper we establish the well posedness and optimal trajectory regularity for the solution of stochastic evolution equations with generalized lipschitz type coefficients driven by general multiplicative noises to ensure the well posedness of the problem the linear operator of the equations is only need to be a generator of a c 0 semigroup and the proposed noises are quite general which include space time white noise and rougher noises when the linear operator generates an analytic c 0 semigroup we derive the optimal trajectory regularity of the solution through a generalized criterion of factorization method

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

corrigendum to necessary and sufficient conditions for oscillation of second order neutral differential equations j math anal appl 252 2000 342 352

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

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

stability properties of stochastic maximal l p regularity in this paper we consider lp regularity estimates for solutions to stochastic evolution equations which is called stochastic maximal lp regularity our aim is to find a theory which is analogously to dore s theory for deterministic evolution equations he has shown that maximal lp regularity is independent of the length of the time interval implies analyticity and exponential stability of the semigroup is stable under perturbation and many more properties we show that the stochastic versions of these results hold

linear codes with covering radius 2 3 and saturating sets in projective geometry infinite families of linear codes with covering radius r 2 3 and codimension tr 1 are constructed on the base of starting codes with codimension 3 and 4 parity check matrices of the starting codes are treated as saturating sets in projective geometry that are obtained by computer search using projective properties of objects upper bounds on the length function and on the smallest sizes of saturating sets are given

oscillation of second order nonlinear neutral differential equations abstract in this paper some sufficient conditions for oscillation and nonoscillation are obtained for the second order nonlinear neutral differential equation x t p t x t q t f x t 0 t 0 where 0 p t 1 q t 0 0

maximal regularity for stochastic convolutions driven by levy processes we generalize the maximal regularity result from da prato and lunardi atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 9 1 25 29 1998 to stochastic convolutions driven by time homogenous poisson random measures and cylindrical infinite dimensional wiener processes