generalization of the norse bounds to codes of higher strength the norse bounds state that all codes of strength 1 and length n have covering radius at most n 2 and all self complementary codes of strength 2 and length n have covering radius at most n square root n 2 this is generalized to arbitrary even values of strength still assuming self complementarity and to odd strengths without this hypothesis the proof techniques used are probabilistic

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

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

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

maximal space regularity for abstract linear non autonomous parabolic equations the linear non autonomous evolution equation u t a t u t f t t e 0 t with the initial datum u o x in the space c o t e where e is a banach space and a t is a family of infinitesimal generators of bounded analytic semigroups is considered the domains d t are supposed constant in t and possibly not dense in e maximal regularity of the strict and classical solutions i e regularity of u and a u with values in the interpolation spaces d o l co and da b between d o and e is studied a characterization of such spaces in a concrete case is also given 0 1985 academic press inc

on the peak to mean envelope power ratio of phase shifted binary codes the peak to mean envelope power ratio pmepr of a code employed in orthogonal frequency division multiplexing ofdm systems can be reduced by permuting its coordinates and by rotating each coordinate by a fixed phase shift motivated by some previous designs of phase shifts using suboptimal methods the following question is considered in this paper for a given binary code how much pmepr reduction can be achieved when the phase shifts are taken from a 2h ary phase shift keying 2h psk constellation a lower bound on the achievable pmepr is established which is related to the covering radius of the binary code generally speaking the achievable region of the pmepr shrinks as the covering radius of the binary code decreases the bound is then applied to some well understood codes including nonredundant bpsk signaling bch codes and their duals reed muller codes and convolutional codes it is demonstrated that most presumably not optimal phase shift designs from the literature attain or approach our bound

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

on polynomial approximation of circular arcs and helices we present a simple method for polynomial approximation of circular arcs and helices by expressing the trigonometric functions using the two point taylor expansion we obtain the degree 2n 1 polynomial for the approximation problem in an efficient way which is very convenient to increase the degree of polynomial by adding new terms an upper bound on the approximation error is available so that we can obtain the lowest degree polynomial curve that can approximate a circular arc or helix segment within any user prescribed error tolerance

the best g 1 cubic and g 2 quartic bezier approximations of circular arcs we obtain cubic and quartic bezier approximations of circular arcs that respectively satisfy g 1 and g 2 end point interpolation conditions we identify the necessary and sufficient conditions for such approximations to be the best in the sense that they have the minimum hausdorff distance to the circular arc we then establish the existence and uniqueness of these best approximations and present practical methods to calculate them which are verified by examples

an arc spline approximation to a clothoid the clothoid is a spiral used in highway and railway route design clothoids are transcendental functions and so have been approximated by polynomials by power series and continued fractions and by rational functions here the clothoid is approximated by an arc spline the chief advantage in doing so is that arc splines are very easy to lay out and to offset examples show that the approximation is of extremely high accuracy it is proved that if the arc spline has n arcs then the error in the approximation is of order o 1 n2

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

weight distribution of cosets of small codes with good dual properties the bilateral minimum distance of a binary linear code is the maximum d such that all nonzero codewords have weights between d and n d let q subset 0 1 n be a binary linear code whose dual has bilateral minimum distance at least d where d is odd roughly speaking we show that the average l infty distance and consequently the l 1 distance between the weight distribution of a random cosets of q and the binomial distribution decays quickly as the bilateral minimum distance d of the dual of q increases for d theta 1 it decays like n theta d on the other d theta n extreme it decays like and e theta d it follows that almost all cosets of q have weight distributions very close to the to the binomial distribution in particular we establish the following bounds if the dual of q has bilateral minimum distance at least d 2t 1 where t geq 1 is an integer then the average vphantom sum r r l infty distance is at most min e ln n 2t t 2t n t 2 sqrt 2 e t 10 for the average l 1 distance we conclude the bound min 2t 1 e ln n 2t t 2t n t 2 1 sqrt 2 n 1 e t 10 which gives nontrivial results for t geq 3 we give applications to the weight distribution of cosets of extended hadamard codes and extended dual bch codes our argument is based on fourier analysis linear programming and polynomial approximation techniques

the covering radius of cyclic codes of length up to 31 corresp the covering radius is given for all binary cyclic codes of length less than or equal to 31 many of these codes are optimal in the sense of having the smallest possible covering radius of any linear code of that length and dimension

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

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

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

bounded solutions of linear periodic abstract parabolic equations on etudie le comportement asymptotique de l operateur d evolution parabolique g t s associe a une famille de generateurs de sous groupes analytiques a t t r dans un espace de banach general sous l hypothese de periodicite a t a t t on trouve de nouveaux resultats de regularite maximale pour les solutions bornees d equations paraboliques non homogenes abstraites dans des intervalles de temps non bornes

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

quintic polynomial approximation of log aesthetic curves by curvature deviation log aesthetic curves lacs possessing monotone curvature and including many classical curves have been widely used to describe fair shapes in geometric modeling however they are generally represented in non polynomial form and are thus not compatible with current cad systems in this paper we present quintic polynomial approximation of lac segments for a given lac segment a quintic g 2 interpolating bezier curve is obtained by minimizing a curvature based error metric with the advantage of being more likely to preserve the monotone curvature property numerical experiments demonstrate that our method can usually generate better results than the previous methods in terms of the deviation in positions and curvatures

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

cα regularity for non autonomous linear integrodifferential equations of parabolic type abstract linear integrodifferential equations in general banach space are studied and applications are given to linear integrodifferential partial differential equations

asymptotic bounds on the covering radius of binary codes asymptotic covering properties of families of binary codes are studied a gaussian approximation to the weight distribution of translates for codes of high strength is used from this an upper bound on the covering radius of these codes is deduced applications include reed muller codes quadratic residue codes and bch codes sufficient conditions for a family of codes to have best possible covering radius asymptotically perfect codes are derived

estimates for the range of binomiality in codes spectra we derive new estimates for the range of binomiality in a code s spectra where the distance distribution of a code is upperbounded by the corresponding normalized binomial distribution the estimates depend on the code s dual distance

planar two point g1 hermite interpolating log aesthetic spirals log aesthetic spirals are currently being studied as fair curves that can be used in computer aided design a family of planar log aesthetic spirals that include a point of zero curvature is used in this paper the two point g 1 hermite data that is considered has some restrictions on the angles this paper proves that for any member of the family a unique segment of that spiral can be found that matches given two point g 1 hermite data