improved derivative bounds of the rational quadratic bezier curves a new derivative bound of the rational quadratic bezier curves is obtained it is an improvement over the bound which was regarded as a sharp bound in a recent paper also the new bound is superior to the existing bounds given by some other authors

total positivity and its applications preface i spline functions total positivity of the spline kernel and its applications b bojanov total positivity properties of lb splines t lyche l l schumaker total positivity and splines k morken totally positive systems and interpolation by subspaces of generalized splines m sommer h strauss ii matrix theory preservation of total nonnegativity under the hadamard product and related topics j garloff d g wagner vertex implications for totally nonnegative under the hadamard product and related topics j garloff on factorizations of totally positive matrices j garloff on factorizations of totally positive matrices m gasca j m pena iii geometric modeling total positivity and optimal bases j m carnicer j m pena total positivity and the shape of curves t n t goodman tchebycheff curves m l mazure h pottmann iv probability and mathematical biology analytical and probabilistic methods for totally positive matrices g s goodman some inequalities of total positivity in pure and applied mathematics s karlin z ziegler new directions and problems of multivariate total positivity motivated by genetic recombination probabilities s karlin v approximation theory tchebycheff spaces and total positivity j m carnicer j m pena cebysev and weak cebysev systems r a zalik vi complex analysis zero sets of entire fourier transforms of multiply positive density functions o m katkova vii statistics total nonnegativity of moment matrices and its application to e optimal designs in polynomial regression b heiligers analysis of regression dependence in 2 xn bivariate distributions and some applications in contingency tables s kasala m b rao viii real analysis total positivity and nonlinear analysis r q jia solutions of certain functional equations s karlin alternance and snakes in the nonlinear finite moment problem k tatalian ix combinatorics the applications of total positivity to combinatorics and conversely f brenti x integral equations spectral properties of totally positive kernels and matrices a pinkus index

varieties with quadratic entry locus i we introduce and study l qel manifolds x subset mathbb p n of type a class of projective varieties whose extrinsic and intrinsic geometry is very rich especially when 0 we prove a strong divisibility property for lqel manifolds of type 3 allowing the classification of those of type delta geq frac dim x 2 in particular we obtain a new and very short proof that severi varieties have dimension 2 4 8 or 16 and also an almost self contained proof of their classification due to zak we also provide the classification of special cremona transformations of type 2 3 and 2 5

generalized hamming weights of nonlinear codes and the relation to the z sub 4 linear representation we give a new definition of generalized hamming weights of nonlinear codes and a new interpretation connected with it these generalized weights are determined by the entropy length profile of the code we show that this definition characterizes the performance of nonlinear codes on the wire tap channel of type ii the new definition is invariant under translates of the code it satisfies the property of strict monotonicity and the generalized singleton bound we check the relations between the generalized weight hierarchies of z sub 4 linear codes and their binary image under the gray map we also show that the binary image of a z sub 4 linear code is a symmetric not necessarily rectangular code moreover if this binary image is a linear code then it admits a twisted squaring construction

on manifolds swept out by high dimensional quadrics we give a completely different much shorter proof of a substantial generalization of the main result from kachi and sato mem am math soc 160 763 2002 it states that embedded projective n folds swept out by quadrics of dimension at least left frac n2 right 2 are either scrolls or hyperquadric fibrations which are also mori contractions

a characterization of certain griesmer codes mmd codes in a more general sense let c be an n k d sub q linear code the defect of c is the parameter s s c n k 1 d if k spl ges m 1 spl ges 2 then by the griesmer bound d spl les q sup m q 1 q sub m 1 s m the author s interest is in those linear codes having the maximum minimum distance i e d q sup m q 1 q sup m 1 s m for m 1 we have d q s 1 and the codes are maximum minimum distance mmd codes in the sense of faldum and willems see ibid vol 44 p 1555 58 1998 thus we consider mmd codes in a more general sense we refer to them simply as mmd codes all mmd codes with m 1 are known up to formal equivalence note that two codes are formally equivalent if they have the same weight distribution the author classifies up to formal equivalence the mmd codes with m spl ges 2

the duals of mmd codes are proper for error detection a linear code when used for error detection on a symmetric channel is said to be proper if the corresponding undetected error probability increases monotonically in spl epsiv the symbol error probability of the channel such codes are generally considered to perform well in error detection a number of well known classes of linear codes are proper e g the perfect codes mds codes macdonald s codes mmd codes and some near mds codes the aim of this work is to show that also the duals of mmd codes are proper

generalized hamming weights of linear codes the generalized hamming weight d sub r c of a binary linear code c is the size of the smallest support of any r dimensional subcode of c the parameter d sub r c determines the code s performance on the wire tap channel of type ii bounds on d sub r c and in some cases exact expressions are derived in particular a generalized griesmer bound for d sub r c is presented and examples are given of codes meeting this bound with equality

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

generalized bezier curves and surfaces based on lupas q analogue of bernstein operator in this paper a new generalization of bezier curves with one shape parameter is constructed it is based on the lupas q analogue of bernstein operator which is the first generalized bernstein operator based on the q calculus the new curves have some properties similar to classical bezier curves moreover we establish degree evaluation and de casteljau algorithms for the generalization furthermore we construct the corresponding tensor product surfaces over the rectangular domain and study the properties of the surfaces as well as the degree evaluation and de casteljau algorithms compared with q bezier curves and surfaces based on phillips q bernstein polynomials our generalizations show more flexibility in choosing the value of q and superiority in shape control of curves and surfaces the shape parameters provide more convenience for the curve and surface modeling

relative generalized hamming weights of one point algebraic geometric codes security of linear ramp secret sharing schemes can be characterized by the relative generalized hamming weights of the involved codes 26 24 in this paper we elaborate on the implication of these parameters and we devise a method to estimate their value for general one point algebraic geometric codes as it is demonstrated for hermitian codes our bound is often tight furthermore for these codes the relative generalized hamming weights are often much larger than the corresponding generalized hamming weights

parameters of goppa codes revisited we discuss parameters of goppa 1970 codes such as minimum distance covering radius distance distribution and generalized hamming weights by a variation on the exponential sums method and combinatorial arguments we sharpen known bounds on these parameters

the mmd codes are proper for error detection the undetected error probability of a linear code used to detect errors on a symmetric channel is a function of the symbol error probability spl epsi of the channel and involves the weight distribution of the code the code is proper if the undetected error probability increases monotonously in spl epsi proper codes are generally considered to perform well in error detection we show in this correspondence that maximum minimum distance mmd codes are proper

extending fundamental formulas from classical b splines to quantum b splines we derive a collection of fundamental formulas for quantum b splines analogous to known fundamental formulas for classical b splines starting from known recursive formulas for evaluation and quantum differentiation along with quantum analogues of the marsden identity we derive quantum analogues of the de boor fix formula for the dual functionals explicit formulas for the quantum b splines in terms of divided differences of truncated power functions formulas for computing divided differences of arbitrary functions by quantum integrating certain quantum derivatives of these functions with respect to the quantum b splines closed formulas for the quantum integral of the quantum b splines over their support and finally a 1 q convolution formula for uniform q b splines

optimal bases for a class of mixed spaces and their associated spline spaces we consider spaces of the form span where the functions u i i 1 4 are algebraic polynomials or trigonometric or hyperbolic functions we find intervals 0 a where we can guarantee that the spaces possess normalized totally positive bases and so shape preserving representations we construct their normalized b bases with optimal shape preserving properties we also present a unified approach to deal with the associated spline spaces a recursive procedure to obtain bases with optimal shape preserving and stability properties is presented

inductive characterizations of hyperquadrics we give two characterizations of hyperquadrics one as non degenerate smooth projective varieties swept out by large dimensional quadric subvarieties passing through a point the other as lqel manifolds with large secant defects

the leech lattice and the golay code bounded distance decoding and multilevel constructions multilevel constructions of the binary golay code and the leech lattice are described both constructions are based upon the projection of the golay code and the leech lattice onto the 6 3 4 hexacode over gf 4 however unlike the previously reported constructions the new multilevel constructions make the three levels independent by way of using a different set of coset representatives for one of the quaternary coordinates based upon the multilevel structure of the golay code and the leech lattice efficient bounded distance decoding algorithms are devised the bounded distance decoder for the binary golay code requires at most 431 operations as compared to 651 operations for the best known maximum likelihood decoder efficient bounded distance decoding of the leech lattice is achieved by means of partitioning it into four cosets of q sub 24 beyond the conventional partition into two h sub 24 cosets the complexity of the resulting decoder is only 953 real operations on the average and 1007 operations in the worst case as compared to about 3600 operations for the best known in maximum likelihood decoder it is shown that the proposed algorithms decode correctly at least up to the guaranteed error correction radius of the maximum likelihood decoder thus the loss in coding gain is due primarily to an increase in the effective error coefficient which is calculated exactly for both algorithms furthermore the performance of the leech lattice decoder on the awgn channel is evaluated experimentally by means of a comprehensive computer simulation the results show a loss in coding gain of less than 0 1 db relative to the maximum likelihood decoder for ber ranging from 10 sup 1 to 10 sup 7

improved bounds on the magnitude of the derivative of rational bezier curves abstract in this paper we derive some new derivative bounds of rational bezier curves according to some existing identities and inequalities the comparison of the new bounds with some existing ones is also presented

the preparata and goethals codes trellis complexity and twisted squaring constructions the trellis complexity of the preparata and goethals codes is examined it is shown that at least for a given set of permutations these codes are rectangular upper bounds on the state complexity profiles of the preparata and goethals codes are given the upper bounds on the state complexity of the preparata and goethals codes are determined by the dimension length profiles dlp of the extended primitive double and triple error correcting bch codes respectively a twisted squaring construction for the preparata and goethals codes is given based on the double and triple error correcting extended primitive bch codes respectively

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

convexity of rational curves and total positivity we analyze convexity preserving properties of curves from a geometric point of view we also characterize totally positive systems of functions in terms of geometric convexity preserving properties of the rational curves rational bezier and nonuniform rational b spline curves are included in this setting

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

density length profiles and trellis complexity of lattices the density length profile dcp of a lattice spl lambda is analogous to the dimension length profile of a linear code the dlp is a geometrical invariant of spl lambda that includes the coding gain of spl lambda duality results analogous to those of linear block codes are derived for lattices bounds on the dlp may be derived from bounds on hermite s constants these hold with equality for many dense lattices in turn the dlp lowerbounds the state complexity profile of a minimal trellis diagram for spl lambda in any coordinate system it is shown that this bound can be met for the e sub 8 lattice by a laminated lattice construction with a novel trellis diagram bounds and constructions for other important low dimensional lattices are given two laminated lattice constructions of the leech lattice yield trellis diagrams with maximum state space sizes 1024 and 972

on the voronoi neighbor ratio for binary linear block codes soft decision decoding of block codes is regarded as the geometrical problem of identifying the voronoi region within which a given input vector lies a measure called the neighbor ratio is proposed to characterize how many facets a voronoi region has theory and algorithms are presented to determine the neighbor ratio for binary linear block codes and results are given for several types of codes an asymptotic analysis for long codes reveals that the neighbor ratio depends on whether the code rate is less than 1 2 or not for rates below this threshold all pairs of codewords tend to share a voronoi facet for higher rates a relatively small fraction of them do

accurate computations with lupa matrices lupa q analogues of the bernstein functions play an important role in approximation theory and computer aided geometric design their collocation matrices are called lupa matrices in this paper we provide algorithms for computing the bidiagonal decomposition of these matrices and their inverses to high relative accuracy it is also shown that these algorithms can be used to perform to high relative accuracy several algebraic calculations with these matrices such as the calculation of their inverses their eigenvalues or their singular values numerical experiments are included

an extension of fujita s non extendability theorem for grassmannians in this paper we study smooth complex projective varieties x containing a grassmannian of lines mathbb g 1 r which appears as the zero locus of a section of a rank two nef vector bundle e among other things we prove that the bundle e cannot be ample

bounds on the minimum support weights the minimum support weight d sub r c of a linear code c over gf q is the minimal size of the support of an r dimensional subcode of c a number of bounds on d sub r c are derived generalizing the plotkin bound and the griesmer bound as well as giving two new existential bounds as the main result it is shown that there exist codes of any given rate r whose ratio d sub rd sub 1 is lower bounded by a number ranging from q sup r 1 q sup r q sup r 1 to r depending on r

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

some remarks on weighted lupas q bezier curves abstract recently han et al 2016 introduced weighted lupas q bezier curves as a novel generalization of the classical rational bezier curves in this note we make some remarks on their article in particular we revisit the rational quadratic case for an illustrative exposition

lexicographic codes error correcting codes from game theory lexicographic codes or lexicodes are defined by various versions of the greedy algorithm the theory of these codes is closely related to the theory of certain impartial games which leads to a number of surprising properties for example lexicodes over an alphabet of size b 2 a are closed under addition while if b 2 2 a the lexicodes are closed under multiplication by scalars where addition and multiplication are in the nim sense explained in the text hamming codes and the binary golay codes are lexieodes remarkably simple constructions are given for the steiner systems s 5 6 12 and s 5 8 24 several record breaking constant weight codes are also constructed

on the bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

the leech lattice the octacode and decoding algorithms new multilevel constructions of the golay code and the leech lattice are presented these are derived from the turyn construction and the holy construction with the octacode as the glue code further we show that the holy construction of the leech lattice with the octacode as the glue code is essentially different from the permuted turyn construction although both constructions rely on the octacode the turyn construction is based on an odd type of the octacode whereas any type of the octacode can be used in the holy construction moreover the multilevel representation of the holy construction leads to a novel lattice partition chain based on these structures we derive new bounded distance decoders for the golay code and the leech lattice whose effective error coefficient is smaller than that of any previously known bounded distance decoder we provide a general theorem for computing the effective error coefficient of coset decoding with bounded distance decoding for the subcode

parallel subspace correction methods for nearly singular systems abstract in this paper we consider the parallel subspace correction psc methods for the nearly singular systems we apply the psc methods as the preconditioners when we solve the nearly singular systems by the conjugate gradient methods our focus is to estimate the condition number of the preconditioned systems we deduce the parameter independent estimates on the psc preconditioners for the nearly singular systems under appropriate assumptions on subspace decomposition the main assumption is that the kernel of the singular part of the system can be decomposed into a sum of local kernel subspaces our estimates can be applied into actual problems and two examples are analyzed in this paper one is the elliptic problem with large jumps in the coefficients the other is the planar nearly incompressible elasticity problem with the scott vogelius finite element discretization we prove that the related parallel multilevel methods for both examples are convergent uniformly with respect to the coefficients and the mesh size

trellis decoding complexity of linear block codes in this partially tutorial paper we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity maximum state and edge dimensions total span length and total vertices edges and mergers we obtain bounds on these complexities as extensions of well known dimension length profile dlp bounds codes meeting these bounds minimize all the complexity measures simultaneously conversely a code attaining the bound for total span length vertices or edges must likewise attain it for all the others we define a notion of uniform optimality that embraces different domains of optimization such as different permutations of a code or different codes with the same parameters and we give examples of uniformly optimal codes and permutations we also give some conditions that identify certain cases when no code or permutation can meet the bounds in addition to dlp based bounds we derive new inequalities relating one complexity measure to another which can be used in conjunction with known bounds on one measure to imply bounds on the others as an application we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths

on the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes it was shown earlier that for a punctured reed muller rm code or a primitive bch code which contains a punctured rm code of the same minimum distance as a large subcode the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions the problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized hamming weight hierarchy of a code and it is shown that for rm codes the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to v k wei 1991 the state complexity of the trellis diagram for the extended and permuted 64 24 bch code is discussed

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

trellis complexity and minimal trellis diagrams of lattices this paper presents results on trellis complexity and low complexity trellis diagrams of lattices we establish constructive upper bounds on the trellis complexity of lattices these bounds both improve and generalize the similar results of tarokh and vardy see ibid vol 43 p 1294 1300 1997 we also construct trellis diagrams with minimum number of paths for some important lattices such trellises are called minimal the constructed trellises which are novel in many cases can be employed to efficiently decode the lattices via the viterbi algorithm in particular a general structure for minimal trellis diagrams of d sub n lattices is obtained this structure corresponds to a new code formula for d sub n moreover we develop some important duality results which are used in both deriving the upper bounds and finding the minimal trellises all the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases

chains demi matroids and profiles it is shown that each chain of linear codes has an associated demi matroid a combinatorial structure that extends the notion of a vector matroid of a linear code these demi matroids are proven to determine important properties of the chain and it is shown that linear code chain duality is represented by demi matroid duality in a natural and yet surprising way profiles are defined for arbitrary demi matroids and thus for linear code chains which generalizes previous results in the literature and provides simple and transparent proofs thereof finally applications are given to encryption of messages through wire tap channels of type ii

on the bounds of the derivative of rational bezier curves according to some identities and inequalities about the intermediate weights and control points of the de casteljau algorithm for rational bezier curve we derive a new bound of the derivative of rational bezier curves the comparison of the new bounds with some existing ones is also presented based on some previous bounds the bound presented in this paper and many numerical examples we propose a conjecture of the bound for the derivative of rational bezier curves

on the lupas q transform the lupasq transform emerges in the study of the limit q lupas operator the latter comes out naturally as a limit for a sequence of the lupas q analogues of the bernstein operator lately it has been studied by several authors from different perspectives in mathematical analysis and approximation theory this operator is closely related to the q deformed poisson probability distribution which is used widely in the q boson operator calculus given email protected 0 1 email protected c 0 1 the q lupas transform of f is defined by l qf z 1 z q email protected email protected k 0 f 1 q k q k k 1 2 q q kz k in this paper we study some analytic properties of l qf z in particular we examine the conditions under which l qf can either be an entire function or a rational one

generalized hamming weights for linear codes motivated by cryptographical applications the algebraic structure of linear codes from a new perspective is studied by viewing the minimum hamming weight as a certain minimum property of one dimensional subcodes a generalized notion of higher dimensional hamming weights is obtained these weights characterize the code performance on the wire tap channel of type ii basic properties of generalized weights are derived the values of these weights for well known classes of codes are determined and lower bounds on code parameters are obtained several open problems are also listed

normalized b basis of the space of trigonometric polynomials and curve design a normalized b basis of the space of trigonometric polynomials of degree n is presented some interesting properties of the basis functions are given based on the basis symmetric trigonometric polynomial curves like bezier curves are constructed the trigonometric polynomial curves present the shape of their control polygons well thus the theoretics and methods are proposed for curve representation of the trigonometric polynomial space by adding additional control points the given graph examples show that the trigonometric polynomial curves are nearer their control polygons than the bezier curves for the same parametric variable and the same degree

complex and integral laminated lattices in an earlier paper we studied real laminated lattices or z modules a where a is the lattice of even integers and a is obtained by stacking layers of a suitable n l dimensional lattice a as densely as possible keeping the same minimal norm the present paper considers two generalizations a complex and quaternionic lattices obtained by replacing z module by module where j may be the eisenstein gaussian or hurwitzian integers etc and b integral laminated lattices in which a is required to be an integral lattice with the prescribed minimal norm this enables us to give a partial answer to a question of j g thompson on integral lattices and to explain some of the computer generated results of plesken and pohst on this problem also a number of familiar lattices now arise in a canonical way for example the coxeter todd lattice is the 6 dimensional integral laminated lattice over z to of minimal norm 2 the paper includes tables of the best real integral lattices in up to 24 dimensions

q bernstein polynomials and bezier curves we define q bernstein polynomials which generalize the classical bernstein polynomials and show that the difference of two consecutive q bernstein polynomials of a function f can be expressed in terms of second order divided differences of f it is also shown that the approximation to a convex function by its q bernstein polynomials is one sided a parametric curve is represented using a generalized bernstein basis and the concept of total positivity is applied to investigate the shape properties of the curve we study the nature of degree elevation and degree reduction for this basis and show that degree elevation is variation diminishing as for the classical bernstein basis

weighted lupas q bezier curves this paper is concerned with a new generalization of rational bernstein bezier curves involving q integers as shape parameters a one parameter family of rational bernstein bezier curves weighted lupas q bezier curves is constructed based on a set of lupas q analogue of bernstein functions which is proved to be a normalized totally positive basis the generalized rational bezier curve is investigated from a geometric point of view the investigation provides the geometric meaning of the weights and the representation for conic sections we also obtain degree evaluation and de casteljau algorithms by means of homogeneous coordinates numerical examples show that weighted lupas q bezier curves have more modeling flexibility than classical rational bernstein bezier curves and lupas q bezier curves and meanwhile they provide better approximations to the control polygon than rational phillips q bezier curves

on the derivative formula of a rational bezier curve at a corner in this paper we develop the exact arbitrary order derivative formula of a rational bezier curve at its corner some examples are presented to demonstrate the computation of the exact derivatives

classification of non degenerate projective varieties with non zero prolongation and application to target rigidity the prolongation mathfrak g k of a linear lie algebra mathfrak g subset mathfrak gl v plays an important role in the study of symmetries of g structures cartan and kobayashi nagano have given a complete classification of irreducible linear lie algebras mathfrak g subset mathfrak gl v with non zero prolongations

a new sphere packing in 20 dimensions we construct a new nonlattice sphere packing in 20 dimensions which is denser than any 20 dimensional packing previously known properties of the new sphere packing are investigated

bezier curves based on lupas p q analogue of bernstein functions in cagd abstract this paper deals with the extension of rational lupas bernstein functions lupas bezier curves and surfaces involving p q integers as shape parameters for all p 0 and q 0 two different techniques such as de casteljau s algorithm and korovkin s type approximation based on p q integers are used a two parameter family for lupas p q bernstein functions is constructed and their degree elevation and reduction properties have been studied for lupas p q bezier curves some of their basic properties as well as degree elevation and de casteljau algorithm have been discussed the new curves have some properties similar to lupas q bezier curves similarly the corresponding tensor product for lupas bezier surfaces over the rectangular domain u v 0 1 0 1 depending on four parameters is constructed the de casteljau algorithm and degree evaluation properties of the surfaces for these generalizations and some fundamental properties are discussed we get lupas q bezier surfaces for u v 0 1 0 1 when we set the parameter p 1 p 2 1 with the help of rational lupas bernstein functions p q lupas bernstein operators are constructed based on korovkin s type approximation it has been shown that the sequence of p q analogue of lupas bernstein operators l p n q n n f x converges uniformly to f x c 0 1 if and only if 0 q n p n 1 such that lim n q n 1 lim n p n 1 and lim n p n n 1 lim n q n n 1 on the other hand for any p 0 fixed and p 1 the sequence l p q n f x converges uniformly to f x c 0 1 if and only if f x a x b for some a b r furthermore in comparison to q bezier curves and surfaces based on lupas q bernstein rational functions our generalization gives us more flexibility in controlling the shapes of curves and surfaces

suboptimal decoding of linear codes partition technique general symmetric channels are introduced and near maximum likelihood decoding in these channels is studied first we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis it is proved that the decoding error probability of suboptimal decoding is bounded above for any q ary code of length n and code rate r by twice the error probability of its maximum likelihood decoding and tends to the latter as n grows second we design a suboptimal trellis like algorithm which reduces the known decoding complexity of the order of q sup n min r 1 r operations to that of q sup nr i r operations for all cyclic codes and virtually all long linear codes we also consider the corresponding bounds for concatenated codes an important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability

a characterization of mmd codes let c be a linear n k d code over gf q with k spl ges 2 if s n k 1 d denotes the defect of c then by the griesmer bound d spl les s 1 q now for obvious reasons we are interested in codes of given defect s for which the minimum distance is maximal i e d s 1 q we classify up to formal equivalence all such linear codes over gf q remember that two codes over gf q are formally equivalent if they have the same weight distribution it turns out that for k spl ges 3 such codes exist only in dimension 3 and 4 with the ternary extended golay code the ternary dual golay code and the binary even weight code as exceptions in dimension 4 they are related to ovoids in pg 3 q except the binary extended hamming code and in dimension 3 to maximal arcs in pg 2 q

derivative bound estimations on rational conic bezier curves abstract in this paper we derive a bound on derivatives of rational conic bezier curves both theoretical analysis and numerical examples show that our bound is sharper than some existed ones discussions on optimal bounds for the derivatives of rational conic bezier curves are also provided

bernstein schurer kantorovich operators based on q integers in this paper we introduce a new kantorovich type generalization of the q bernstein schurer operators defined in muraru 2011 first we give the basic convergence theorem and then obtain the local direct results for these operators estimating the rate of convergence by using the modulus of smoothness and the lipschitz type maximal function respectively we also obtain a voronovskaja type theorem and investigate the statistical approximation properties of these operators with the help of a korovkin type statistical approximation theorem given in duman 2008

new minimal bounds for the derivatives of rational bezier paths and rational rectangular bezier surfaces new minimal bounds are derived for the magnitudes of the derivatives of the rational bezier paths and the rational rectangular bezier surface patches of arbitrary degree which improve previous work of this type in many cases moreover our new bounds are explicitly given by simple and closed form expressions an important advantage of the closed form expressions is that they allow us to prove that our bounds are sharp under certain well defined conditions some numerical examples highlighting the potential of the new bounds in providing improved estimates are given in an appendix

codes satisfying the chain condition the authors considered weight hierarchies of codes satisfying the chain condition they called these chain good first they gave a set of simple necessary conditions for a sequence to be chain good they proved that given one chain good sequence there is an infinite set of chain good sequences that can be constructed from this one sequence finally they used this result to completely describe the sets of chain good sequences of dimensions up to 5

the weight hierarchies of q ary codes of dimension 4 the weight hierarchy of a linear n k q code c over gf q is the sequence d sub 1 d sub 2 d sub k where d sub spl tau is the smallest support of an spl tau dimensional subcode of c the possible weight hierarchies of n 4 q codes are studied in particular the possible weight hierarchies of n 4 3 codes are determined

minimum support weights of binary codes some relations between the minimum support weights are discussed in particular the possible weight hierarchies of codes of dimension 4 are determined

a novel generalization of bezier curve and surface a new formulation for the representation and designing of curves and surfaces is presented it is a novel generalization of bezier curves and surfaces firstly a class of polynomial basis functions with n adjustable shape parameters is present it is a natural extension to classical bernstein basis functions the corresponding bezier curves and surfaces the so called quasi bezier i e q bezier for short curves and surfaces are also constructed and their properties studied it has been shown that the main advantage compared to the ordinary bezier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters when the control polygon is maintained the q bezier curve and surface inherit the most properties of bezier curve and surface and can be more approximated to the control polygon it is visible that the properties of end points on q bezier curve and surface can be locally controlled by these shape parameters some examples are given by figures

the construction of λµ b spline curves and its application to rotational surfaces in order to provide more flexible approaches for designers the b spline curves are constructed as a generalization of the traditional cubic uniform b spline curves possessing multiple local shape control parameters b spline curves not only inherit the properties of cubic uniform b spline curves but also exhibit better performance when adjusting its local shapes through two local shape control parameters particularly to adjust and control the shapes of rotational surfaces more elegantly the b spline rotational surfaces with two local shape parameters are presented and utilized a rotational surface is produced by combining b spline with a transfinite vector valued rational interpolation function further the properties of rotational surfaces as well as its applications in rotational surface designs are explored finally the modeling examples are supplied to illustrate the proposed method in admitting the easy control of the shape of a surface which suggest the much wider applications to the pattern design system of apparel cad cam