the degree theorem in higher rank let n be any closed riemannian manifold in this paper we prove that for most locally symmetric nonpositively curved riemannian manifolds m and for every continuous map f n m the map f is homotopic to a smooth map with jacobian bounded by a universal constant depending as it must only on ricci curvature bounds of n from this we deduce an extension of gromov s volume comparison theorem for negatively curved manifolds to most nonpositively curved locally symmetric manifolds

simplical volume of closed locally symmetric spaces of non compact type we show that closed locally symmetric spaces of non compact type have positive simplicial volume this gives a positive answer to a question that was first raised by gromov in inst hautes etudes sci publ math 56 5 1982

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

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

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

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

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

erratum for the degree theorem in higher rank the purpose of this erratum is to correct a mistake in the proof of theorem 4 1 of cf in this note we fix a mistake in theorem 4 1 of cf this error was pointed out to us by inkang kim and sungwoon kim to whom we are extremely grateful the error occurs in the final step of the proof of theorem 4 4 in cf the stated angle inequality should hold not just for a subspace v but for each individual vector in an orthonormal k frame the problem with the proof in cf occurs at the very end of 5 at the top of page 52 lemma 5 4 in cf applies to all of v k but one needs to justify that this lemma applies to the subspace w the fix throughout the present paper we use the notation and terminology of cf the setup is as follows let x g k be a symmetric space of noncompact type with no local r h2 or sl3 r so 3 factors we also assume cf 4 2 of cf that x is irreducible we fix a point x x and a maximal flat f through x x the stabilizer of x in g is after conjugation k and k acts by the derivative action on the tangent space txx which we identify as a subspace p of the lie algebra g p k of g endowed with the standard inner product coming from the killing form b we identify f and f with their corresponding tangent spaces in txx as in 4 4 of cf define the angle between two subspaces v w txx as v w inf dso txx i p p so h with pv w or pw v we do not see how to prove theorem 4 4 of cf called there the eigenvalue matching theorem as stated we instead prove the following result call a set of vectors w1 wk is a orthonormal k frame if wi wj for all 1 i j k theorem 0 1 weak eigenvalue matching for each symmetric space x as above there are constants c1 and c so that the following holds given any 1 rank x 1 2 for any orthonormal k frame v1 vk in txx with k rank x whose span v satisfies v f there is a c1eorthonormal 2k frame given by vectors v 1 v 1 v k v k such that for i 1 k 0 1 hv i f c hvi f

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

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

matrix characterization of mds linear codes over modules abstract let r be a commutative ring with identity n be an r module and m aij r k be a matrix over r a linear code c of length n over n is defined to be a submodule of nn it is shown that a linear code c k r with parity check matrix m i r is maximum distance separable mds iff the determinant of every h h submatrix h 1 2 min k r of m is not an annihilator of any nonzero element of n this characterization is used to derive some results for group codes over abelian groups

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

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

on line versus off line learning from random examples general results i propose a general model of on line learning from random examples which when applied to a smooth realizable stochastic rule yields the same asymptotic generalization error rate as optimal batch algorithms the approach is based on an iterative gaussian approximation to the posterior gibbs distribution of rule parameters

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

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

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

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

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

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

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

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

proportionality principle for the simplicial volume of families of mathbb q rank 1 locally symmetric spaces we establish the proportionality principle between the riemannian volume and locally finite simplicial volume for mathbb q rank 1 locally symmetric spaces covered by products of hyperbolic spaces giving the first examples for manifolds whose cusp groups are not necessarily amenable also we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of mathbb q rank 1 locally symmetric spaces with amenable cusp groups established by loh and sauer 26

learning dynamics on different timescales the special character of certain degrees of freedom in two layered neural networks is investigated for on line learning of realizable rules our analysis shows that the dynamics of these degrees of freedom can be put on a faster timescale than those remaining with the profit of speeding up the overall adaptation process this is shown for two groups of degrees of freedom second layer weights and bias weights for the former case our analysis provides a theoretical explanation of phenomenological findings the resulting learning algorithm is compared with natural gradient descent in order to check whether the proposed scaling can be naturally derived from that type of learning rule

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

adaptive and self averaging thouless anderson palmer mean field theory for probabilistic modeling we develop a generalization of the thouless anderson palmer tap mean field approach of disorder physics which makes the method applicable to the computation of approximate averages in probabilistic models for real data in contrast to the conventional tap approach where the knowledge of the distribution of couplings between the random variables is required our method adapts to the concrete set of couplings we show the significance of the approach in two ways our approach reproduces replica symmetric results for a wide class of toy models assuming a nonglassy phase with given disorder distributions in the thermodynamic limit on the other hand simulations on a real data model demonstrate that the method achieves more accurate predictions as compared to conventional tap approaches

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

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

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

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

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

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

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

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

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

simplicial volume barycenter method and bounded cohomology we prove that the locally finite simplicial volume of most mathbb q rank 1 locally symmetric spaces is positive which has been open for many years as a corollary we improve the degree theorem of connell and farb for mathbb q rank 1 locally symmetric spaces during the course of a proof we show that codimension one dimensional jacobian of the barycentric straightening map is uniformly bounded for most higher rank symmetric spaces we also address the issue of surjectivity of the comparison map for semisimple lie groups of rank 2

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

dimension and rigidity of quasi fuchsian representations let fo c so n 1 n 2 be a cocompact lattice and p fo f be an injective representation into a convex cocompact discrete isometric subgroup of a noncompact rank 1 symmetric space the hausdorff dimension 6 f of the limit set of f p fo satisfies 6 f 6 fo n 1 we prove that equality holds if and only if p is a fuchsian representation i e f preserves a totally geodesic copy of hri in hr this generalizes the result of 2 and settles a question raised by tukia 43 p 428 actually we prove a more general result in the context of variable negative curvature strikingly there are no quasi fuchsian representations at least for the lower codimensional case in complex hyperbolic geometry that is for a cocompact lattice fo c su n 1 n 2 and an injective representation p fo su m 1 n m 2n 1 with f p fo convex cocompact we prove that one always has 6 f 6 fo and moreover f must stabilize a totally geodesic copy of h in h this can be viewed as a global generalization of goldman and millson s local rigidity theorem see 20 another global generalization was obtained by k corlette 5 various other related rigidity results are also obtained

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

the ergodic theory of discrete isometry groups on manifolds of variable negative curvature this paper studies the ergodic theory at infinity of an arbitrary discrete isometry group acting on any hadamard manifold h of pinched variable negative curvature most of the results obtained by sullivan in the constant curvature case are generalized to the case of variable curvature we describe connections between measures supported on the limit set of dynamics of the geodesic flow and the geometry of m h we explore the relationship between the growth exponent of the group the hausdorff dimension of the limit set and the topological entropy of the geodesic flow the equivalence of various descriptions of an analogue of the hopf dichotomy is proved as applications we settle a question of j feldman and m ratner about the horocycle flow on a finite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points at the end of this paper we apply our results to the study of some rigidity problems more applications to mostow rigidity for discrete subgroups of rank 1 noncompact semisimple lie groups with infinite covolume will be published in subsequent papers by the author notations h is a hadamard manifold with pinched sectional curvature k 2 k k 1 k2 k1 0 is a non elementary discrete isometry group acting on h freely and properly discontinuously which is called fuchsian h is the ideal boundary of h m h is a complete riemannian manifold sh or sm is the unit tangent bundle g is the geodesic flow on sh or sm v t is the unique geodesic in sh resp sm with the initial velocity v 0 v sh resp sm v resp v is the asymptotic class of the geodesic v t resp v t l is the limit set of on h l is the radical limit set of in l is the nonwandering set of the geodesic flow on sm received by the editors may 7 1995 1991 mathematics subject classification primary 58f17 secondary 58f11 58f15 20h10 research at msri supported by nsf grant dms 8505550 also partially supported by nsf grant dms 9403870 and sfb 170 c 1996 american mathematical society

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

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

dynamics of batch training in a perceptron early stopping and weight decay are studied in a linear perceptron using a new simplified approach to the dynamics in the thermodynamical limit the approach is directly deduced from the gradient descent weight update it allows an exact description of the dynamics of the batch training process the results are compared with a recent study of early stopping and weight decay based on the equilibrium statistical mechanics approach it is shown that the equilibrium results for early stopping are good approximations and are exact for weight decay furthermore in the dynamical approach it is possible to determine the necessary number of training steps to fulfil certain termination conditions it can be shown that asymptotically i e if the number of examples is large only two batch steps are required to reach the optimal convergence if the learning rate is optimally chosen

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

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

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

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

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

sparse on line gaussian processes we develop an approach for sparse representations of gaussian process gp models which are bayesian types of kernel machines in order to overcome their limitations for large data sets the method is based on a combination of a bayesian on line algorithm together with a sequential construction of a relevant subsample of the data that fully specifies the prediction of the gp model by using an appealing parameterization and projection techniques in a reproducing kernel hilbert space recursions for the effective parameters and a sparse gaussian approximation of the posterior process are obtained this allows for both a propagation of predictions and bayesian error measures the significance and robustness of our approach are demonstrated on a variety of experiments

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

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

new short coverings of fq3 from pairwise weakly linearly independent sets abstract pairwise weakly linearly independent pwli sets were introduced in order to compute sharp cyclic coverings of a finite module over a finite ring in this work new relationships between pwli sets and short coverings of the linear space f q 3 are explored as a consequence we investigate the minimum cardinality of a short covering of f q 3 when q is odd in order to improve a previous upper bound

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

superefficiency in blind source separation blind source separation is the problem of extracting independent signals from their mixtures without knowing the mixing coefficients nor the probability distributions of source signals and may be applied to eeg and meg imaging of the brain it is already known that certain algorithms work well for the extraction of independent components the present paper is concerned with superefficiency of these based on the statistical and dynamical analysis in a statistical estimation using t examples the covariance of any two extracted independent signals converges to 0 of the order of 1 t on line dynamics shows that the covariance is of the order of spl eta when the learning rate spl eta is fixed to a small constant in contrast with the above general properties a surprising superefficiency holds in blind source separation under certain conditions where superefficiency implies that covariance decreases in the order of 1 t sup 2 or of spl eta sup 2 the paper uses the natural gradient learning algorithm and method of estimating functions to obtain superefficient procedures for both batch estimation and on line learning a standardized estimating function is introduced to this end superefficiency does not imply that the error variances of the extracted signals decrease in the order of 1 t sup 2 or spl eta sup 2 but implies that their covariances and independencies do