optimal information bit decoding of linear block codes we consider the problem of decoding a linear block code used over a binary symmetric channel with crossover probability p when the goal of the decoding is to minimize the probability of an information bit error in general there will be several different decoding strategies each corresponding to the optimal rule over some range p spl isin p sub i 1 p sub i each of these strategies can be implemented by standard array with the appropriate choice of coset leaders that are not in general of minimum weight we give optimal rules for selecting coset leaders in the first region 0 p sub 1 and the region p sub max 1 2 for any linear mapping between information bits and codewords the determination of the optimal rule has been an open problem for small p the rule for p p sub max is important because for many codes when p is fixed the optimal rule for any coset corresponds to either the optimal rule for small p or the optimal rule for p near 1 2 the optimal decoding strategy varies with the generator matrix used in general but it is demonstrated that the optimal generator matrix is separable when d sup spl perp 2 the results are shown to extend in a natural way to the case where the goal is to minimize the probability that arbitrary collections of information bits are in error we examine some specific codes and compare the performance under optimal decoding to the performance of previously suggested strategies

topological gauge theories and group cohomology we show that three dimensional chern simons gauge theories with a compact gauge groupg not necessarily connected or simply connected can be classified by the integer cohomology grouph4 bg z in a similar way possible wess zumino interactions of such a groupg are classified byh3 g z the relation between three dimensional chern simons gauge theory and two dimensional sigma models involves a certain natural map fromh4 bg z toh3 g z we generalize this correspondence to topological spin theories which are defined on three manifolds with spin structure and are related to what might be calledz2 graded chiral algebras or chiral superalgebras in two dimensions finally we discuss in some detail the formulation of these topological gauge theories for the special case of a finite group establishing links with two dimensional holomorphic orbifold models

subvarieties of suc 2 and 2theta divisors in the jacobian let suc 2 l denote the projective moduli variety of semistable rank 2 vector bundles with determinant l pic c on a smooth curve c of genus g 2 and suppose that deg l is even it is well known that on the one hand the singular locus of suc 2 l is isomorphic to the kummer variety of the jacobian and on the other hand that when c is nonhyperelliptic suc 2 o has an injective morphism into the linear series 2 on the jacobian jg 1 c which restricts to the kummer embedding a 7 a a on the singular locus dually suc 2 k injects into the linear series l on j c where l o 2 for any theta characteristic and again this map restricts to the kummer map jg 1 c 2 l on the singular locus this map to projective space the two cases are of course isomorphic comes from the complete series on the ample generator of the picard group and at least for a generic curve is an embedding of the moduli space moreover its image contains much of the geometry studied in connection with the schottky problem notably the configuration of prym kummer varieties in this paper we explore a little of the interplay via this embedding between the geometry of vector bundles and the geometry of 2 divisors on the vector bundle side we are principally concerned with the brill noether loci wr suc 2 k defined by the condition h e r on stable bundes e these are analogous to the very classical varieties w r g 1 jg 1 c unlike the line bundle theory however general results connectedness dimension smoothness and so on are not known for the varieties wr see 6 on the 2 side we shall consider the fay trisecants of the 2 embedded kummer variety and the subseries p 00 l consisting of divisors having multiplicity 4 at the origin this subseries is known to be important in the study of principally polarised abelian varieties 10 in the jacobian of a curve its base locus is the surface c c j c plus a pair of isolated points

heat kernel expansion on a generalized cone it is shown that the short time expansion of the integrated heat kernel on a locally flat generalized cone c n as defined by cheeger consists of just the weyl volume term and the constant term this latter is explicitly evaluated when n is a lens space sd zm for odd d elliptic space sd z2 for all d and any of the three dimensional homogeneous space forms s3 agreement is found with the corresponding expansion on the orbifold version t4 z2 of the k3 surface and in fact with all td z2

rate gains in block coded modulation systems with interblock memory this paper examines the performance gains achievable by adding interblock memory to and altering the mapping of coded bits to symbols in block coded modulation systems the channel noise considered is additive gaussian and the twin design goals are to maximize the asymptotic coding gain and to minimize the number of apparent nearest neighbors in the case of the additive white gaussian noise channel these goals translate into the design of block codes of a given weighted or normalized distance whose rate is as high as possible and whose number of codewords at minimum normalized distance is low the effect of designing codes for normalized distance rather than hamming distance is to ease the problem of determining the best codes for given parameters in the cases of greatest interest and many such best codes are given

a conic bundle degenerating on the kummer surface let c be a genus 2 curve and mathcal su c 2 the moduli space of semi stable rank 2 vector bundles on c with trivial determinant in bolognesi adv geom 7 1 113 144 2007 we described the parameter space of non stable extension classes of the canonical sheaf of c by 1 in this paper we study the classifying rational map varphi mathbb p ext 1 omega omega 1 cong mathbb p 4 dashrightarrow mathcal su c 2 cong mathbb p 3 that sends an extension class to the corresponding rank two vector bundle moreover we prove that if we blow up mathbb p 4 along a certain cubic surface s and mathcal su c 2 at the point p corresponding to the bundle mathcal o oplus mathcal o then the induced morphism tilde varphi bl s rightarrow bl p mathcal su c 2 defines a conic bundle that degenerates on the blow up at p of the kummer surface naturally contained in mathcal su c 2 furthermore we construct the mathbb p 2 bundle that contains the conic bundle and we discuss the stability and deformations of one of its components

gromov invariants for holomorphic maps from riemann surfaces to grassmannians two compactifications of the space of holomorphic maps of fixed degree from a compact riemann surface to a grassmannian are studied it is shown that the uhlenbeck compactification has the structure of a projective variety and is dominated by the algebraic compactification coming from the grothendieck quot scheme the latter may be embedded into the moduli space of solutions to a generalized version of the vortex equations studied by bradlow this gives an effective way of computing certain intersection numbers known as gromov invariants on the space of holomorphic maps into grassmannians we carry out these computations in the case where the riemann surface has genus one department of mathematics university of utah salt lake city utah 84112 e mail address bertram math utah edu department of mathematics brown university providence rhode island 02912 e mail address daskal gauss math brown edu department of mathematics university of california irvine california 92717 e mail address raw math uci edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use

renormalization group approach to random walks on disordered lattices one dimensional random walks with static disorder are analyzed using a real space renormalization group procedure the presence of disorder leads to a non markovian description of the macroscopic behavior of the random walk we consider random walks with nearest neighbor hopping described by a master equation with both on site and site to site disorder in the transition matrix site to site disorder leads to a generalized diffusion coefficient with at 3 2 long time tail whereas on site disorder leads to a generalized burnett coefficient with at 1 2 long time tail

regularity and normality of the secant variety to a projective curve abstract for a smooth curve of genus g embedded by a line bundle of degree at least 2 g 3 we show that the ideal sheaf of the secant variety is 5 regular this bound is sharp with respect to both the degree of the embedding and the bound on the regularity further we show that the secant variety is projectively normal for the generic embedding of degree at least 2 g 3

diffusion on a quasiperiodic chain an exact solution to the problem of diffusion on a quasiperiodic chain is provided by a decimation method involving scaling by an irrational length factor

secant varieties and birational geometry we show how to use information about the equations defining secant varieties to smooth projective varieties in order to construct a natural collection of birational transformations these were first constructed as flips in the case of curves by m thaddeus via geometric invariant theory and the first flip in the sequence was constructed by the author for varieties of arbitrary dimension in an earlier paper we expose the finer structure of a second flip again for varieties of arbitrary dimension we also prove a result on the cubic generation of the secant variety and give some conjectures on the behavior of equations defining the higher secant varieties

singularities and syzygies of secant varieties of nonsingular projective curves in recent years the equations defining secant varieties and their syzygies have attracted considerable attention the purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies the main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus g is greater than 2g 2k p for nonnegative integers k and p then the k th secant variety of the curve has normal du bois singularities is arithmetically cohen macaulay and satisfies the property n k 2 p in addition the singularities of the secant varieties are further classified according to the genus of the curve and the castelnuovo mumford regularities are also obtained as well as one of the main technical ingredients we establish a vanishing theorem on the cartesian products of the curve which may have independent interests and may find applications elsewhere

long time tails in stationary random media ii applications in a previous paper we developed a mode coupling theory to describe the long time properties of diffusion in stationary statistically homogeneous random media here the general theory is applied to deterministic and stochastic lorentz models and several hopping models the mode coupling theory predicts that the amplitudes of the long time tails for these systems are determined by spatial fluctuations in a coarse grained diffusion coefficient and a coarse grained free volume for one dimensional models these amplitudes can be evaluated and the mode coupling theory is shown to agree with exact solutions obtained for these models for higher dimensional lorentz models the formal theory yields expressions which are difficult to evaluate for these models we develop an approximation scheme based upon projecting fluctuations in the diffusion coefficient and free volume onto fluctuations in the density of scatterers

flat connections and geometric quantization using the space of holomorphic symmetric tensors on the moduli space of stable bundles over a riemann surface we construct a projectively flat connection on a vector bundle over teichmuller space the fibre of the vector bundle consists of the global sections of a power of the determinant bundle on the moduli space both dolbeault and cech techniques are used

groupe de picard des varietes de modules de fibres semi stables sur les courbes algebriques soient x une courbe algebrique projective lisse de genre g 2 sur r d des entiers avec r 2 on note u r d la variete de modules des fibres algebriques semi stables sur x de rang r et de degre d us r d l ouvert de u r d correspondant aux fibres stables on sait que u r d est une variete algebrique projective irreductible et normale si r et d ne sont pas premiers entre eux u r d n est pas lisse sauf dans une exception le cas ou g 2 r 2 et d est pair on supposera dans cet article qu on n est pas dans ce cas on a alors codimu r d u r d us r d 2 et u r d us r d est le lieu des points singuliers de u r d si l est un fibre en droites de degre d sur x on note u r l la sous variete fermee de u r d correspondant aux fibres vectoriels de determinant isomorphe a l on etudie dans cet article les groupes de picard pic u r d et pic u r l on montre d abord que les varietes u r d et u r l sont localement factorielles on calcule ensuite pic u r d et pic u r l qui sont decrits au moyen des diviseurs thetas generalises

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

random walks on site disordered lattices random walk models with disordered waiting times site disorder are analysed using a real space renormalisation group method anomalous diffusion is found in all dimensions whenever the first moment of the distribution of waiting times diverges above two dimensions the fixed points are described by continuous time random walks with non analytic waiting time functions below two dimensions the fixed points are themselves disordered random walks where the disorder is characterised by a stable probability distribution a fractal dimension is defined for site disordered lattices and below two dimensions a hyperscaling law is found which relates the fractal dimension the euclidean dimension and the exponent describing the displacement of the walker

the nordstrom robinson code representation over gf 4 and efficient decoding we show that the nordstrom robinson 1968 code may be represented as the union of binary images of two isomorphic linear 4 2 3 codes over gf 4 certain symmetries of the unique quaternary 4 2 3 quadracode are discussed it is shown how the properties of the nordstrom robinson code itself such as weight distribution distance invariance and the well known representation as the union of eight cosets of the 16 5 8 reed muller code may be easily rederived from these symmetries in addition we introduce a decoding algorithm for the nordstrom robinson code which involves projecting its codewords onto the codewords of the quadracode the algorithm is simple enough to enable hard decision decoding of the nordstrom robinson code by hand furthermore we present an algorithm for maximum likelihood soft decision decoding of the nordstrom robinson code based on its representation over gf 4 the complexity of the proposed algorithm is at most 205 real operations this is to the best of our knowledge less than the complexity of any existing decoder and twice as efficient as the fast maximum likelihood decoder of adoul 1987 we also investigate several related topics such as bounded distance decoding sphere packings and the 20 2048 6 code

su 2 wzw theory at higher genera we compute by free field techniques the scalar product of thesu 2 chern simons states on genus 1 surfaces the result is a finite dimensional integral over positions of screening charges and one complex modular parameter it uses an effective description of the cs states closely related to the one worked out by bertram 1 the scalar product formula allows to express the higher genus partition functions of the wzw conformal field theory by finite dimensional integrals it should provide the hermitian metric preserved by the knizhnik zamolodchikov bernard connection describing the variations of the cs states under the change of the complex structure of the surface

on the normality of secant varieties abstract in this paper we show that the secant variety to a smooth projective variety embedded by a sufficiently positive line bundle is normal as an application we deduce that the secant variety to a general canonical curve of genus at least 7 is normal

orbit and coset analysis of the golay and related codes let b be a code of length n over a field f with automorphism group g b sub w denotes the subset of codewords of weight w the goal is to classify the vectors of f sup n into orbits under g and to determine their distances from the various subcodes b sub w this is done for the first order reed muller nordstrom robinson and hamming codes of length 16 the golay and shortened golay codes of lengths 22 23 24 and the ternary golay code of length 12

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

on the stopping distance and the stopping redundancy of codes it is now well known that the performance of a linear code copf under iterative decoding on a binary erasure channel and other channels is determined by the size of the smallest stopping set in the tanner graph for copf several recent papers refer to this parameter as the stopping distance s of copf this is somewhat of a misnomer since the size of the smallest stopping set in the tanner graph for copf depends on the corresponding choice of a parity check matrix it is easy to see that s les d where d is the minimum hamming distance of copf and we show that it is always possible to choose a parity check matrix for copf with sufficiently many dependent rows such that s d we thus introduce a new parameter the stopping redundancy of copf defined as the minimum number of rows in a parity check matrix h for copf such that the corresponding stopping distance s h attains its largest possible value namely s h d we then derive general bounds on the stopping redundancy of linear codes we also examine several simple ways of constructing codes from other codes and study the effect of these constructions on the stopping redundancy specifically for the family of binary reed muller codes of all orders we prove that their stopping redundancy is at most a constant times their conventional redundancy we show that the stopping redundancies of the binary and ternary extended golay codes are at most 34 and 22 respectively finally we provide upper and lower bounds on the stopping redundancy of mds codes

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

diffusion in random one dimensional systems diffusion on the one dimensional lattice is described by a master equation with nearest neighbor transfer rates symmetric or asymmetric the transfer rates associated with bonds are assumed to be independent equally distributed random variables under various conditions on their common distribution the large time behavior of averaged site probabilities and or related quantities is exhibited

scaling of conformal blocks and generalized theta functions over overline mathcal m _ g n by way of intersection theory on overline mathcal m g n we show that geometric interpretations for conformal blocks as sections of ample line bundles over projective varieties do not have to hold at points on the boundary we show such a translation would imply certain recursion relations for first chern classes of these bundles while recursions can fail geometric interpretations are shown to hold under certain conditions

on the hitchin system 1 1 what is known as the hitchin system is a completely integrable hamiltonian system cihs involving vector bundles over algebraic curves identified by hitchin in h1 h2 it was recently generalized by faltings f in this paper we only consider the case of ranktwo vector bundles with trivial determinant in that case the hitchin system corresponding to a curve c of genus g is obtained as follows let

2 1 d topological quantum field theory and 2 d conformal field theory we describe the relation between three dimensional topological quantum field theory and two dimensional conformal field theory some applications to quantum knot invariants leading to the equivalence of chern simons witten and kohno s approaches are outlined

a new table of constant weight codes a table of binary constant weight codes of length nl28 is presented explicit constructions are given for most of the 600 codes in the table the majority of these codes are new the known techniques for constructing constant weight codes are surveyed and a table of unrestricted binary codes of length nl28 is given

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

on the covering radius of z sub 4 codes and their lattices in this correspondence we investigate the covering radius of codes over z sub 4 for the lee and euclidean distances in relation with those of binary nonlinear codes and lattices obtained by the gray map and construction a sub 4 respectively we give several upper and lower bounds on covering radii including z sub 4 analogs of the sphere covering bound the packing radius bound the delsarte bound and the redundancy bound we show that any euclidean optimal type ii code of length 24 has covering radius 8 with respect to the euclidean distance we determine the covering radius of the klemm codes with respect to the lee distance we derive lower bounds on the covering radii of the niemeier lattices