closed subsets in duals of commutative table algebras a lower bound on the number of elements outside a closed subset of a c basis of a commutative c algebra with dual a table algebra is derived as is an equivalent condition for when the lower bound is met as corollaries lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative imprimitive association scheme and for a given normal subgroup n of a finite group g on the number of irreducible characters of g whose kernels do not contain n and on the number of conjugacy classes of g not contained in n also found are equivalent conditions for when these lower bounds are met

permutation modules over table algebras and applications to association schemes abstract inspired by the notion of an action of a finite hypergroup on a finite set we introduce the more general concept of a permutation module over a table algebra it is easy to see that each permutation module over a table algebra is a direct sum of transitive permutation modules among the transitive permutation modules the w maximal and the maximal permutation modules are the most interesting ones we give various characterizations of these modules and we shall see that a standard table algebra admits a maximal transitive permutation module if and only if it arises from a finite association scheme we also show that the regular module of a standard table algebra is isomorphic to a direct summand of each w maximal transitive permutation module as a consequence one obtains 1 m for any irreducible character of the bose mesner algebra of a finite association scheme and its multiplicity m

threshold for monotone symmetric properties through a logarithmic sobolev inequality threshold phenomena are investigated using a general approach following talagrand ann probab 22 1994 1576 1587 and friedgut and kalai proc amer math soc 12 1999 1017 10541 the general upper bound for the threshold width of symmetric monotone properties is improved this follows from a new lower bound on the maximal influence of a variable on a boolean function the method of proof is based on a well known logarithmic sobolev inequality on 0 1 n this new bound is shown to be asymptotically optimal

some remarks on the stability of time domain electromagnetic simulations aspects of the stability of time domain update schemes for electromagnetic problems are examined based on the consideration of system amplification matrices arising from the discretization of maxwell s equations in the sequence space we discuss time domain stability for a given discretization scale and under three different assumptions of decreasing generality general updates general leapfrog updates and reciprocal leapfrog updates

the generalized terwilliger algebra and its finite dimensional modules when d 2 terwilliger j algebraic combin 1 1992 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations are often called the triple product relations they determine much of the structure of t though not all of it in general to illuminate the role these relations play the current author introduced j algebra233 2000 213 252 a generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c the dimensions of c and c are equal let d 1 denote this common dimension intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s triple product relations t is infinite dimensional and noncommutative in general in this paper we study t and its finite dimensional modules when d 2 and t has no extra vanishing intersection numbers or dual intersection numbers in this case we show t is c algebra isomorphic to m3 c a where m3 c denotes the c algebra consisting of all 3 by 3 matrices with entries in c and a denotes the associative c algebra with identity generated by the symbols e and f subject to the relations e2 e and f2 f we find a basis for a and we determine the center of a we classify the finite dimensional indecomposable a modules up to isomorphism there are four such a modules in every odd dimension and in every even dimension these modules are parameterized by a single complex number we also classify the finite dimensional irreducible a modules up to isomorphism using our results concerning a we find a basis for t we describe the center of t and we classify both the finite dimensional indecomposable and the finite dimensional irreducible t modules up to isomorphism

a general approach to extend berenger s absorbing boundary condition to anisotropic and dispersive media previous studies have focused on the extension of the berenger s 1994 absorbing boundary condition abc the perfectly matched layer pml to anisotropic and dispersive media we describe how the pml can be extended for anisotropic and or dispersive media or even bianisotropic media with little analytical effort in a simple and general conceptual setting by using an analytical continuation of the coordinate space of the maxwell s equations to a complex variables domain using this approach there is no conceptual distinction between the derivation of pml for more complex media and the derivation for isotropic dispersionless media

efficient erasure correcting codes we introduce a simple erasure recovery algorithm for codes derived from cascades of sparse bipartite graphs and analyze the algorithm by analyzing a corresponding discrete time random process as a result we obtain a simple criterion involving the fractions of nodes of different degrees on both sides of the graph which is necessary and sufficient for the decoding process to finish successfully with high probability by carefully designing these graphs we can construct for any given rate r and any given real number spl epsiv a family of linear codes of rate r which can be encoded in time proportional to ln 1 spl epsiv times their block length n furthermore a codeword can be recovered with high probability from a portion of its entries of length 1 spl epsiv rn or more the recovery algorithm also runs in time proportional to n ln 1 spl epsiv our algorithms have been implemented and work well in practice various implementation issues are discussed

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

a stable fdtd algorithm for non diagonal anisotropic dielectrics a stable fdtd algorithm is developed for simulating maxwell s equations in anisotropic dielectric materials with principal axes not aligned with the grid the algorithm is stable because the finite difference operator that converts d to e is symmetric and positive semidefinite for contrast a previously developed asymmetric algorithm is shown to suffer from late time instabilities the presented algorithm has second order error for continuous dielectric materials and the error can be reduced to third order by richardson extrapolation applied to dielectrics with sharp interfaces the algorithm has first order error even when averaging the dielectric within partially filled grid cells however averaging the dielectric permits richardson extrapolation to obtain second order error

a fast near to far field transformation in body of revolution finite difference time domain method we present an efficient near to far field transformation technique to be used in conjunction with the body of revolution finite difference time domain bor fdtd algorithm the efficiency is gained by factoring out the azimuthal angle spl phi from the two dimensional 2d far field integration using an analytical approach thereby reducing the 2d integral on the surface of the huygens box to a 1d version in the process to illustrate the application of the proposed algorithm the potter horn is analyzed the numerical results show good agreement with those obtained by using either analytical where possible or 2d integration schemes

weight distribution and list decoding size of reed muller codes the weight distribution and list decoding size of reed muller codes are studied in this work given a weight parameter we are interested in bounding the number of reed muller codewords with weight up to the given parameter and given a received word and a distance parameter we are interested in bounding the size of the list of reed muller codewords that are within that distance from the received word obtaining tight bounds for the weight distribution of reed muller codes has been a long standing open problem in coding theory dating back to 1976 in this work we make a new connection between computer science techniques used to study low degree polynomials and these coding theory questions this allows us to resolve the weight distribution and list decoding size of reed muller codes for all distances previous results could only handle bounded distances azumi kasami and tokura gave bounds on the weight distribution which hold up to 2 5 times the minimal distance of the code and gopalan klivans and zuckerman gave bounds on the list decoding size which hold up to the johnson bound

about products of irreducible characters and products of conjugacy classes in finite groups recently the research on products of ordinary irreducible characters and products of conjugacy classes in finite groups developed considerably in many articles analogous theorems for these two kinds of products were proved each acm af ah ahs al b bc ch ma for example it was proved in acm that a finite group

the gleason prange theorem the gleason prange theorem describes a nontrivial automorphism of an extended quadratic residue code by using the fourier transform in an appropriate extension field methods of digital signal processing are used to make the proof transparent and the automorphism intuitive the theorem is proved both in a field of characteristic tow and in an arbitrary field and is stated simply as a general property of the discrete fourier transform

on primitive commutative association schemes with a nonsymmetric relation of valency 4 let x g be a primitive commutative association scheme if g g is nonsymmetric of valency 4 then the graph x g is uniquely determined up to isomorphism in particular the cardinality of x is the cube of an odd prime

locally conformal fdtd for anisotropic conductive interfaces locally conformal discretizations have been developed for the finite difference time domain fdtd method to mitigate staircasing errors that arise in the modeling of arbitrary e g curved slanted interfaces by regular orthogonal grids in particular locally conformal algorithms based on the computation of effective material parameters in partially filled fdtd cells are attractive because they are simple to implement and maintain the basic conditional stability of fdtd with only minor changes on the courant limit in this paper we propose a new locally conformal implementation for fdtd tailored for anisotropic conductive interfaces instead of employing arithmetic or geometrical averages we invoke the quasi static relationship between the current density and the electric field to derive the effective conductivities of partially filled fdtd cells

categorical aspects of generating functions i exponential formulas and krull schmidt categories abstract in this paper we study formal power series with exponents in a category for example the generating function of a category e with finite hom sets is defined by e t tx aut x where the summation is taken over all isomorphism classes of objects of e we can use such power series to enumerate the number of e structures along a faithful functor theorem 4 6 our theory is closely related to the theory of species joyal 1981 a species can be identified with a faithful functor from a groupoid to the category of finite sets theorem 3 6 we use mainly the concept of faithful functors with finite fibers instead of species so that we can separate the roles categories and functors play for example the exponential formula e t exp con e t means the unique coproduct decomposition property theorem 5 8 in the final section we give some applications of our theory to rather classical enumerations

order evaluation of products of subsets in finite groups and its applications ii in this paper we give a new estimate of the cardinality of the product of subsets ab in a finite non abelian simple group where a is normal and b is arbitrary this estimate improves the one given in j algebra 182 1996 577 603

finite difference time domain modeling of curved surfaces em scattering the finite difference time domain fdtd method is generalized to include the accurate modeling of curved surfaces this generalization the contour path cp method accurately models the illumination of bodies with curved surfaces yet retains the ability to model corners and edges cp modeling of two dimensional electromagnetic wave scattering from objects of various shapes and compositions is presented

the category of linear codes slepian 1960 introduced a structure theory for linear binary codes and proved that every such code was uniquely the sum of indecomposable codes he had hoped to produce a canonical form for the generator matrix of an indecomposable code so that he might read off the properties of the code from such a matrix but such a program proved impossible we here work over an arbitrary field and define a restricted class of indecomposable codes which we call critical for these codes there is a quasicanonical form for the generator matrix every indecomposable code has a generator matrix that is obtained from the generator matrix of a critical indecomposable code by augmentation as an application of the this quasicanonical form we illuminate the perfect linear codes giving for example a canonical form for the generator matrix of the ternary golay 1949 code

on the solution of a class of large body problems with full or partial circular symmetry by using the finite difference time domain fdtd method this paper presents an efficient method to accurately solve large body scattering problems with partial circular symmetry the method effectively reduces the computational domain from three to two dimensions by using the reciprocity theorem it does so by dividing the problem into two parts a larger 3 d region with circular symmetry and a smaller 2 d region without circular symmetry an finite difference time domain fdtd algorithm is used to analyze the circularly symmetric 3 d case while a method of moments mom code is employed for the nonsymmetric part of the structure the results of these simulations are combined via the reciprocity theorem to yield the radiation pattern of the composite system the advantage of this method is that it achieves significant savings in computer storage and run time in performing an equivalent 2 d as opposed to a full 3 d fdtd simulation in addition to enhancing computational efficiency the fdtd algorithm used in this paper also features one improvement over conventional fdtd methods a conformal approach for improved accuracy in modeling curved dielectric and conductive surfaces the accuracy of the method is validated via a comparison of simulated and measured results

a generalization of the terwilliger algebra abstract p m terwilliger 1992 j algebraic combin 1 363 388 considered the c algebra generated by a given bose mesner algebra m and the associated dual bose mesner algebra m this algebra is now known as the terwilliger algebra and is usually denoted by t terwilliger showed that each vanishing intersection number and krein parameter of m gives rise to a relation on certain generators of t these relations determine much of the structure of t thought not all of it in general to illuminate the role these relations play we consider a certain generalization t of t to go from t to t we replace m and m with a pair of dual character algebras c and c we define t by generators and relations intuitively t is the associative c algebra with identity generated by c and c subject to the analogues of terwilliger s relations t is infinite dimensional and noncommutative in general we construct an irreducible t module which we call the primary module the dimension of this module is the same as that of c and c we find two bases of the primary module one diagonalizes c and the other diagonalizes c we compute the action of the generators of t on these bases we show t is a direct sum of two sided ideals t 0 and t 1 with t 0 isomorphic to a full matrix algebra we show that the irreducible module associated with t 0 is isomorphic to the primary module we compute the central primitive idempotent of t associated with t 0 in terms of the generators of t

on non solvable camina pairs in this paper we study non solvable and non frobenius camina pairs g n it is known d chillag a mann c scoppola generalized frobenius groups ii israel j math 62 1988 269 282 that in this case n is a p group our first result theorem 1 3 shows that the solvable residual of g op g is isomorphic either to sl 2 pe p is a prime or to sl 2 5 sl 2 13 with p 3 or to sl 2 5 with p 7 our second result provides an example of a non solvable and non frobenius camina pair g n with op g 55 and g op g sl 2 5 note that g has a character which is zero everywhere except on two conjugacy classes groups of this type were studies by s m gagola s m gagola characters vanishing on all but two conjugacy classes pacific j math 109 1983 363 385 to our knowledge this group is the first example of a gagola group which is non solvable and non frobenius

soft decision decoding of reed muller codes recursive lists recursive list decoding is considered for reed muller rm codes the algorithm repeatedly relegates itself to the shorter rm codes by recalculating the posterior probabilities of their symbols intermediate decodings are only performed when these recalculations reach the trivial rm codes in turn the updated lists of most plausible codewords are used in subsequent decodings the algorithm is further improved by using permutation techniques on code positions and by eliminating the most error prone information bits simulation results show that for all rm codes of length 256 and many subcodes of length 512 these algorithms approach maximum likelihood ml performance within a margin of 0 1 db as a result we present tight experimental bounds on ml performance for these codes

une theorie combinatoire des series formelles abstract this paper presents a combinatorial theory of formal power series the combinatorial interpretation of formal power series is based on the concept of species of structures a categorical approach is used to formulate it a new proof of cayley s formula for the number of labelled trees is given as well as a new combinatorial proof due to g labelle of lagrange s inversion formula polya s enumeration theory of isomorphism classes of structures is entirely renewed recursive methods for computing cycle index polynomials are described a combinatorial version of the implicit function theorem is stated and proved the paper ends with general considerations on the use of coalgebras in combinatorics

iterative decoding threshold analysis for ldpc convolutional codes an iterative decoding threshold analysis for terminated regular ldpc convolutional ldpcc codes is presented using density evolution techniques the convergence behavior of an iterative belief propagation decoder is analyzed for the binary erasure channel and the awgn channel with binary inputs it is shown that for a terminated ldpcc code ensemble the thresholds are better than for corresponding regular and irregular ldpc block codes

decomposition of products in nilpotent table algebras abstract two theorems on the decomposition of products in nilpotent commutative table algebras are established this generalizes several results of adan bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p groups and extends these results to fusion rule algebras and association schemes

any code of which we cannot think is good a central paradox of coding theory concerns the existence and construction of the best codes virtually every linear code is good in the sense that it meets the gilbert varshamov bound on distance versus redundancy despite the sophisticated constructions for codes derived over the years however no one has succeeded in demonstrating a constructive procedure that yields such codes over arbitrary symbol fields using the theory of kolmogorov complexity it is shown that this statement holds true in a rigorous mathematical sense any linear code that is truly random in the sense that there is no concise way of specifying the code is good furthermore random selection of a code that does contain some constructive pattern results with probability bounded away from zero in a code that does not meet the gilbert varshamov bound regardless of the block length of the code in contrast to the situation for linear codes it is shown that there are effectively random nonlinear codes which have no guarantee on distance in addition it is shown that the techniques of kolmogorov complexity can be used to derive typical properties of classes of codes in a novel way

an improved thin wire representation for fdtd computations we have shown that finite difference time domain fdtd electromagnetic computations for a conductor system having a radius smaller than 0 15deltar or larger than 0 65 deltar deltar is the lateral side length of cells employed modeled using arbitrary radius wire representations proposed so far with a time increment determined from the upper limit of courant s stability condition result in numerical instability a primary factor causing this numerical instability is that the speed of waves propagating in the radial direction from the wire in the immediate vicinity of the wire exceeds the speed of light and therefore courant s condition is not satisfied there it is further shown that for these cases the arbitrary radius wire representation can be improved by modifying the material parameters for the axial field components closest to the wire as well as those for the radial electric and circulating magnetic field components the improved wire representation is effective in representing a wire whose radius ranges from 0 0001deltar to 0 9deltar

soft decision decoding of reed muller codes a simplified algorithm soft decision decoding is considered for general reed muller rm codes of length n and distance d used over a memoryless channel a recursive decoding algorithm is designed and its decoding threshold is derived for long rm codes the algorithm has complexity of order nlnn and corrects most error patterns of the euclidean weight of order radicn lnn instead of the decoding threshold radicd 2 of the bounded distance decoding also for long rm codes of fixed rate r the new algorithm increases 4 pi times the decoding threshold of its hard decision counterpart

on wohlfahrt series and wreath products abstract suppose that a group a contains only a finite number of subgroups of index d for each positive integer d let g s n be the wreath product of a finite group g with the symmetric group s n on 1 n for each positive integer n let k n be a subgroup of g s n containing the commutator subgroup of g s n if the sequence k n 0 satisfies a certain compatible condition then the exponential generating function n 0 hom a k n x n g n n of the sequence hom a k n 0 takes the form of a sum of exponential functions

lattice electromagnetic theory from a topological viewpoint the language of differential forms and topological concepts are applied to study classical electromagnetic theory on a lattice it is shown that differential forms and their discrete counterparts cochains provide a natural bridge between the continuum and the lattice versions of the theory allowing for a natural factorization of the field equations into topological field equations i e invariant under homeomorphisms and metric field equations the various potential sources of inconsistency in the discretization process are identified distinguished and discussed a rationale for a consistent extension of the lattice theory to more general situations such as to irregular lattices is considered

on planar algebras arising from hypergroups abstract let a be an associative algebra with identity and with trace we study the family of planar algebras on 1 boxes that arise from a in the work of jones but with the added assumption that the labels on the 1 boxes come from a discrete hypergroup in the sense of sunder this construction equips the algebra pna with a canonical basis b n a which turns out to be a tabular basis we examine special cases of this construction to exhibit a close connection between such bases and kazhdan lusztig bases of hecke algebras of types a b h or i

a symmetric function generalization of the chromatic polynomial of a graph abstract for a finite graph g with d vertices we define a homogeneous symmetric function xg of degree d in the variables x1 x2 if we set x1 xn 1 and all other xi 0 then we obtain g n the chromatic polynomial of g evaluated at n we consider the expansion of xg in terms of various symmetric function bases the coefficients in these expansions are related to partitions of the vertices into stable subsets the mobius function of the lattice of contractions of g and the structure of the acyclic orientations of g the coefficients which arise when xg is expanded in terms of elementary symmetric functions are particularly interesting and for certain graphs are related to the theory of hecke algebras and kazhdan lusztig polynomials

some structure theory of table algebras and applications to association schemes abstract in this paper we prove some basic structure theorems for arbitrary table algebras in particular we prove a krull schmidt theorem for arbitrary table algebras and give applications to association schemes

reed muller codes achieve capacity on erasure channels we introduce a new approach to proving that a sequence of deterministic linear codes achieves capacity on an erasure channel under maximum a posteriori decoding rather than relying on the precise structure of the codes our method exploits code symmetry in particular the technique applies to any sequence of linear codes where the blocklengths are strictly increasing the code rates converge and the permutation group of each code is doubly transitive in other words we show that symmetry alone implies near optimal performance an important consequence of this result is that a sequence of reed muller codes with increasing blocklength and converging rate achieves capacity this possibility has been suggested previously in the literature but it has only been proven for cases where the limiting code rate is 0 or 1 moreover these results extend naturally to all affine invariant codes and thus to extended primitive narrow sense bch codes this also resolves in the affirmative the existence question for capacity achieving sequences of binary cyclic codes the primary tools used in the proof are the sharp threshold property for symmetric monotone boolean functions and the area theorem for extrinsic information transfer functions

nilpotency property in table algebras abstract table algebras form an important class of c algebras the dual of a table algebra may not be a table algebra but just a c algebra it is not known under what conditions the dual of a table algebra is also a table algebra in this paper we prove that if a table algebra has nilpotency property then its dual also is a table algebra finally we conclude that if a b is a nilpotent table algebra then its dual also is a nilpotent table algebra

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

good codes can be produced by a few permutations our main result is that good codes even those meeting the random coding bound can be produced with relatively few linear in the block length permutations from a single codeword this cutdown in complexity may be of practical importance the motivation for looking at such codes came from ahlswede s covering lemma which makes it possible to build correlated source codes from channel codes via permutations in appendix i we show that the problem of finding the best error exponents for coding sources with full side information at the decoder which has received attention in the recent literature can easily be reduced to the familiar one for the discrete memoryless channel dmc finally in appendices ii and iii we give rather precise double exponentially small bounds on the probabilities that a randomly chosen code will fail to meet the random coding or expurgated bound for the dmc according to these results good codes are hard to miss if selected at random this also explains why good codes of a iow complexity such as those produced by

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

on cryptographic properties of the cosets of r 1 m we introduce a new approach for the study of weight distributions of cosets of the reed muller code of order 1 our approach is based on the method introduced by kasami 1968 using pless 1963 identities by interpreting some equations we obtain a necessary condition for a coset to have a high minimum weight most notably we are able to distinguish such cosets which have three weights only we then apply our results to the problem of the nonlinearity of boolean functions we particularly study the links between this criterion and the propagation characteristics of a function

conformal fdtd methods to avoid time step reduction with and without cell enlargement during the last decades there have been considerable efforts to develop accurate and yet simple conformal methods for modelling curved boundaries within the finite difference time domain fdtd algorithm in an earlier publication we proposed the uniformly stable conformal usc approach as a general three dimensional extension of fdtd without the need to reduce the maximum stable time step the main idea of usc is the usage of virtually enlarged cells near to the boundary leading to an increased implementation effort in this paper we review the usc method and introduce a new simple and accurate conformal scheme which does not use such enlarged cells this simplified conformal sc scheme has the same number of operations and algorithmic logic as the standard staircase method and thus is easily realizable in existing fdtd codes like usc it leads to accurate results without time step reduction showing a nearly second order convergence in practice the method is verified and compared to other approaches by means of several numerical 2d and 3d examples

a new upper bound on the block error probability after decoding over the erasure channel motivated by cryptographic applications we derive a new upper bound on the block error probability after decoding over the erasure channel the bound works for all linear codes and is in terms of the generalized hamming weights it turns out to be quite useful for reed muller codes for which all the generalized hamming weights are known whereas the full weight distribution is only partially known for these codes the error probability is related to the cryptographic notion of algebraic immunity we use our bound to show that the algebraic immunity of a random balanced m variable boolean function is of order m 2 1 o 1 with probability tending to 1 as m goes to infinity

the permutation group of affine invariant extended cyclic codes the extended cyclic codes of length p sup m p is a prime which are invariant under the affine group acting on f sub pm are called affine invariant codes following results of berger 1996 1994 we present the formal expression of the permutation group of these codes afterwards we give several tools in order to determine effectively the group of a given code or of some infinite class of codes we next prove by studying some examples that our tools are efficient in the end we give our main application the permutation group of primitive bch codes defined on any prime field

semisimple commutative algebras with positive bases abstract algebras that serve as models for concurrent studying of certain aspects of both the algebra of ordinary characters and the center of the group algebra of a finite group have been considered by various authors in this article we offer another such model the main differences between our model and the known ones are 1 our model includes brauer characters and principal indecomposable characters as special cases 2 our emphasis is on the eigenvalues of the regular representation of the algebra elements an approach that gives results on values of characters ordinary central brauer principal indecomposable as well as on their products

decomposable functors and the exponential principle abstract we develop a combinatorial theory for the operation of forming the logarithm of generating functions associated with covariant functors on the category of finite sets and bijective maps this leads to a rather general and flexible exponential principle for labeled combinatorial structures we also introduce the concept of a topos like category which turns out to be a convenient framework for the discussion of a wide range of applications a few of which are studied in detail

tabular algebras and their asymptotic versions we introduce tabular algebras which are simultaneous generalizations of cellular algebras in the sense of graham lehrer and table algebras in the sense of arad blau we show that if a tabular algebra is equipped with a certain kind of trace map then the algebra has a corresponding asymptotic version whose structure can be explicitly determined we also study various natural examples of tabular algebras

the generalized area theorem and some of its consequences there is a fundamental relationship between belief propagation bp and maximum a posteriori decoding the case of transmission over the binary erasure channel was investigated in detail in a companion paper c meacuteasson a montanari and r urbanke maxwell s construction the hidden bridge between iterative and maximum a posteriori decoding ieee transactions on information theory submitted for publication this paper investigates the extension to general memoryless channels paying special attention to the binary case an area theorem for transmission over general memoryless channels is introduced and some of its many consequences are discussed we show that this area theorem gives rise to an upper bound on the maximum a posteriori threshold for sparse graph codes in situations where this bound is tight the extrinsic soft bit estimates delivered by the bp decoder coincide with the correct a posteriori probabilities above the maximum a posteriori threshold more generally it is conjectured that the fundamental relationship between the maximum a posteriori probability map and the bp decoder which was observed for transmission over the binary erasure channel carries over to the general case we finally demonstrate that in order for the design rate of an ensemble to approach the capacity under bp decoding the component codes have to be perfectly matched a statement which is well known for the special case of transmission over the binary erasure channel

fusion rings with few degrees abstract the main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p a well known theorem of isaacs and passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary numerically compatible nc fusion rings are defined as a potential generalization of the grothendieck rings of fusion categories and the nc fusion rings of order p p 2 p 3 pq and p q 2 where p q are primes are classified in terms of certain standard table algebras as another application of the main theorem

low frequency finite difference time domain modeling of a pec sphere based on a quasi analytical coupled dipole approximation a computational formulation is presented for the low frequency single cell finite difference time domain fdtd modeling of a perfectly electric conducting pec sphere the approach is based on the fact that the scattered field from electrically small objects can be expressed in terms of an electric and magnetic dipole these dipoles can be decomposed with respect to the dipole moments that can be defined along the discrete field components that comprise the cell wherein the pec sphere is inscribed the dipole moment components couple to each other and this mechanism is quantified by a quasi analytical coupled dipole approximation cda the quasi analyticity requires to substitute the involved dyadic green s function dgf terms in the cda formula by their numerically computed fdtd compatible equivalents the material properties of the equivalent electric and magnetic spheres are derived using the quasi analytical cda that leads to expressions that resemble the claussius mossotti mixing formula the theoretically derived results are supported by numerical simulations

on the hardness of approximating minimum vertex cover we prove the minimum vertex cover problem to be np hard to approximate to within a factor of 1 3606 extending on previous pcp and hardness of approximation technique to that end one needs to develop a new proof framework and to borrow and extend ideas from several fields

generating functions for permutation representations abstract we discuss the categorical approach to representations in wreath products and generalize the wohlfahrt formula of the exponential generating function for the number of permutation representations

threshold saturation via spatial coupling why convolutional ldpc ensembles perform so well over the bec convolutional low density parity check ldpc ensembles introduced by felstrom and zigangirov have excellent thresholds and these thresholds are rapidly increasing functions of the average degree several variations on the basic theme have been proposed to date all of which share the good performance characteristics of convolutional ldpc ensembles we describe the fundamental mechanism that explains why convolutional like or spatially coupled codes perform so well in essence the spatial coupling of individual codes increases the belief propagation bp threshold of the new ensemble to its maximum possible value namely the maximum a posteriori map threshold of the underlying ensemble for this reason we call this phenomenon threshold saturation this gives an entirely new way of approaching capacity one significant advantage of this construction is that one can create capacity approaching ensembles with an error correcting radius that is increasing in the blocklength although we prove the threshold saturation only for a specific ensemble and for the binary erasure channel bec empirically the phenomenon occurs for a wide class of ensembles and channels more generally we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly this might give rise to improved algorithms and new techniques for analysis

reed muller codes for random erasures and errors this paper studies the parameters for which binary reed muller rm codes can be decoded successfully on the binary erasure channel and binary symmetry channel and in particular when can they achieve capacity for these two classical channels necessarily this paper also studies the properties of evaluations of multivariate gf 2 polynomials on the random sets of inputs for erasures we prove that rm codes achieve capacity both for very high rate and very low rate regimes for errors we prove that rm codes achieve capacity for very low rate regimes and for very high rates we show that they can uniquely decode at about the square root of the number of errors at capacity the proofs of these four results are based on different techniques which we find interesting in their own right in particular we study the following questions about e m r the matrix whose rows are the truth tables of all the monomials of degree leq r in m variables what is the most resp least number of random columns in e m r that define a submatrix having full column rank resp full row rank with high probability we obtain tight bounds for very small resp very large degrees r which we use to show that rm codes achieve capacity for erasures in these regimes our decoding from random errors follows from the following novel reduction for every linear code c of sufficiently high rate we construct a new code c obtained by tensorizing c such that for every subset s of coordinates if c can recover from erasures in s then c can recover from errors in s specializing this to the rm codes and using our results for erasures imply our result on the unique decoding of the rm codes at high rate finally two of our capacity achieving results require tight bounds on the weight distribution of rm codes we obtain such bounds extending the recent bounds from constant degree to linear degree polynomials

maxwell construction the hidden bridge between iterative and maximum a posteriori decoding there is a fundamental relationship between belief propagation and maximum a posteriori decoding a decoding algorithm which is called the maxwell decoder is introduced and provides a constructive description of this relationship both the algorithm itself and the analysis of the new decoder are reminiscent of the maxwell construction in thermodynamics this paper investigates in detail the case of transmission over the binary erasure channel while the extension to general binary memoryless channels is discussed in a companion paper

spatially coupled ensembles universally achieve capacity under belief propagation we investigate spatially coupled code ensembles for transmission over the binary erasure channel it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a priori threshold of the underlying component ensemble this explains why convolutional ldpc ensembles originally introduced by felstrom and zigangirov perform so well over this channel we show that the equivalent result holds true for transmission over general binary input memoryless output symmetric channels more precisely given a desired error probability and a gap to capacity we can construct a spatially coupled ensemble that fulfills these constraints universally on this class of channels under belief propagation decoding in fact most codes in this ensemble have this property the quantifier universal refers to the single ensemble code that is good for all channels but we assume that the channel is known at the receiver the key technical result is a proof that under belief propagation decoding spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble we conclude by discussing some interesting open problems

on russo s approximate zero one law consider the product measure mu p on 0 1 n when 0 operatorname resp 1 is given weight 1 p operatorname resp p consider a monotone subset a of 0 1 n we give a precise quantitative form to the following statement if a does not depend much on any given coordinate d mu p a dp is large thus in that case there is a threshold effect and mu p a jumps from near 0 to near 1 in a small interval

the burnside ring of profinite groups and the witt vector construction abstract the classical construction of witt vectors can be understood as a special case of a construction which can be defined relative to any given profinite group g in terms of its completed burnside ring lde g it specializes to the classical situation in case g is infinite cyclic

powers of characters of finite groups letg be a finite group and a complex character ofg define irr to be the set of all irreducible constituents of andirr g to be the set of all irreducible characters ofg thecharacter covering number of a finite groupg ccn g is defined as the smallest positive integer m such thatirr m irr g for all irr g 1g if no such positive integer exists we say that the character covering number ofg is infinite in this article we show that a finite nontrivial groupg has a finite character covering number if and only ifg is simple and non abelian and ifg is a nonabelian simple group thenccn g k2 3k 4 wherek is the number of conjugacy classes ofg then we show using the classification of the finite simple groups that the only finite group with a character covering number equal to two is the smallest janko s group j1 these results are analogous to results obtained previously concerning the covering of groups by powers of conjugacy classes other related results are shown

near to far field transformation by a time domain spherical multipole analysis the contribution deals with a new time domain near to far field nff transformation which is particularly suited for the finite difference time domain fdtd method the far field is derived from the tangential field components on a surface enclosing the scatterer by employing a time domain spherical multipole representation the necessary time domain convolution is performed as on the fly in parallel with the fdtd time stepping algorithm the efficiency of the method is improved by incorporating a temporal linear interpolation of the near field data with the once obtained multipole amplitudes an analytic series representation of the time domain far field is achieved which allows a physical interpretability of the result moreover this expansion serves as an ideal basis for a systematic and efficient post processing the proposed technique may also be useful for other numerical and for asymptotic methods

the character values of commutative quasi thin schemes the main result of this article shows that all character values of commutative quasi thin schemes are cyclotomic algebraic integers in particular all of the eigenvalues of the adjacency matrices corresponding to relations in these association schemes are cyclotomic algebraic integers

axiomatic theory of burnside rings i abstract in this paper we study abstract burnside rings of essentially finite categories under unique epi mono factorization property and the existence of coequalizers for some kind we prove the existence of a fundamental exact sequence for abr furthermore an abr can be embedded into a direct product of rational character rings

channel polarization a method for constructing capacity achieving codes for symmetric binary input memoryless channels a method is proposed called channel polarization to construct code sequences that achieve the symmetric capacity i w of any given binary input discrete memoryless channel b dmc w the symmetric capacity is the highest rate achievable subject to using the input letters of the channel with equal probability channel polarization refers to the fact that it is possible to synthesize out of n independent copies of a given b dmc w a second set of n binary input channels wn i 1 les i les n such that as n becomes large the fraction of indices i for which i wn i is near 1 approaches i w and the fraction for which i wn i is near 0 approaches 1 i w the polarized channels wn i are well conditioned for channel coding one need only send data at rate 1 through those with capacity near 1 and at rate 0 through the remaining codes constructed on the basis of this idea are called polar codes the paper proves that given any b dmc w with i w 0 and any target rate r i w there exists a sequence of polar codes cfrn nges1 such that cfrn has block length n 2n rate ges r and probability of block error under successive cancellation decoding bounded as pe n r les o n 1 4 independently of the code rate this performance is achievable by encoders and decoders with complexity o n logn for each

algebraic immunity for cryptographically significant boolean functions analysis and construction recently algebraic attacks have received a lot of attention in the cryptographic literature it has been observed that a boolean function f used as a cryptographic primitive and interpreted as a multivariate polynomial over f sub 2 should not have low degree multiples obtained by multiplication with low degree nonzero functions in this paper we show that a boolean function having low nonlinearity is also weak against algebraic attacks and we extend this result to higher order nonlinearities next we present enumeration results on linearly independent annihilators we also study certain classes of highly nonlinear resilient boolean functions for their algebraic immunity we identify that functions having low degree subfunctions are weak in terms of algebraic immunity and we analyze some existing constructions from this viewpoint further we present a construction method to generate boolean functions on n variables with highest possible algebraic immunity spl lceil n 2 spl rceil this construction first presented at the 2005 workshop on fast software encryption fse 2005 has been the first one producing such functions these functions are obtained through a doubly indexed recursive relation we calculate their hamming weights and deduce their nonlinearities we show that they have very high algebraic degrees we express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables we deduce a very fast way of computing the output to these functions given their input

the burnside ring of the infinite cyclic group and its relations to the necklace algebra λ rings and the universal ring of witt vectors abstract it is shown that well known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the burnside ring of the infinite cyclic group on the one hand and grothendieck s ring of formal power series with constant term 1 as well as the universal ring of witt vectors on the other hand