a lower bound on the undetected error probability and strictly optimal codes error detection is a simple technique used in various communication and memory systems to enhance reliability we study the probability that a q ary linear or nonlinear block code of length n and size m fails to detect an error a lower bound on this undetected error probability is derived in terms of q n and m the new bound improves upon other bounds mentioned in the literature even those that hold only for linear codes block codes whose undetected error probability equals the new lower bound are investigated we call these codes strictly optimal codes and give a combinatorial characterization of them we also present necessary and sufficient conditions for their existence in particular we find all values of n and m for which strictly optimal binary codes exist and determine the structure of all of them for example we construct strictly optimal binary coded decimal codes of length four and five and we show that these are the only possible lengths of such codes

eisenstein series on loop groups maass selberg relations 2 we prove that truncated pseudo eisenstein series have compact support on the arithmetic quotient

lower bounds on the quantum capacity and highest error exponent of general memoryless channels tradeoffs between the information rate and fidelity of quantum error correcting codes are discussed quantum channels to be considered are those subject to independent errors and modeled as tensor products of copies of a general completely positive cp linear map where the dimension of the underlying hilbert space is a prime number on such a quantum channel the highest fidelity of a quantum error correcting code of length n and rate r is proven to be lower bounded by 1 exp ne r o n for some function e r the e r is positive below some threshold r sub 0 a direct consequence of which is that r sub 0 is a lower bound on the quantum capacity this is an extension of the author s earlier result while the earlier work states the result for the depolarizing channel and a slight generalization of it pauli channels the result of this work applies to general discrete memoryless channels including channel models derived from a physical law of time evolution

optimal codes for error detection the probability of undetected error for codes over gf 2 sup m having a generator matrix over gf 2 is studied the optimal codes of dimensions four or less are determined for higher dimensions some properties of optimal codes are determined

krawtchouk polynomials and universal bounds for codes and designs in hamming spaces universal bounds for the cardinality of codes in the hamming space f sub r sup n with a given minimum distance d and or dual distance d are stated a self contained proof of optimality of these bounds in the framework of the linear programming method is given the necessary and sufficient conditions for attainability of the bounds are found the parameters of codes satisfying these conditions are presented in a table a new upper bound for the minimum distance of self dual codes and a new lower bound for the crosscorrelation of half linear codes are obtained

growth processes related to the dispersionless lax equations this paper is a short review of the connection between certain types of growth processes and the integrable systems theory written from the viewpoint of the latter starting from the dispersionless lax equations for the 2d toda hierarchy we interpret them as evolution equations for conformal maps in the plane this provides a unified approach to evolution of smooth domains such as laplacian growth and growth of slits we show that the lowner differential equation for a parametric family of conformal maps of slit domains arises as a consistency condition for reductions of the dispersionless toda hierarchy it is also demonstrated how the both types of growth processes can be simulated by the large n limit of the dyson gas picture for the model of normal random matrices

improved upper bounds for codes with unequal error protection asymptotic nonexistence bounds for unequal error protecting codes with two protection levels are considered we show that the improved estimates on the possible distance distributions for codes may sometimes yield sharper upper bounds than the previously known ones on the higher significance protection level of both nonlinear and linear codes having two protection levels

reductions and hodograph solutions of the dispersionless kp hierarchy a general scheme for analysing reductions of dispersionless integrable hierarchies is presented it is based on a method for determining the s function by means of a system of first order differential equations compatibility systems of nonlinear partial differential equations of bourlet type characterizing both reductions and hodograph solutions of the dkp hierarchy are obtained wide classes of illustrative explicit examples are exhibited

new upper bounds on error exponents we derive new upper bounds on the error exponents for the maximum likelihood decoding and error detecting in the binary symmetric channels this is an improvement on the best earlier known bounds by shannon gallager berlekamp 1967 and mceliece omura 1977 for the probability of undetected error the new bounds are better than the bounds by levenshtein 1978 1989 and the bound by abdel ghaffar see ibid vol 43 p 1489 502 1997 moreover we further extend the range of rates where the undetected error exponent is known to be exact the new bounds are based on an analysis of possible distance distributions of the codes along with some inequalities relating the distance distributions to the error probabilities

eisenstein series on affine kac moody groups over function fields in his pioneering work 4 h garland constructed eisenstein series on affine kac moody groups over the field of real numbers he established the almost everywhere convergence of these series obtained a formula for their constant terms and proved a functional equation for the constant terms in his subsequent paper 5 the convergence of the eisenstein series was obtained in this paper we define eisenstein series on affine kac moody groups over global function fields using an adelic approach in the course of proving the convergence of these eisenstein series we also calculate a formula for the constant terms and prove their convergence and functional equations

a direct geometrical method for bounding the error exponent for any specific family of channel codes i cutoff rate lower bound for block codes a direct general and conceptually simple geometrical method for determining lower and upper bounds on the error exponent of any specific family of channel block codes is presented it is considered that a specific family of codes is characterized by a unique distance distribution exponent the tight linear lower bound of slope 1 on the code family error exponent represents the code family cutoff rate bound it is always a minimum of a sum of three functions the intrinsic asymptotic properties of channel block codes are revealed by analyzing these functions and their relationships it is shown that the random coding technique for lower bounding the channel error exponent is a special case of this general method the requirements that a code family should meet in order to have a positive error exponent and at best attain the channel error exponent are stated in a clear way using the direct distance distribution method presented

theta functions and weil representations of loop symplectic groups we introduce weil representations for loop symplectic groups and prove the convergence and modularity of the related theta functions

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

conformal maps and integrable hierarchies we show that conformal maps of simply connected domains with an analytic boundary to a unit disk have an intimate relation to the dispersionless 2d toda integrable hierarchy the maps are determined by a particular solution to the hierarchy singled out by the conditions known as string equations the same hierarchy locally solves the 2d inverse potential problem i e reconstruction of the domain out of a set of its harmonic moments this is the same solution which is known to describe 2d gravity coupled to c matter we also introduce a concept of the function for analytic curves

linear time encodable decodable codes with near optimal rate we present an explicit construction of linear time encodable and decodable codes of rate r which can correct a fraction 1 r spl epsiv 2 of errors over an alphabet of constant size depending only on spl epsiv for every 0 0 the error correction performance of these codes is optimal as seen by the singleton bound these are near mds codes such near mds linear time codes were known for the decoding from erasures our construction generalizes this to handle errors as well concatenating these codes with good constant sized binary codes gives a construction of linear time binary codes which meet the zyablov bound and also the more general blokh zyablov bound by resorting to multilevel concatenation our work also yields linear time encodable decodable codes which match forney s error exponent for concatenated codes for communication over the binary symmetric channel the encoding decoding complexity was quadratic in forney s result and forney s bound has remained the best constructive error exponent for almost 40 years now in summary our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding but in addition have linear time encoding and decoding algorithms

improved upper bounds on the reliability function of the gaussian channel a new lower bound on the distance distribution of spherical codes is derived this yields two new upper bounds on the reliability function of the gaussian channel these bounds outperform previously known bounds and imply a new range of rates for which the exact value of the reliability function is known

new upper bounds on generalized weights we derive new asymptotic upper bounds on the generalized weights of a binary linear code of a given size we also prove some asymptotic results on the distance distribution of binary codes

automorphic forms with degenerate fourier coefficients the main theme of this paper is that singular automorphic forms on classical groups are given by theta series liftings we establish several inequalities relating the automorphic multiplicities of a given representation and that of its abstract theta lift our methods allow one to lift noncuspidal square integrable automorphic forms when the dual pair involved is in the stable range in this way we construct new families of singular automorphic forms many of which are clearly unipotent in fact starting from one dimensional representations and repeating the procedure of lifting in the stable range one may obtain all automorphic forms which are quadratic unipotent in the sense of moeglin

on the probability of undetected error for linear block codes the problem of computing the probability of undetected error is considered for linear block codes used for error detection the recent literature is first reviewed and several results are extended it is pointed out that an exact calculation can be based on either the weight distribution of a code or its dual using the dual code formulation the probability of undetected error for the ensemble of all nonbinary linear block codes is derived as well as a theorem that shows why the probability of undetected error may not be a monotonic function of channel error rate for some poor codes several bounds on the undetected error probability are then presented we conclude with detailed examples of binary and nonbinary codes for which exact results can be obtained an efficient technique for measuring an unknown weight distribution is suggested and exact results are compared with experimental results

s functions reductions and hodograph solutions of the rth dispersionless modified kp and dym hierarchies we introduce an s function formulation for the recently found rth dispersionless modified kp and rth dispersionless dym hierarchies giving also a connection of these s functions with the orlov functions of the hierarchies then we discuss a reduction scheme for the hierarchies that together with the s function formulation leads to hodograph systems for the associated solutions we consider also the connection of these reductions with those of the dispersionless kp hierarchy and with hydrodynamic type systems in particular for the one component and two component reduction we derive for both hierarchies ample sets of examples of explicit solutions

holomorphic mappings between hyperquadrics with small signature difference in this paper we study holomorphic mappings sending a hyperquadric of signature ell in bbb c n into a hyperquadric of signature ell in bbb c n we show theorem 1 1 that if the signature difference ell ell is not too large then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and in particular the image of the mapping is contained in a complex plane of a dimension that depends only on ell and ell and not on the target dimension n we also prove a hopf lemma type result theorem 1 3 for such mappings

on the structure of correlation functions in the normal matrix model we study the structure of the normal matrix model nmm we show that all correlation functions of the model with an axially symmetric potential can be expressed in terms of a single holomorphic function of one variable this observation is used to demonstrate the exact solvability of the model the two point correlation function is calculated in the continuum limit the answer is proven to be universal i e potential independent up to a change of the scale in connection with nmm with a general polynomial potential we have developed a two dimensional free fermion formalism and constructed a family of completely integrable hierarchies of non linear differential equations which we call the extended kp n hierarchies the well known kp hierarchy is a lower dimensional reduction of this family the extended kp 1 hierarchy contains the 2 1 dimensional burgers equations the partition function of the n n nmm is the function of the extended kp n hierarchy which is invariant with respect to a subalgebra of an algebra of all infinitesimal diffeomorphisms of the plane

entirety of cuspidal eisenstein series on loop groups in this paper we prove the entirety of loop group eisenstein series induced from cusp forms on the underlying finite dimensional group by demonstrating their absolute convergence on the full complex plane this is quite in contrast to the finite dimensional setting where such series only converge absolutely in a right half plane and have poles elsewhere coming from l functions in their constant terms our result is the bbb q analog of a theorem of a braverman and d kazhdan from the function field setting who previously showed the analogous eisenstein series there are finite sums

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

on the siegel weil formula for unitary groups following s s kudla and s rallis we extend the siegel weil formula for unitary groups which relates a value of a siegel eisenstein series to the convergent integral of a theta function

tight performance bounds for permutation invariant binary linear block codes over symmetric channels random coding performance bounds for l list permutation invariant binary linear block codes transmitted over output symmetric channels are presented under list decoding double and single exponential bounds are deduced by considering permutation ensembles of the above codes and exploiting the concavity of the double exponential function over the region of erroneous received vectors the proposed technique specifies fixed list sizes l for specific codes under which the corresponding list decoding error probability approaches zero in a double exponential manner the single exponential bound constitutes a generalization of shulman feder bound and allows the treatment of codes with rates below the cutoff limit numerical examples of the new bounds for the specific category of codes are presented

quantum error detection ii bounds for pt i see ibid vol 46 no 3 p 778 88 2000 in part i of this paper we formulated the problem of error detection with quantum codes on the depolarizing channel and gave an expression for the probability of undetected error via the weight enumerators of the code in this part we show that there exist quantum codes whose probability of undetected error falls exponentially with the length of the code and derive bounds on this exponent the lower existence bound is proved for stabilizer codes by a counting argument for classical self orthogonal quaternary codes upper bounds are proved by linear programming first we formulate two linear programming problems that are convenient for the analysis of specific short codes next we give a relaxed formulation of the problem in terms of optimization on the cone of polynomials in the krawtchouk basis we present two general solutions of the problem together they give an upper bound on the exponent of undetected error the upper and lower asymptotic bounds coincide for a certain interval of code rates close to 1

bounds on the reliability function of typewriter channels new lower and upper bounds on the reliability function of typewriter channels are given our lower bounds improve upon the multiletter expurgated bound of gallager furnishing a new and simple counterexample to a conjecture made in 1967 by shannon gallager and berlekamp on its tightness the only other known counter example is due to katsman tsfasman and vl du who used algebraic geometric codes on a q ary symmetric channels q geq 49 here we prove by introducing dependence between codewords of a random ensemble that the conjecture is false even for a typewriter channel with q 4 inputs in the process we also demonstrate that lovasz s proof of the capacity of the pentagon was implicitly contained but unnoticed in the works of jelinek and gallager on the expurgated bound done at least ten years before lovasz in the opposite direction new upper bounds on the reliability function are derived for channels with an odd number of inputs by using an adaptation of delsarte s linear programming bound first we derive a bound based on the minimum distance which combines lovasz s construction for bounding the graph capacity with the mceliece rodemich rumsey welch construction for bounding the minimum distance of codes in the hamming space then for the particular case of cross over probability 1 2 we derive an improved bound by also using the method of kalai and linial to study the spectrum distribution of codes

sphere packing bounds revisited for moderate block lengths the main reference of this paper is the sphere packing bound of 1967 sp67 derived by shannon gallager and berlekamp it offers a lower bound on the decoding error probability over a very large variety of channels if it has failed so far to provide any usable material in practical implementation of telecommunication systems it is due to an original focus on asymptotic results making it inapplicable for moderate code lengths and to the difficulty of the involved methods which makes the derivation of sp67 quite hermetic and uninspiring for further research the purpose of this paper is two fold 1 to stir up some renewed interest in the topic on which shannon concluded his career in information theory thanks to a qualitative rather than technical review of the derivation of sp67 introduced by a review of the simpler sphere packing bound derived by shannon in 1959 2 to prove the practical interest of sp67 by extending its field of application to continuous output channels and particularly the additive white gaussian noise awgn channel used with any particular modulation scheme and by improving its lower bound for the moderate code length case so that it becomes the best lower bound for most iteratively decodable codes turbo codes low density parity check ldpc codes repeat accumulate ra codes etc of usual lengths

upper bound on list decoding radius of binary codes consider the problem of packing hamming balls of a given relative radius subject to the constraint that they cover any point of the ambient hamming space with multiplicity at most l for odd l ge 3 an asymptotic upper bound on the rate of any such packing is proved the resulting bound improves the best known bound due to blinovsky 1986 for rates below a certain threshold the method is a superposition of the linear programming idea of ashikhmin barg and litsyn that was previously used to improve the estimates of blinovsky for l 2 and a ramsey theoretic technique of blinovsky as an application it is shown that for all odd l the slope of the rate radius tradeoff is zero at zero rate

the weight distribution of cosets sullivan s 1967 inequality between the weight distribution function of a binary linear code and the weight distribution function of a proper subset of the code is generalized to linear codes over arbitrary finite fields

the bee identification problem bounds on the error exponent consider the problem of identifying a massive number of bees uniquely labeled with barcodes using noisy measurements we formally introduce this bee identification problem define its error exponent and derive efficiently computable upper and lower bounds for this exponent we show that joint decoding of barcodes provides a significantly better exponent compared to separate decoding followed by permutation inference for low rates we prove that the lower bound on the bee identification exponent obtained using typical random codes trc is strictly better than the corresponding bound obtained using a random code ensemble rce further as the rate approaches zero we prove that the upper bound on the bee identification exponent meets the lower bound obtained using trc with joint barcode decoding

conformal mappings and dispersionless toda hierarchy ii general string equations in this article we classify the solutions of the dispersionless toda hierarchy into degenerate and non degenerate cases we show that every non degenerate solution is determined by a function mathcal h z 1 z 2 of two variables we interpret these non degenerate solutions as defining evolutions on the space mathfrak d of pairs of conformal mappings g f where g is a univalent function on the exterior of the unit disc f is a univalent function on the unit disc normalized such that g f 0 0 and f 0 g 1 for each solution we show how to define the natural time variables t n n in mathbb z as complex coordinates on the space mathfrak d we also find explicit formulas for the tau function of the dispersionless toda hierarchy in terms of mathcal h z 1 z 2 imposing some conditions on the function mathcal h z 1 z 2 we show that the dispersionless toda flows can be naturally restricted to the subspace of mathfrak d defined by f w 1 overline g 1 bar w this recovers the result of zabrodin theor math phy 12 1511 1525 2001

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

lower bounds on the error probability of block codes based on improvements on de caen s inequality new lower bounds on the error probability of block codes with maximum likelihood decoding are proposed the bounds are obtained by applying a new lower bound on the probability of a union of events derived by improving on de caen s lower bound the new bound includes an arbitrary function to be optimized in order to achieve the tightest results since the optimal choice of this function is known but leads to a trivial and useless identity we find several useful approximations for it each resulting in a new lower bound for the additive white gaussian noise awgn channel and the binary symmetric channel bsc the optimal choice of the optimization function is stated and several approximations are proposed when the bounds are further specialized to linear codes the only knowledge on the code used is its weight enumeration the results are shown to be tighter than the latest bounds in the current literature such as those by seguin 1998 and by keren and litsyn 2001 moreover for the bsc the new bounds widen the range of rates for which the union bound analysis applies thus improving on the bound to the error exponent compared with the de caen based bounds

the bee identification error exponent with absentee bees the bee identification problem was formally defined by tandon tan and varshney ieee trans commun vol 67 2019 and the error exponent was studied this work extends the results for the absentee bees scenario where a fraction of the bees are absent in the beehive image used for identification for this setting we present an exact characterization of the bee identification error exponent and show that independent barcode decoding is optimal i e joint decoding of the bee barcodes does not result in a better error exponent relative to independent decoding of each noisy barcode this is in contrast to the result without absentee bees where joint barcode decoding results in a significantly higher error exponent than independent barcode decoding we also define and characterize the capacity for the bee identification problem with absentee bees and prove a strong converse for the same

integrable structure of the dirichlet boundary problem in two dimensions we study how the solution of the two dimensional dirichlet boundary problem for smooth simply connected domains depends upon variations of the data of the problem we show that the hadamard formula for the variation of the dirichlet green function under deformations of the domain reveals an integrable structure the independent variables corresponding to the infinite set of commuting flows are identified with harmonic moments of the domain the solution to the dirichlet boundary problem is expressed through the tau function of the dispersionless toda hierarchy we also discuss a degenerate case of the dirichlet problem on the plane with a gap in this case the tau function is identical to the partition function of the planar large n limit of the hermitian one matrix model

laplacian growth in a channel and hurwitz numbers we study the integrable structure of the 2d laplacian growth problem with zero surface tension in an infinite channel with periodic boundary conditions in a transverse direction similarly to the laplacian growth in radial geometry this problem can be embedded into the 2d toda lattice hierarchy in the zero dispersion limit however the relevant solution to the hierarchy is different we characterize this solution by the string equations and construct the corresponding dispersionless tau function this tau function is shown to coincide with the genus zero part of the generating function for double hurwitz numbers

generalizations of the korzhik bound corresp the probability of undetected error is studied when a code is used both for error correction and error detection a number of generalizations is given of an upper bound of korzhik on the minimal probability of undetected error for an n k code

eisenstein series on rank 2 hyperbolic kac moody groups we define eisenstein series on rank 2 hyperbolic kac moody groups over mathbb r induced from quasi characters we prove convergence of the constant term and hence the almost everywhere convergence of the eisenstein series we define and calculate the degenerate fourier coefficients we also consider eisenstein series induced from cusp forms and show that these are entire functions

distance distribution of binary codes and the error probability of decoding we address the problem of bounding below the probability of error under maximum likelihood decoding of a binary code with a known distance distribution used on a binary symmetric channel bsc an improved upper bound is given for the maximum attainable exponent of this probability the reliability function of the channel in particular we prove that the random coding exponent is the true value of the channel reliability for codes rate r in some interval immediately below the critical rate of the channel an analogous result is obtained for the gaussian channel

double affine hecke algebras and 2 dimensional local fields the concept of an n dimensional local field was introduced by a n parshin pal with the aim of generalizing the classical adelic formalism to absolutely ndimensional schemes by definition a 0 dimensional local field is just a finite field and an n dimensional local field n 0 is a complete discrete valued field whose residue field is n 1 dimensional local thus for n 1 we get locally compact fields such as qp fq t and for n 2 we get fields such as qp t fq ti t2 etc in representation theory harmonic analysis on reductive groups over 0and 1dimensional local fields leads in particular to consideration of the finite and affine hecke algebras hq hq associated to aniy finite root system r and anly q e c these algebras can be defined in several ways one being by generators and relations another as the convolution algebra with respect to the haar measure of functions on the group bi invariant with respect to an appropriate subgroup i e as the algebra of double cosets harmonic analysis on groups over 2 dimensional local fields has not been developed the main difficulty being the infinite dimensionality absense of local compactness of such fields however the double affine hecke

using codes for error correction and detection corresp a linear code c over gf q is good for t error correction and error detection if p c t epsilon leq p c t q 1 q for all epsilon 0 leq epsilon leq q 1 q where p c t epsilon is the probability of an undetected error after a codeword in c is transmitted over a q ary symmetric channel with error probability epsilon and correction is performed for all error patterns with t or fewer errors a sufficient condition for a code to be good is derived this sufficient condition is easy to check and examples to illustrate the method are given

on the theta functional of weil representations for symplectic loop groups abstract we prove the convergence of the theta functional for loop group weil representation under certain natural conditions and discuss the modularity

algebraic theory of dicritical divisors the topological theory of dicritical divisors is algebracized which makes it valid for nonzero characteristic as well as mixed characteristic

the hodge conjecture and arithmetic quotients of complex balls let s be a closed shimura variety uniformized by the complex n ball associated with a standard unitary group the hodge conjecture predicts that every hodge class in h 2k s mathbb q k 0 dots n is algebraic we show that this holds for all degrees k away from the neighborhood bigl tfrac13n tfrac23n bigr of the middle degree we also prove the tate conjecture for the same degrees as the hodge conjecture and the generalized form of the hodge conjecture in degrees away from an interval depending on the codimension c of the subvariety centered at the middle dimension of s these results are derived from a general theorem that applies to all shimura varieties associated with standard unitary groups of any signature the proofs make use of arthur s endoscopic classification of automorphic representations of classical groups as such our results rely on the stabilization of the trace formula for the disconnected groups gl n rtimes langle theta rangle associated with base change

a lower bound on the probability of undetected error for binary constant weight codes in this correspondence we study the probability of undetected error for binary constant weight codes first we derive a new lower bound on the probability of undetected error for binary constant weight codes next we show that this bound is tight if and only if the binary constant weight codes are generated from certain t designs in combinatorial design theory this means that these binary constant weight codes generated from certain t designs are uniformly optimal for error detection along the way we determine the distance distributions of such binary constant weight codes in particular it is shown that binary constant weight codes generated from steiner systems are uniformly optimal for error detection thus we prove a conjecture of xia fu jiang and ling furthermore the distance distribution of a binary constant weight code generated from a steiner system is determined finally we study the exponent of the probability of undetected error for binary constant weight codes we derive some bounds on the exponent of the probability of undetected error for binary constant weight codes these bounds enable us to extend the region in which the exponent of the probability of undetected error is exactly determined

metaplectic covers of kac moody groups and whittaker functions starting from some linear algebraic data a weyl group invariant bilinear form and some arithmetic data a bilinear steinberg symbol we construct a central extension of a kac moody group generalizing the work of matsumoto specializing our construction over non archimedean local fields for each positive integer n we obtain the notion of n fold metaplectic covers of kac moody groups in this setting we prove a casselman shalika type formula for whittaker functions

a regularized siegel weil formula for unitary groups for the quasi split unitary group u n n we prove a refined analogue of the result of s s kudla and s rallis on an identity a regularized siegel weil formula between a residue of an eisenstein series and the integral of a certain theta function

linear block codes for error detection corresp the probability of undetected error of linear block codes for use on a binary symmetric channel is investigated upper hounds are derived several classes of linear block codes are proved to have good error detecting capability

faltings heights of cm cycles and derivatives of l functions we study the faltings height pairing of arithmetic special divisors and cm cycles on shimura varieties associated to orthogonal groups we compute the archimedean contribution to the height pairing and derive a conjecture relating the total pairing to the central derivative of a rankin l function we prove the conjecture in certain cases where the shimura variety has dimension 0 1 or 2 in particular we obtain a new proof of the gross zagier formula

source and channel rate allocation for channel codes satisfying the gilbert varshamov or tsfasman vladut zink bounds we derive bounds for optimal rate allocation between source and channel coding for linear channel codes that meet the gilbert varshamov or tsfasman vladut zink 1984 bounds formulas giving the high resolution vector quantizer distortion of these systems are also derived in addition we give bounds on how far below the channel capacity the transmission rate should be for a given delay constraint the bounds obtained depend on the relationship between channel code rate and relative minimum distance guaranteed by the gilbert varshamov bound and do not require sophisticated decoding beyond the error correction limit we demonstrate that the end to end mean squared error decays exponentially fast as a function of the overall transmission rate which need not be the case for certain well known structured codes such as hamming codes

on the distance distribution of codes the distinct distribution of a binary code c is the sequence b sub i sub i 0 sup n defined as follows let b sub i w be the number of codewords at distance i from the codeword w and let b sub i be the average of b sub i w over all w in c in this correspondence we study the distance distribution for codes of length n and minimal distance spl delta n with spl delta 0 fixed and n spl rarr spl infin our main aim is to relate the size of the code with the distribution of distances near the minimal distance

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

a lower bound on the quantum capacity of channels with correlated errors the highest fidelity of quantum error correcting codes of length n and rate r is proven to be lower bounded by 1 exp ne r o n for some function e r on noisy quantum channels that are subject to not necessarily independent errors the e r is positive below some threshold r0 which implies r0 is a lower bound on the quantum capacity this work is an extension of the author s previous works m hamada phys rev a 65 052305 2002 quant ph 0109114 lanl 2001 m hamada quant ph 0112103 lanl 2001 which presented the bound for channels subject to independent errors or channels modeled as tensor products of copies of a completely positive linear map the relation of the channel class treated in this article to those in the previous works is similar to that of markov chains to sequences of independent identically distributed random variables

fingering in a channel and tripolar loewner evolutions a class of laplacian growth models in the channel geometry is studied using the formalism of tripolar loewner evolutions in which three points namely the channel corners and the point at infinity are kept fixed initially the problem of fingered growth where growth takes place only at the tips of slitlike fingers is revisited and a class of exact solutions of the corresponding loewner equation is presented for the case of stationary driving functions a model for interface growth is then formulated in terms of a generalized tripolar loewner equation and several examples are presented it is shown that the growing interface evolves into a steadily moving finger and that tip competition arises for nonsymmetric initial configurations with multiple tips

laplacian growth and whitham equations of soliton theory the laplacian growth the hele shaw problem of multiply connected domains in the case of zero surface tension is proven to be equivalent to an integrable system of whitham equations known in soliton theory the whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations through this connection the laplacian growth is understood as a flow in the moduli space of riemann surfaces

eisenstein series on loop groups maass selberg relations 3 in this paper we prove the maass selberg relations for the inner product of truncated pseudo eisenstein series one can then extract the maass selberg relations for truncated eisenstein series themselves from those for truncated pseudo eisenstein series a task which we shall complete in a subsequent paper