asymmetric evaluations of erasure and undetected error probabilities the problem of channel coding with the erasure option is revisited for discrete memoryless channels the interplay between the code rate the undetected and total error probabilities is characterized using the information spectrum method a sequence of codes of increasing blocklengths n is designed to illustrate this tradeoff furthermore for additive discrete memoryless channels with uniform input distribution we establish that our analysis is tight with respect to the ensemble average this is done by analyzing the ensemble performance in terms of a tradeoff between the code rate the undetected and total errors this tradeoff is parameterized by the threshold in a generalized likelihood ratio test two asymptotic regimes are studied first the code rate tends to the capacity of the channel at a rate slower than n 1 2 corresponding to the moderate deviations regime in this case both error probabilities decay subexponentially and asymmetrically the precise decay rates are characterized second the code rate tends to capacity at a rate of n 1 2 in this case the total error probability is asymptotically a positive constant while the undetected error probability decays as exp bn 1 2 for some b 0 the proof techniques involve the applications of a modified or shifted version of the gartner ellis theorem and the type class enumerator method to characterize the asymptotic behavior of a sequence of cumulant generating functions

variable length lossy source code using a constrained random number generator a variable length lossy source code is introduced with a rate distortion pair close to the rate distortion function random numbers that satisfy a condition specified by a function and its value are used to construct a stochastic encoder the proof of the theorem is based on the balanced coloring property of an ensemble of functions since an ensemble of systematic sparse matrices has this property we can construct a tractable code for a memoryless source some algorithms for implementing the code are introduced and compared by simulation

achievable error exponents for channels with side information erasure and list decoding we consider a decoder with an erasure option and a variable size list decoder for channels with non casual side information at the transmitter first a universally achievable region of error exponents is offered for decoding with an erasure option using a parameterized decoder in the spirit of csiszar and korner s decoder then the proposed decoding rule is generalized by extending the range of its parameters to allow variable size list decoding this extension gives a unified treatment for erasure list decoding an achievable region of exponential bounds on the probability of list error and the average number of incorrect messages on the list are given relations to forney s and csiszar and korner s decoders for discrete memoryless channel are discussed these results are obtained by exploring a random binning code with conditionally constant composition codewords proposed by moulin and wang but with a different decoding rule and a modified analysis

composite scheme lr th for decoding with erasures and its effective equivalence to forney s rule for decoding with erasures forney s scheme is known to be optimal in the sense that no other scheme can make the erasure probability p sub ers and undetected error probability p sub uer simultaneously smaller we propose a scheme for erasure decision which tests the likelihood ratio as well as the likelihood itself and show that the attainable upper bounds on p sub ers and p sub uer are the same as those proved for the optimal scheme up to a constant factor we also show that the scheme gives when applied to convolutional codes a bound which is related to the block coding bound via forney s inverse concatenation construction we show that this bound is the same as the one which naturally arises when we apply raghavan and baum s 1998 optimal scheme to convolutional code

variable length coding over an unknown channel burnashev in 1976 gave an exact expression for the reliability function of a discrete memoryless channel dmc with noiseless feedback a coding scheme that achieves this exponent needs in general to know the statistics of the channel suppose now that the coding scheme is designed knowing only that the channel belongs to a family q of dmcs is there a coding scheme with noiseless feedback that achieves burnashev s exponent uniformly over q at a nontrivial rate we answer the question in the affirmative for two families of channels binary symmetric and z for these families we show that for any given fraction there is a feedback coding strategy such that for any member of the family i guarantees this fraction of its capacity as rate and ii guarantees the corresponding burnashev s exponent therefore for these families in terms of delay and error probability the knowledge of the channel becomes asymptotically irrelevant in feedback code design there are blind schemes that perform as well as the best coding scheme designed with the foreknowledge of the channel under use however a converse result shows that in general even for families that consist of only two channels such blind schemes do not exist

a computer assisted verification of hyperchaos in the saito hysteresis chaos generator this paper presents a computer assisted verification of hyperchaos in the well known saito hysteresis chaos generator shcg by virtue of topological horseshoe theory by means of interval analysis we find two disjoint compact subsets in a carefully chosen 3d cross section that can guarantee the existence of a topological horseshoe for the corresponding third return poincare map numerical studies show that the poincare map expands in two directions it justifiably indicates that there exists hyperchaos in the shcg

canonical singularities of orders over surfaces we define and study canonical singularities of orders over surfaces these are non commutative analogues of kleinian singularities that arise naturally in the minimal model program for orders over surfaces d chan and c ingalls invent math 161 2005 427 452 we classify canonical singularities of orders using their minimal resolutions which we define we describe them explicitly as invariant rings for the action of a finite group on a full matrix algebra over a regular local ring we also prove that they are gorenstein describe their auslander reiten quivers and note a simple version of the mckay correspondence

del pezzo orders on projective surfaces abstract in this paper we generalise the notion of del pezzo surfaces to orders on surfaces we show that these del pezzo orders have del pezzo centre if the centre is normal gorenstein and the order has finite representation type we proceed to classify these del pezzo orders the main result is that if the centre is not p 1 p 1 or the quadric cone then these del pezzo orders can be obtained from del pezzo orders on p 2 finally we classify del pezzo orders on p 2 p 1 p 1 and the quadric cone

the minimal model program for orders over surfaces we develop the minimal model program for orders over surfaces and so establish a noncommutative generalisation of the existence and uniqueness of minimal algebraic surfaces we define terminal orders and show that they have unique etale local structures this shows that they are determined up to morita equivalence by their centre and algebra of quotients this reduces our problem to the study of pairs z consisting of a surface z and an element of the brauer group brk z we then extend the minimal model program for surfaces to such pairs combining these results yields a noncommutative version of resolution of singularities and allows us to show that any order has either a unique minimal model up to morita equivalence or is ruled or del pezzo

quantizable orders over surfaces abstract infinitesimal deformations of maximal orders over smooth algebraic surfaces are studied we classify which maximal orders may admit deformations that are not finite over their centres the classification is in terms of the algebraic surface that corresponds to the centre of the order and the ramification divisor of the order

detecting unstable periodic orbits in chaotic continuous time dynamical systems we extend the recently developed method for detecting unstable periodic points of chaotic time discrete dynamical systems to find unstable periodic orbits in time continuous systems given by a set of ordinary differential equations this is achieved by the reduction of the continuous flow to a poincar e map which is then searched for periodic points the algorithm has global convergence properties and needs no a priori knowledge of the system it works well for both dissipative and hamiltonian dynamical systems which is demonstrated by exploring the lorenz system and the hydrogen atom in a strong magnetic field the advantages and general features of the approach are discussed in detail

universal decoding for finite state channels universal decoding procedures for finite state channels are discussed although the channel statistics are not known universal decoding can achieve an error probability with an error exponent that for large enough block length or constraint length in case of convolutional codes is equal to the random coding error exponent associated with the optimal maximum likelihood decoding procedure for the given channel the same approach is applied to sequential decoding yielding a universal sequential decoding procedure with a cutoff rate and an error exponent that are equal to those achieved by the classical sequential decoding procedure

variational method for finding periodic orbits in a general flow a variational principle is proposed and implemented for determining unstable periodic orbits of flows as well as unstable spatiotemporally periodic solutions of extended systems an initial loop approximating a periodic solution is evolved in the space of loops toward a true periodic solution by a minimization of local errors along the loop the newton descent partial differential equation that governs this evolution is an infinitesimal step version of the damped newton raphson iteration the feasibility of the method is demonstrated by its application to the henon heiles system the circular restricted three body problem and the kuramoto sivashinsky system in a weakly turbulent regime

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

large deviations behavior of the logarithmic error probability of random codes this work studies the deviations of the error exponent of the constant composition code ensemble around its expectation known as the error exponent of the typical random code trc in particular it is shown that the probability of randomly drawing a codebook whose error exponent is smaller than the trc exponent is exponentially small upper and lower bounds for this exponent are given which coincide in some cases in addition the probability of randomly drawing a codebook whose error exponent is larger than the trc exponent is shown to be double exponentially small upper and lower bounds to the double exponential exponent are given the results suggest that codebooks whose error exponent is larger than the error exponent of the trc are extremely rare the key ingredient in the proofs is a new large deviations result of type class enumerators with dependent variables

graph decomposition a new key to coding theorems a new and simple method is proposed for finding good encoders both for channels and for sources with side information this method relies on the continuous version of a graph decomposition result of lovasz the presently known best exponential error bounds for both problems follow in a unified manner with an improvement on the source coding bound the previous bounds for universal codes of the authors and marton are also improved

minimax universal decoding with an erasure option motivated by applications of rateless coding decision feedback and automatic repeat request arq we study the problem of universal decoding for unknown channels in the presence of an erasure option specifically we harness the competitive minimax methodology developed in earlier studies in order to derive a universal version of forney s classical erasure list decoder which in the erasure case optimally trades off between the probability of erasure and the probability of undetected error the proposed universal erasure decoder guarantees universal achievability of a certain fraction xi of the optimum error exponents of these probabilities in a sense to be made precise in the sequel a single letter expression for xi which depends solely on the coding rate and the neyman pearson threshold to be defined is provided the example of the binary symmetric channel is studied in full detail and some conclusions are drawn

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

universal composite hypothesis testing a competitive minimax approach a novel approach is presented for the long standing problem of composite hypothesis testing in composite hypothesis testing unlike in simple hypothesis testing the probability function of the observed data given the hypothesis is uncertain as it depends on the unknown value of some parameter the proposed approach is to minimize the worst case ratio between the probability of error of a decision rule that is independent of the unknown parameters and the minimum probability of error attainable given the parameters the principal solution to this minimax problem is presented and the resulting decision rule is discussed since the exact solution is in general hard to find and a fortiori hard to implement an approximation method that yields an asymptotically minimax decision rule is proposed finally a variety of potential application areas are provided in signal processing and communications with special emphasis on universal decoding

existence of terminal resolutions of geometric brauer pairs of arbitrary dimension abstract a geometric brauer pair is a pair x where x is a smooth quasi projective variety over an algebraically closed field and is an element in the 2 torsion part of the brauer group of the function field of x a geometric brauer pair y is a terminal pair if the brauer discrepancy of y is positive we show that given a geometric brauer pair x there is a terminal pair y with a birational morphism y x in short any geometric brauer pair admits a terminal resolution

erasure list random coding error exponents are not universally achievable we study the problem of universal decoding for unknown discrete memoryless channels in the presence of erasure list option at the decoder in the random coding regime in particular we harness a universal version of forney s classical erasure list decoder developed in earlier studies which is based on the competitive minimax methodology and guarantees universal achievability of a certain fraction of the optimum random coding error exponents in this paper we derive an exact single letter expression for the maximum achievable fraction examples are given in which the maximal achievable fraction is strictly less than unity which imply that in general there is no universal erasure list decoder which achieves the same random coding error exponents as the optimal decoder for a known channel this is in contrast to the situation in ordinary decoding without the erasure list option where optimum exponents are universally achievable as is well known it is also demonstrated that previous lower bounds derived for the maximal achievable fraction are not tight in general we then analyze a generalized random coding ensemble which incorporate a training sequence in conjunction with a suboptimal practical decoder plug in decoder which first estimates the channel using the available training sequence and then decodes the remaining symbols of the codeword using the estimated channel one of the implications of our results is setting the stage for a reasonable criterion of optimal training finally we compare the performance of the plug in decoder and the universal decoder in terms of the achievable error exponents and show that the latter is noticeably better than the former

errors and erasures decoding for block codes with feedback inner and outer bounds are derived on the optimal performance of fixed length block codes on discrete memoryless channels with feedback and errors and erasures decoding first an inner bound is derived using a two phase encoding scheme with communication and control phases together with the optimal decoding rule for the given encoding scheme among decoding rules that can be represented in terms of pairwise comparisons between the messages then an outer bound is derived using a generalization of the straight line bound to errors and erasures decoders and the optimal error exponent tradeoff of a feedback encoder with two messages in addition upper and lower bounds are derived for the optimal erasure exponent of error free block codes in terms of the rate finally a proof is provided for the fact that the optimal tradeoff between error exponents of a two message code does not improve with feedback on discrete memoryless channels dmcs

error exponents of erasure list decoding revisited via moments of distance enumerators the analysis of random coding error exponents pertaining to erasure list decoding due to forney is revisited instead of using jensen s inequality as well as some other inequalities in the derivation we demonstrate that an exponentially tight analysis can be carried out by assessing the relevant moments of certain distance enumerators the resulting bound has the following advantages i it is at least as tight as forney s bound ii under certain symmetry conditions associated with the channel and the random coding distribution it is simpler than forney s bound in the sense that it involves an optimization over one parameter only rather than two and iii in certain special cases like the binary symmetric channel bsc the optimum value of this parameter can be found in closed form and so there is no need to conduct a numerical search we have not found yet a numerical example where this new bound is strictly better than forney s bound and this may provide an additional evidence to support forney s conjecture that his bound is tight for the average code however when applying the proposed analysis technique to a certain universal decoder with erasures we demonstrate that it may yield significantly tighter exponential error bounds we believe that this technique can be useful in simplifying and improving exponential error bounds in other problem settings as well

the error exponent for the noiseless encoding of finite ergodic markov sources a new approach to the classical fixed length noiseless source coding problem is proposed for the case of finite ergodic markov sources this approach is based on simple counting arguments the central notion of markov type a set containing all the source sequences having the same transition counts from letter to letter is introduced and the cardinality of such a set is evaluated via graph theoretical tools the error exponent is shown to be a weighted average of informational divergences and the universal character of the result is stressed as a corollary the classical source coding theorem determining the achievable rates is rederived

universal decoding for arbitrary channels relative to a given class of decoding metrics we consider the problem of universal decoding for arbitrary finite alphabet unknown channels in the random coding regime for a given random coding distribution and a given class of metric decoders we propose a generic universal decoder whose average error probability is within a subexponential multiplicative factor no larger than that of the best decoder within this class of decoders since the optimum maximum likelihood ml decoder of the underlying channel is not necessarily assumed to belong to the given class of decoders this setting suggests a common generalized framework for 1 mismatched decoding 2 universal decoding for a given family of channels and 3 universal coding and decoding for deterministic channels using the individual sequence approach the proof of our universality result is fairly simple and it is demonstrated how some earlier results on universal decoding are obtained as special cases we also demonstrate how our method extends to more complicated scenarios like incorporation of noiseless feedback the multiple access channel and continuous alphabet channels

on universal hypotheses testing via large deviations a prototype problem in hypotheses testing is discussed the problem of deciding whether an i i d sequence of random variables has originated from a known source p sub 1 or an unknown source p sub 2 is considered the exponential rate of decrease in type ii probability of error under a constraint on the minimal rate of decrease in type i probability of error is chosen for a criterion of optimality using large deviations estimates a decision rule that is based on the relative entropy of the empirical measure with respect to p sub 1 is proposed in the case of discrete random variables this approach yields weaker results than the combinatorial approach used by hoeffding 1965 however it enables the analysis to be extended to the general case of r sup n valued random variables finally the results are extended to the case where p sub 1 is an unknown parameter dependent distribution that is known to belong to a set of distributions p sup 0 sub 1 theta in theta

error exponents of optimum decoding for the interference channel exponential error bounds for the finite alphabet interference channel ifc with two transmitter receiver pairs are investigated under the random coding regime our focus is on optimum decoding as opposed to heuristic decoding rules that have been used in previous works like joint typicality decoding decoding based on interference cancellation and decoding that considers the interference as additional noise indeed the fact that the actual interfering signal is a codeword and not an independent and identically distributed i i d noise process complicates the application of conventional techniques to the performance analysis of the optimum decoder using analytical tools rooted in statistical physics we derive a single letter expression for error exponents achievable under optimum decoding and demonstrate strict improvement over error exponents obtainable using suboptimal decoding rules but which are amenable to more conventional analysis

linear codes for sources and source networks error exponents universal coding for slepian wolf source networks the error exponents obtained by korner marton and the author are shown to be universally attainable by linear codes also improved exponents are derived for linear codes with large rates specializing the results to simple discrete memoryless sources reveals their relationship to the random coding and expurgated bounds for channels with additive noise one corollary is that there are universal linear codes for this class of channels which attain the random coding error exponent for each channel in the class the combinatorial approach of csiszar korner marton is used in particular all results are derived from a lemma specifying good encoders in terms of purely combinatorial properties

exact random coding exponents for erasure decoding random coding of channel decoding with an erasure option is studied by analyzing the large deviations behavior of the code ensemble we obtain exact single letter formulas for the error exponents in lieu of forney s lower bounds the analysis technique we use is based on an enhancement and specialization of tools for assessing the statistical properties of certain distance enumerators we specialize our results to the setup of the binary symmetric channel case with uniform random coding distribution and derive an explicit expression for the error exponent which unlike forney s bounds does not involve optimization over two parameters we also establish the fact that for this setup the difference between the exact error exponent corresponding to the probability of undetected decoding error and the exponent corresponding to the erasure event is equal to the threshold parameter numerical calculations indicate that for this setup as well as for a z channel forney s bound coincides with the exact random coding exponent

exact correct decoding exponent of the wiretap channel decoder the performance of the achievability scheme for wyner s wiretap channel model is examined from the perspective of the probability of correct decoding pc at the wiretap channel decoder in particular for finite alphabet memoryless channels the exact random coding exponent of pc is derived as a function of the total coding rate r1 and the rate of each subcode r2 two different representations are given for this function and its basic properties are provided we also characterize the region of pairs of rates r1 r2 of full security in the sense of the random coding exponent of pc in other words the region where the exponent of this achievability scheme is the same as that of blind guessing at the eavesdropper side finally an analogous derivation of the correct decoding exponent is outlined for the case of the gaussian channel

a neyman pearson approach to universal erasure and list decoding when information is to be transmitted over an unknown possibly unreliable channel an erasure option at the decoder is desirable using constant composition random codes we propose a generalization of csiszar and korner s maximum mutual information mmi decoder with an erasure option for discrete memoryless channels the new decoder is parameterized by a weighting function that is designed to optimize the fundamental tradeoff between undetected error and erasure exponents for a compound class of channels the class of weighting functions may be further enlarged to optimize a similar tradeoff for list decodersin that case undetected error probability is replaced with average number of incorrect messages in the list explicit solutions are identified the optimal exponents admit simple expressions in terms of the sphere packing exponent at all rates below capacity for small erasure exponents these expressions coincide with those derived by forney 1968 for symmetric channels using maximum a posteriori decoding thus for those channels at least ignorance of the channel law is inconsequential conditions for optimality of the csiszar korner rule and of the simpler empirical mutual information thresholding rule are identified the error exponents are evaluated numerically for the binary symmetric channel

the generalized random energy model and its application to the statistical physics of ensembles of hierarchical codes in an earlier work the statistical physics associated with finite temperature decoding of code ensembles along with the relation to their random coding error exponents were explored in a framework that is analogous to derrida s random energy model rem of spin glasses according to which the energy levels of the various spin configurations are independent random variables the generalized rem grem extends the rem in that it introduces correlations between energy levels in an hierarchical structure in this paper we explore some analogies between the behavior of the grem and that of code ensembles which have parallel hierarchical structures in particular in analogy to the fact that the grem may have different types of phase transition effects depending on the parameters of the model then the above mentioned hierarchical code ensembles behave substantially differently in the various domains of the design parameters of these codes we make an attempt to explore the insights that can be imported from the statistical mechanics of the grem and be harnessed to serve for code design considerations and guidelines

achievable error exponents for the private fingerprinting game fingerprinting systems in the presence of collusive attacks are analyzed as a game between a fingerprinter and a decoder on the one hand and a coalition of two or more attackers on the other hand the fingerprinter distributes to different users different fingerprinted copies of a host data covertext drawn from a memoryless stationary source embedded with different fingerprints the coalition members create a forgery of the data while aiming at erasing the fingerprints in order not to be detected their action is modeled by a multiple access channel mac we analyze the performance of two classes of decoders associated with different kinds of error events the decoder of the first class aims at detecting the entire coalition whereas the second is satisfied with the detection of at least one member of the coalition both decoders have access to the original covertext data and observe the forgery in order to identify member s of the coalition motivated by a worst case approach we assume that the coalition of attackers is informed of the hiding strategy taken by the fingerprinter and the decoder while they are uninformed of the attacking scheme achievable single letter expressions for the two kinds of error exponents are obtained single letter lower bounds are also derived for the subclass of constant composition codes these lower and the upper bounds coincide for the error exponent of the first class further for the error of the first kind a decoder that is optimal is introduced and the worst case attack channel is characterized

lorenz like systems and classical dynamical equations with memory forcing an alternate point of view for singling out the origin of chaos an alternate view for the emergence of chaos in lorenz like systems is presented in this paper for such purpose the lorenz problem is reformulated in a classical mechanical form and it turns out to be equivalent to the problem of a damped and forced one dimensional motion of a particle in a two well potential with a forcing term depending on the memory of the particle past motion the dynamics of the original lorenz system in the proposed particle phase space can then be rewritten in terms of a one dimensional first exit time problem the emergence of chaos turns out to be due to the discontinuous solutions of the transcendental equation ruling the time for the particle to cross the intermediate potential wall the whole problem is tackled analytically deriving a piecewise linearized lorenz like system that preserves all the essential properties of the original model