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

node and element resequencing using the laplacian of a finite element graph part i general concepts and algorithm a finite element graph feg is defined here as a nodal graph g a dual graph g or a communication graph g associated with a generic finite element mesh the laplacian matrix l g l g or l g used for the study of spectral properties of an feg is constructed from usual vertex and edge connectivities of a graph an automatic algorithm based on spectral properties of an feg g g or g is proposed to reorder the nodes and or elements of the associated finite element mesh the new algorithm is called spectral peg resequencing sfr this algorithm uses global information in the graph it does not depend on a pseudoperipheral vertex in the resequencing process and it does not use any kind of level structure of the graph moreover the sfr algorithm is of special advantage in computing environments with vector and parallel processing capabilities nodes or elements in the mesh can be reordered depending on the use of an adequate graph representation associated with the mesh if g is used then the nodes in the mesh are properly reordered for achieving profile and wavefront reduction of the finite element stiffness matrix if either g or g is used then the elements in the mesh are suitably reordered for a finite element frontai solver a unified approach involving fegs and finite element concepts is presented conclusions are inferred and possible extensions of this research are pointed out in part ii of this work 1 the computational implementation of the sfr algorithm is described and several numerical examples are presented the examples emphasize important theoretical numerical and practical aspects of the new resequencing method

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

an algorithm for profile and wavefront reduction of sparse matrices summary an algorithm for reducing the profile and wavefront of a sparse matrix is described the scheme is applicable to any sparse matrix which has a symmetric pattern of zeros and may be used to generate efficient labellings for finite element grids in particular it is suitable for generating efficient labellings for profile and frontal solution schemes empirical evidence obtained from analysis of the 30 test problems collected by everstine suggests that the new algorithm is superior to existing methods for profile and wavefront reduction it is fast requires only a small amount of memory and is simple to program

detection power estimation efficiency and predictability in event related fmri experimental designs for event related functional magnetic resonance imaging can be characterized by both their detection power a measure of the ability to detect an activation and their estimation efficiency a measure of the ability to estimate the shape of the hemodynamic response randomized designs offer maximum estimation efficiency but poor detection power while block designs offer good detection power at the cost of minimum estimation efficiency periodic single trial designs are poor by both criteria we present here a theoretical model of the relation between estimation efficiency and detection power and show that the observed trade off between efficiency and power is fundamental using the model we explore the properties of semirandom designs that offer intermediate trade offs between efficiency and power these designs can simultaneously achieve the estimation efficiency of randomized designs and the detection power of block designs at the cost of increasing the length of an experiment by less than a factor of 2 experimental designs can also be characterized by their predictability a measure of the ability to circumvent confounds such as habituation and anticipation we examine the relation between detection power estimation efficiency and predictability and show that small increases in predictability can offer significant gains in detection power with only a minor decrease in estimation efficiency

analysis of fmri time series revisited again friston et al 1995 neuroimage 2 45 53 presented a method for detecting activations in fmr1 time series based on the general linear model and a heuristic analysis of the effective degrees of freedom in this communication we present corrected results that replace those of the previous paper and solve the same problem without recourse to heuristic arguments specifically we introduce a proper and unbiased estimator for the error terms and provide a more generally correct expression for the effective degrees of freedom the previous estimates of error variance were biased and in some instances could have led to a 10 20 overestimate of z values although the previous results are almost correct for the random regressors chosen for validation the present theoretical results are exact for any covariate or waveform we comment on some aspects of experimental design and data analysis in the light of the theoretical framework discussed here

the use of profile reduction algorithms with a frontal code we study profile reduction algorithms when used to order the elements for the frontal solution of a system of linear equations with a symmetric sparsity pattern we consider two distinct procedures for producing an efficient element ordering one based on assembling the pattern of the finite element matrix reordering the variables and using the new variable order to resequence the elements and the other based on generating adjacency lists for the elements themselves and reordering the elements directly we compare the results of using several variants of these algorithms in conjunction with the harwell frontal code ma32 on the cray 2 for a range of practical problems we find that given suitable enhancements both approaches are practical and neither is consistently superior to the other

estimating efficiency a priori a comparison of blocked and randomized designs this technical note deals with a priori estimation of efficiency of functional magnetic resonance imaging fmri designs the efficiency of an estimator is a measure of how reliable it is and depends on error variance the variance not modeled by explanatory variables in the design matrix and the design variance a function of the explanatory variables and the contrast tested changes in the experimental design can induce changes in the variance of estimated responses this translates into changes in the standard error of the response estimate or equivalently into changes in efficiency one consequence is that statistics testing for the same activation in different contexts i e experimental designs can change substantially even if the activation and error variance are exactly the same we demonstrate this effect using an event related fmri study of single word reading during blocked and randomized trial presentations furthermore we show that the error variance can change with the experimental design this highlights a problem with a priori comparison of efficiency for two or more experimental designs which usually assumes identical error variance

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

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

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

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

a two step approach to finite element ordering abstract a two step approach to finite element ordering is introduced the scheme involves ordering of the finite elements first based on their adjacency followed by a local numbering of the nodal variables the ordering of the elements is performed by the cuthill mckee algorithm this approach takes into consideration the underlying structure of the finite element mesh and may be regarded as a natural finite element ordering scheme the experimental results show that this two step scheme is more efficient than the reverse cuthill mckee algorithm applied directly to the nodes in terms of both execution time and the number of fill in entries particularly when higher order finite elements are used in addition to its efficiency the two step approach increases modularity and flexibility in finite element programs and possesses potential application to a number of finite element solution methods author

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

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

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

element resequencing for use with a multiple front algorithm the multiple front algorithm is an extension of the frontal method to allow parallelism to be exploited in the solution process the finite element domain is partitioned into a number of subdomains and a frontal decomposition is performed on each subdomain separately for a given partitioning of the domain the efficiency of the multiple front algorithm depends on the ordering of the elements within each subdomain we look at the limitations of existing element reordering algorithms when applied to a subdomain and consider how these limitations may be overcome extensive numerical experiments are performed on a range of practical problems and on the basis of the results we propose a new element resequencing algorithm for use with a multiple front algorithm