analysis of the block error probability of concatenated polar code ensembles in this paper we provide an analysis of the performance of concatenation of polar codes with outer cyclic redundancy check crc codes separated by an interleaver in the short and moderate block length regimes the analysis addresses maximum likelihood decoding as a proxy to the code performance under successive cancellation list decoding the analysis is carried out by introducing the concatenated polar code cpc ensembles whose distance properties can be analyzed for sufficiently short block lengths by means of the uniform interleaver approach at moderate block lengths we resort to the monte carlo simulations results show that if the inner polar code possesses a low minimum distance and the outer crc code has a sufficiently large amount of redundancy then the choice of the outer code generator polynomial and the interleaver may yield to a large variability in the performance of the resulting cpc

minimal area metrics on the swiss cross and punctured torus the closed string theory minimal area problem asks for the conformal metric of least area on a riemann surface with the condition that all non contractible closed curves have length greater than or equal to an arbitrary constant that can be set to one through every point in such a metric there is a geodesic that saturates the length condition and the saturating geodesics in a given homotopy class form a band the extremal metric is unknown when bands of geodesics cross as it happens for surfaces of non zero genus we use recently proposed convex programs to numerically find the minimal area metric on the square torus with a square boundary for various sizes of the boundary for large enough boundary the problem is equivalent to the swiss cross challenge posed by strebel we find that the metric is positively curved in the two band region and flat in the single band regions for small boundary the metric develops a third band of geodesics wrapping around it and has both regions of positive and of negative curvature this surface can be completed to provide the minimal area metric on a once punctured torus representing a closed string tadpole diagram

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

interleaved polar i polar codes by inserting interleavers between intermediate stages of the polar encoder a new class of polar codes termed interleaved polar i polar codes is proposed by the uniform interleaver assumption we derive the weight enumerating function wef and the input output weight enumerating function iowef averaged over the ensemble of i polar codes the average wef can be used to calculate the upper bound on the average block error rate bler of a code selected at random from the ensemble of i polar codes also we propose a concatenated coding scheme that employs p high rate codes as the outer code and q i polar codes as the inner code with an interleaver in between the average wef of the concatenated code is derived based on the uniform interleaver assumption simulation results show that bler upper bounds can well predict bler performance levels of the concatenated codes the results show that the performance of the proposed concatenated code with p q 2 is better than that of the crc aided i polar code with p q 1 of the same length and code rate at high signal to noise ratios snrs moreover the proposed concatenated code allows multiple decoders to operate in parallel which can reduce the decoding latency and hence is suitable for ultra reliable low latency communications urllc

application of wall functions to generalized nonorthogonal curvilinear coordinate systems a method has been developed for the application of wall functions to generalized curvilinear coordinate systems with nonorthogonal grids two test cases have been computed using this method with the k e turbulence model flow over a flat plate at 0 deg angle of attack using a nonorthogonal grid at the wall and flow over a prolate hemispheroid with a hemispherical nose cap at 0 deg angle of attack all results are compared with experimental data in addition the hemispheroid results are compared with computations using the baldwin lomax algebraic turbulence model and the chien low reynolds number k e turbulence model

list decoding of polar codes we describe a successive cancellation list decoder for polar codes which is a generalization of the classic successive cancellation decoder of arikan in the proposed list decoder l decoding paths are considered concurrently at each decoding stage where l is an integer parameter at the end of the decoding process the most likely among the l paths is selected as the single codeword at the decoder output simulations show that the resulting performance is very close to that of maximum likelihood decoding even for moderate values of l alternatively if a genie is allowed to pick the transmitted codeword from the list the results are comparable with the performance of current state of the art ldpc codes we show that such a genie can be easily implemented using simple crc precoding the specific list decoding algorithm that achieves this performance doubles the number of decoding paths for each information bit and then uses a pruning procedure to discard all but the l most likely paths however straightforward implementation of this algorithm requires omega l n 2 time which is in stark contrast with the o n log n complexity of the original successive cancellation decoder in this paper we utilize the structure of polar codes along with certain algorithmic transformations in order to overcome this problem we devise an efficient numerically stable implementation of the proposed list decoder that takes only o l n log n time and o l n space

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

on the spectrum of distances of a multilevel code decoded by a multistage decoder in this correspondence we are concerned with the error rates contributed by component codes in a multilevel code decoded by a multistage decoder these error rates are required for estimating the performance of the structure but more importantly they can serve as a design tool for achieving high coding gain the latter objective can be reached by approaching a balance among these error rates for evaluating the error rate contributed by a component code it is essential to calculate the corresponding spectrum of euclidean distances a method for computing this spectrum is developed in this correspondence the method is general and applicable to any signal constellation and convolutional as well as block component codes the error coefficient available in the literature for some multilevel codes with an infinite size signal constellation is shown to be significantly different from that of a practical finite constellation note that the complete spectrum is essential when codewords of a component code with hamming distance beyond the minimum distance affect the error rate see e g turbo codes

a criterion for flatness in minimal area metrics that define string diagrams it has been proposed that the string diagrams of closed string field theory be defined by a minimal area problem that requires that all nontrivial homotopy curves have length greater than or equal to 2 consistency requires that the minimal area metric be flat in a neighbourhood of the punctures the theorem proven in this paper yieds a criterion which if satisfied will ensure this requirement the theorem states roughly that the metric is flat in an open set u if there is a unique closed curve of length 2 through every point inu and all of these closed curves are in the same free homotopy class

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

weight distribution for closest coset decoding of u u v constructed codes in this correspondence the exact weight distribution for closest coset decoding of u u v constructed codes is derived the results allow more accurate evaluations of the decoding error probabilities

tadpole graph in covariant closed string field theory a one loop tadpole graph in covariant closed string field theory must generate all inequivalent tori with one puncture in order to determine the region of integration in moduli space we map the tadpole to a two sheeted sphere and use the analyticity of the modular parameter with respect to the complex propagator parameter it is shown that the naive closed string extension of the witten open string vertex fails to give the correct modular region for this one loop amplitude

construction of polar codes for arbitrary discrete memoryless channels it is known that polar codes can be efficiently constructed for binary input channels at the same time existing algorithms for general input alphabets are less practical because of high complexity we address the construction problem for the general case and analyze an algorithm that is based on successive reduction of the output alphabet size of the subchannels in each recursion step for this procedure we estimate the approximation error as o mu 1 q 1 where q is the input alphabet size and mu is the quantization parameter i e the maximum size of the subchannel output alphabet allowed by the algorithm the complexity of the code construction scales as o n mu 2 log mu where n is the length of the code we also show that if the polarizing operation relies on modulo q addition it is possible to merge subsets of output symbols without any loss in subchannel capacity performing this procedure before each approximation step results in a further speed up of the code construction and the resulting codes have smaller gap to capacity we also show that a similar acceleration can be attained for polar codes over finite field alphabets experimentation shows that the suggested construction algorithms can be used to construct long polar codes for alphabets of size q 16 and more with acceptable loss of the code rate for a variety of polarizing transforms

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

relaxed polar codes polar codes are the latest breakthrough in coding theory as they are the first family of codes with explicit construction that provably achieve the symmetric capacity of binary input discrete memoryless channels polar encoding and successive cancellation decoding have the complexities of n log n for code length n although the complexity bound of n log n is asymptotically favorable we report in this work methods to further reduce the encoding and decoding complexities of polar coding the crux is to relax the polarization of certain bit channels without performance degradation we consider schemes for relaxing the polarization of both very good and very bad bit channels in the process of channel polarization relaxed polar codes are proved to preserve the capacity achieving property of polar codes analytical bounds on the asymptotic and finite length complexity reduction attainable by relaxed polarization are derived for binary erasure channels we show that the computation complexity can be reduced by a factor of six while preserving the rate and error performance we also show that relaxed polar codes can be decoded with significantly reduced latency for additive white gaussian noise channels with medium code lengths we show that relaxed polar codes can have lower error probabilities than conventional polar codes while having reduced encoding and decoding computation complexities

implicit multiple grid solution of the compressible navier stokes equations using k epsilon turbulence closure in the present work an algorithm for the numerical solution of the two dimensional mass weighted ensemble averaged compressible navier stokes equations is presented turbulence closure for the reynolds stresses is obtained using a low reynolds n umber two equation k s model which includes near wall effects the same algorithm is used for the integration of both the navier stokes and the turbulence transport equations the algorithm consists of a basic explicit finite volume scheme whose convergence is accelerated using local time step ping and multiple grid techniques convergence is further enhanced by the use of an implicit residual smoothing technique applied both in the fine and the coarser grids the application of the numerical scheme to the solution of the k s equations is examined in detail comparisons of computations with experimental data are presented for three shock wave turbulent boundary layer interaction flows algorithm convergence rate and cpu time require ments are discussed he numerical solution of the navier stokes equations for the simulation of turbulent flowfields is a subject of intense research and code development work 1 interest is focused both on code applicability to variegated config urations and on computational rapidity although direct simulation of turbulent flows2 or large eddy numerical sim ulations using subgrid turbulence modeling 3 4 have appeared in the literature current status of algorithm development and computer power indicate that for practical applications the ensemble averaged navier stokes equations should be used in conjunction with transport equations for turbulence closure 5 7 algebraic or mixing length models may provide satisfactory results for relatively simple flow geometries8 but require the determination of the shear layer thickness and of the distance of each grid point from the solid boundary thus they lack both geometric invariance and locality which are highly desirable properties from a computational viewpoint when addressing complex flowfields with many interacting solid boundaries although research on reynolds stre ss transport closure appears quite promising 9 10 its extension to three dimensions involves considerable increase of the number of transport equations to be solved also further research con cerning wall effects is required the more so since the use of wall functions is considered inadequate 11 on the other hand the use of two equation closures such as a 12 14 is quite widespread notwithstanding the fact that the use of the boussinesq hypothesis15 introduces a question able simplification of the flow physics 16 the major drawback of the k s model is probably the near wall formulation which fails to reproduce correctly the effects of the solid boundary on turbulence several variants of the original jones launder model12 have been developed in an attempt to improve upon near wall modeling some ten such models have been thor oughly examined in a critical review article by patel et al 17 who concluded that the models of launder and sharma 14

randomized polar subcodes with optimized error coefficient a method for construction of polar subcodes with reduced error coefficient is presented the proposed approach relies on explicit enumeration of low weight non zero codewords in a polar code and construction of dynamic freezing constraints which define a subcode not containing most of these codewords the obtained codes provide a large performance gain in the high snr region compared to non optimized polar subcodes and polar codes with crc

multilevel codes theoretical concepts and practical design rules this paper deals with 2 sup l ary transmission using multilevel coding mlc and multistage decoding msd the known result that mlc and msd suffice to approach capacity if the rates at each level are appropriately chosen is reviewed using multiuser information theory it is shown that there is a large space of rate combinations such that mlc and full maximum likelihood decoding mld can approach capacity it is noted that multilevel codes designed according to the traditional balanced distance rule tend to fall in the latter category and therefore require the huge complexity of mld the capacity rule the balanced distances rules and two other rules based on the random coding exponent and cutoff rate are compared and contrasted for practical design simulation results using multilevel binary turbo codes show that capacity can in fact be closely approached at high bandwidth efficiencies moreover topics relevant in practical applications such as signal set labeling dimensionality of the constituent constellation and hard decision decoding are emphasized bit interleaved coded modulation proposed by caire et al see ibid vol 44 p 927 46 1998 is reviewed in the context of mlc finally the combination of signal shaping and coding is discussed significant shaping gains are achievable in practice only if these design rules are taken into account

performance limits and practical decoding of interleaved reed solomon polar concatenated codes a scheme for concatenating the recently invented polar codes with non binary mds codes as reed solomon codes is considered by concatenating binary polar codes with interleaved reed solomon codes we prove that the proposed concatenation scheme captures the capacity achieving property of polar codes while having a significantly better error decay rate we show that for any e 0 and total frame length n the parameters of the scheme can be set such that the frame error probability is less than 2 n 1 e while the scheme is still capacity achieving this improves upon 2 n 0 5 e the frame error probability of arikan s polar codes the proposed concatenated polar codes and arikan s polar codes are also compared for transmission over channels with erasure bursts we provide a sufficient condition on the length of erasure burst which guarantees failure of the polar decoder on the other hand it is shown that the parameters of the concatenated polar code can be set in such a way that the capacity achieving properties of polar codes are preserved we also propose decoding algorithms for concatenated polar codes which significantly improve the error rate performance at finite block lengths while preserving the low decoding complexity

multilevel trellis coded modulations for the rayleigh fading channel the authors present coded 8 phase shift keyed 8 psk modulations for the rayleigh fading channel the schemes are based on multilevel trellis coded modulation constructions and utilize maximum free hamming distance binary convolutional codes as building blocks a suboptimal multistage decoder that utilizes interstage interleaving and iterative decoding is proposed and evaluated examples are constructed to show that the proposed schemes outperform the best modified codes of the ungerboeck type due to significantly higher implicit time diversity yielding seven branches of built in time diversity whereas the ungerboeck code yields four branches of time diversity for a 64 state system the transmission delay is higher however the new schemes can provide three levels of unequal error protection when 8 psk or 8 differential phase shift keying 8 dpsk modulations are used they provide 10 14 db channel signal to noise ratio gain over uncoded 4 dpsk at a bit error rate of 10 sup 3 for a modest decoding complexity

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

on trellis codes with a delay processor and a signal mapper it is already known that a trellis code t which is constructed by using the encoder of a convolutional code c with short constraint length followed by a delay processor and a signal mapper is equivalent to a trellis code with large constraint length in this paper we derive a new lower bound on the free distance of t which in some cases is better than the previously derived bound moreover instead of the decoding used in earlier publications we apply iterative decoding on both tailbiting and zero tail representations of t to take advantage of the new lower bound and in the meantime to decrease the associated error coefficient caused by the decoding used in earlier publications comparisons among various designs of such a trellis code and some well known coding methods are also provided

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

shortened polar codes an optimization algorithm for finding a shortening pattern and a set of frozen symbols for polar codes is proposed the structure of polar codes is exploited to eliminate many equivalent shortening patterns thus reducing the search space a reduced complexity suboptimal algorithm is proposed for finding shortening patterns for long polar codes shortened codes obtained with the proposed method are shown to outperform low density parity check ldpc codes

weight distributions for successive cancellation decoding of polar codes in this paper we derive the exact weight distributions that emerge during each stage of successive cancellation decoding of polar codes though we do not compute the distance spectrum of polar codes the results allow us to get an estimate of the decoding error probability and to show a link between the first nonzero components of the weight distribution and the partial order between the synthetic channels also we establish the minimal distance between two cosets associated with two paths that differ in two positions this can be regarded as a first step toward analyzing the weight distributions for successive cancellation list decoding

on distance properties of convolutional polar codes a lower bound on the minimum distance of convolutional polar codes is provided the bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix moreover a construction of convolutional polar subcodes is proposed which provides improved performance under successive cancellation list decoding for sufficiently large list size the decoding complexity of convolutional polar subcodes appears to be lower compared with arikan polar subcodes with the same performance the error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of arikan polar subcodes with the same list size

prediction of curved channel flow with an extended k epsilon model of turbulence a l aide d une approximation algebrique pour les equations de contraintes de reynolds on calcule une expression plus generale pour le coefficient c de l equation de viscosite turbulente qui rend compte simultanement des effets de courbure de ligne de courant et de la deformation due a la pression dans l ecoulement le dernier incluant les effets d amortissement a la paroi

intersection theory on the moduli space of curves and the matrix airy function we show that two natural approaches to quantum gravity coincide this identity is nontrivial and relies on the equivalence of each approach to kdv equations we also investigate related mathematical problems

how to construct polar codes a method for efficiently constructing polar codes is presented and analyzed although polar codes are explicitly defined straightforward construction is intractable since the resulting polar bit channels have an output alphabet that grows exponentially with the code length thus the core problem that needs to be solved is that of faithfully approximating a bit channel with an intractably large alphabet by another channel having a manageable alphabet size we devise two approximation methods which sandwich the original bit channel between a degraded and an upgraded version thereof both approximations can be efficiently computed and turn out to be extremely close in practice we also provide theoretical analysis of our construction algorithms proving that for any fixed e 0 and all sufficiently large code lengths n polar codes whose rate is within e of channel capacity can be constructed in time and space that are both linear in n

how covariant closed string theory solves a minimal area problem for a given genusg riemann surface withn 0 punctures n 3 forg 0 we consider the problem of finding the metric of minimal area under the condition that the length of any nontrivial closed curve be greater or equal to 2 the minimal area metrics are found for the case of all punctured genus zero surfaces and for many of the higher genus surfaces both with and without punctures these metrics are induced by jenkins strebel quadratic differentials they arise from the string diagrams corresponding to restricted feynman graphs of a closed string field theory action containing classical and quantum restricted polyhedra

limitations of the near wall k epsilon turbulence model comparisons are given at two different reynolds numbers between the measured turbulent kinetic energy in channel flow and the predictions of several near wall variants of the k e closure included among these are new calculations based on a form of the e equation that is consistent with the physically correct boundary condition it is observed that all of the approaches fail to account for the large peak k value in the wall region that is evident in the experimental data by contrasting the computed energy budget with its measured values it is shown that this defect may be attributed to a fundamental inconsistency in the commonly used model for the pressure diffu sion term in the k equation near the boundary he k e closure is widely utilized today in the prediction of the mean properties of turbulent flowfields in recent years extensions of its basic form have been developed to permit treatment of complex fluid motions such as those found in curved pipes 1 compressible boundary layers 2 com bustion chambers 3 etc a strong incentive for the continued development and application of the k e model has been the remarkable success it has achieved in the prediction of mean velocity fields in wall bounded shear flows such as occur in channels pipes and boundary layers 4 6 the success of the k e closure in these relatively simple flows is not complete however since the high accuracy obtained for mean velocity is not duplicated in the case of the turbulent kinetic energy k a notable defect4 7 9 is the 25 or more underprediction of the large peak k value shown by clark 10 and kreplin and eckelmann11 to occur at y 15 in a channel flow for ap plications where predictions of turbulence levels are impor tant such as in internal combustion engine simulations 12 this failure can be of great practical significance it may be expected that major errors in the prediction of k stem from deficiencies in the low turbulent reynolds number forms of the k e equations which have been developed for the region adjacent to boundaries jones and launder4 for mulated the first such model that permitted k and e to be calculated down to a solid wall without the use of wall func tions several more recent studies including those of lam and bremhorst 7 hassid and poreh 8 and chien 9 have at tempted to improve upon the jones and launder approach but these authors report only modest gains in the accuracy of k the object of the present study is to elucidate the cause of this discrepancy by making direct comparisons between the terms in the exact and modeled energy equations to ac complish this a set of new calculations was performed in which e was required to satisfy the physically correct bound ary condition at the wall it was revealed that the failure of the k e model to predict the correct peak k value is most likely due to reliance on a flawed model for the pressure velocity correlation in the k equation this relation will be shown to be inconsistent with the experimentally measured values of the exact term as found by kreplin and eckelmann 11 and may be directly implicated in suppressing the peak k value removal of this limitation should allow for better predictions of k near solid boundaries

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

generalized partial orders for polar code bit channels we study partial orders pos for the synthesized bit channels of polar codes first we give an alternative proof of an existing po for bit channels with the same hamming weight and use the underlying idea to extend the bit channel ordering to some additional cases in particular the bit channel ordering for a given code block length is used to generate additional bit channel ordering relationships for larger block lengths generalizing previously known pos next we consider pos especially for the binary erasure channel bec we identify a symmetry property of the bhattacharyya parameters of complementary bit channel pairs on the bec and provide a condition for the alignment of polarized sets of bit channels for the bec and general binary input memoryless symmetric bms channels numerical examples and further properties about the pos for the bit channels with different hamming weights are provided to illustrate the new pos the bit channels with universal ordering positions which are independent of the channel erasure probability are verified for all of the code block lengths finally we show the threshold behavior of the bhattacharyya parameters of some bit channels by approximating the threshold values the corresponding value for a bit channel can be used to determine whether it is good or bad when the underlying channel is known

dynamic scflip decoding of polar codes this paper proposes a generalization of the recently introduced successive cancellation flip scflip decoding of polar codes characterized by a number of extra decoding attempts where one or several positions are flipped from the standard sc decoding to make such an approach effective we first introduce the concept of higher order bit flips and propose a new metric to determine the bit flips that are more likely to correct the trajectory of the sc decoding we then propose a generalized scflip decoding algorithm referred to as dynamic scflip d scflip which dynamically builds a list of candidate bit flips while guaranteeing that the next attempt has the highest probability of success among the remaining ones simulation results show that d scflip is an effective alternative to sc list decoding of polar codes by providing very good error correcting performance with an average computation complexity close to the one of the sc decoder

a proof that witten s open string theory gives a single cover of moduli space we show that witten s open string diagrams are surfaces with metrics of minimal area under the condition that all nontrivial open jordan curves be longer or equal to the minimal area property is used together with a mini max problem to establish a new existence and uniqueness theorem for quadratic differentials in open riemann surfaces with or without punctures on the boundaries this theorem implies that the feynman rules of open string theory give a single cover of the moduli of open riemann surfaces

efficient design and decoding of polar codes polar codes are shown to be instances of both generalized concatenated codes and multilevel codes it is shown that the performance of a polar code can be improved by representing it as a multilevel code and applying the multistage decoding algorithm with maximum likelihood decoding of outer codes additional performance improvement is obtained by replacing polar outer codes with other ones with better error correction performance in some cases this also results in complexity reduction it is shown that gaussian approximation for density evolution enables one to accurately predict the performance of polar codes and concatenated codes based on them

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

perturbative series and the moduli space of riemann surfaces on utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires f g ou f est une surface et g une epine de f

performance bounds of concatenated polar coding schemes a concatenated coding scheme over binary memoryless symmetric bms channels using a polarization transformation followed by outer sub codes is analyzed achievable error exponents and upper bounds on the error rate are derived the first bound is obtained using outer codes which are typical random linear codes as a byproduct of this bound it determines the rates of the outer codes a lower bound on the error exponent that holds for all bms channels with a given capacity is then derived improved bounds and approximations for finite blocklength codes using channel dispersions normal approximation as well as converse and approximate converse results are also obtained the bounds are compared with actual simulation results from the literature for the cases considered when transmitting over the binary input additive white gaussian noise channel there was only a small gap between the channel dispersion based approximation and the actual error rate of concatenated bch polar codes

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

coded phase frequency modulation channel coding combined with expanded signal sets has been shown 1 to improve error performance over uncoded modulation without expanding the bandwidth of the transmitted signals in this paper new coded modulation formats defined over an expanded set of signals varying both in phase and frequency are presented the new schemes combine fsk and psk modulation and make use of trellis coding and viterbi decoding to improve error performance over uncoded modulation the free euclidean distance is calculated for several classes of codes and upper bounds and simulation results are also presented for some simple codes the spectral characteristics of the proposed coded modulation formats are evaluated and compared to conventional two dimensional modulation formats differential encoding and various extensions of the basic scheme are also discussed

soft decision decoding of reed muller codes as generalized multiple concatenated codes constructs reed muller codes by generalized multiple concatenation of binary block codes of length 2 as a consequence of this construction a new decoding procedure is derived that uses soft decision information the algorithm is designed for low decoding complexity and is applicable to all reed muller codes it gives better decoding performance than soft decision bounded distance decoding its decoding complexity is much lower than that of maximum likelihood trellis decoding of reed muller codes especially for long codes

construction of polar codes with sublinear complexity consider the problem of constructing a polar code of block length n for a given transmission channel w previous approaches require one to compute the reliability of the n synthetic channels and then use only those that are sufficiently reliable however we know from two independent works by schurch and by bardet et al that the synthetic channels are partially ordered with respect to degradation hence it is natural to ask whether the partial order can be exploited to reduce the computational burden of the construction problem we show that if we take advantage of the partial order we can construct a polar code by computing the reliability of roughly a fraction 1 log 3 2 n of the synthetic channels in particular we prove that n log 3 2 n is a lower bound on the number of synthetic channels to be considered and such a bound is tight up to a multiplicative factor log log n this set of roughly n log 3 2 n synthetic channels is universal in the sense that it allows one to construct polar codes for any w and it can be identified by solving a maximum matching problem on a bipartite graph our proof technique consists of reducing the construction problem to the problem of computing the maximum cardinality of an antichain for a suitable partially ordered set as such this method is general and it can be used to further improve the complexity of the construction problem in case a refined partial order on the synthetic channels of polar codes is discovered

minimal area problems and quantum open strings we discuss minimal area problems for surfaces with boundaries and both open and closed string punctures we define open closed string diagrams to be surfaces with metrics of minimal area under the condition that any nontrivial jordan open curve be longer or equal to and any nontrivial jordan closed curve be longer or equal to 2 it is proven that the double of an open closed string diagram is a closed string diagram of covariant closed string field theory

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

polar codes characterization of exponent bounds and constructions polar codes were recently introduced by arikan they achieve the symmetric capacity of arbitrary binary input discrete memoryless channels under a low complexity successive cancellation decoding scheme the original polar code construction is closely related to the recursive construction of reed muller codes and is based on the 2 2 matrix 1 0 1 1 it was shown by arikan telatar that this construction achieves an error exponent of 1 2 i e that for sufficiently large blocklengths the error probability decays exponentially in the square root of the blocklength it was already mentioned by arikan that in principle larger matrices can be used to construct polar codes in this paper it is first shown that any l l matrix none of whose column permutations is upper triangular polarizes binary input memoryless channels the exponent of a given square matrix is characterized upper and lower bounds on achievable exponents are given using these bounds it is shown that there are no matrices of size smaller than 15 15 with exponents exceeding 1 2 further a general construction based on bch codes which for large i achieves exponents arbitrarily close to 1 is given at size 16 16 this construction yields an exponent greater than 1 2

open string diagrams i topological type an arbitrary feynman graph for string field theory interactions is analyzed and the homeomorphism type of the corresponding world sheet surface is completely determined even in the nonorientable cases algorithms are found to mechanically compute the topological characteristics of the resulting surface from the structure of the signed oriented graph whitney s permutation theoretic coding of graphs is utilized

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

polar codes bounds on bhattacharyya parameters and their applications this paper proposes a novel lower bound on the bhattacharyya parameter of synthesized channels in polar codes the proposed bound is a result of a newly defined merge operation that reduces the cardinality of output alphabets of a symmetric binary input discrete memoryless channel by repeated applications of the proposed merge operation we establish a universal bound on the bhattacharyya parameters of synthesized channels that holds for any symmetric binary input discrete memoryless transmission channel we then discuss some applications of the new bound for example using the derived universal lower bound along with the existing upper bound we find a partial order among the bhattacharyya parameters of those synthesized channels that cannot be ordered through existing results this ordering can help efficient design of polar codes in addition some extremal behaviors of the binary symmetric channel are also proved

an approach to ungerboeck coding for rectangular signal sets corresp a method of finding good ungerboeck codes for large rectangular quadrature amplitude modulation qam signal sets is described using the concept of euclidean weights due to ungerboeck we prove that a 2 n point basic constellation may be employed to determine exactly the free distance for an ungerboeck coded rectangular 2 m point set when m n 1 bits are uncoded and the remaining bits pass through a rate n 1 n convolutionai encoder it is shown that rate 2 3 encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered

reduced lists of error patterns for maximum likelihood soft decoding a method whereby a substantially reduced family of error patterns called survivors may be created for maximum likelihood soft decoding is introduced the family of survivor depends on the received word a decoder based on this approach first forms the survivors then scores them rather than obtaining the survivors by online elimination of error patterns the use of predetermined lists that represent all families of survivors is proposed

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

convolutional polar kernels a family of polarizing kernels is presented together with polynomial complexity algorithm for computing scaling exponent the proposed convolutional polar kernels are based on convolutional polar codes also known as b mera codes for these kernels a polynomial complexity algorithm is proposed to find weight spectrum of unrecoverable erasure patterns needed for computing scaling exponent as a result we obtain scaling exponent and polarization rate for convolutional polar kernels of size up to 1024