solving bimatrix games with fuzzy payoffs by introducing nature as a third player the paper elaborates on the fuzzy bimatrix games the payoffs are assumed to be fuzzy intervals with bounded support as defined by dubois and prade a new equilibrium concept is defined for this game by introducing nature as a third player and using a cuts sufficient conditions for its existence and a procedure for its computation are provided

design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

rate compatible puncturing of low density parity check codes in this correspondence we consider puncturing of low density parity check ldpc codes for additive white gaussian noise awgn channels we show that good puncturing patterns exist and that the puncturing can be performed in a rate compatible fashion furthermore rate compatible puncturing results in a small loss of performance with respect to threshold namely the punctured code is good in terms of threshold across a range of rates when compared with the optimal codes for each rate this allows one to implement a single mother encoder and decoder that is good across a wide range of rates

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

on the design of ldpc code ensembles for biawgn channels existing design methods for irregular low density parity check ldpc codes over the additive white gaussian noise channel are based on using asymptotic analysis tools such as density evolution in an optimization process such a process is computationally expensive particularly when a large number of constituent variable node degrees are involved in the design in this paper we propose a systematic approach for the design of irregular ldpc codes the proposed method which is based on a pre computed upper bound on the fraction of edges connected to variable nodes of degree 3 is considerably less complex than the conventional optimization approach through a number of examples we demonstrate that using our method ensembles with performance very close to those devised based on optimization can be designed in addition to having very good performance the number of constituent variable node degrees in the designed ensembles is only three or four this in some cases is much smaller than the corresponding number for optimization based designs with similar performance

hybrid arq in multiple antenna slow fading channels performance limits and optimal linear dispersion code design this paper focuses on studying the fundamental performance limits and linear dispersion code design for the mimo arq slow fading channel the optimal average rate of well known harq protocols is analyzed the optimal design of space time coding for the mimo arq channel is discussed information theoretic measures are used to optimize the rate assignment and derive the optimum design criterion which is then used to evaluate the optimality of existing space time codes a different design criterion which is obtained from the error probability analysis of space time coded mimo harq is presented examples are studied to reveal the gain of arq feedback in space time coded mimo systems

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

on achievable rates and complexity of ldpc codes over parallel channels bounds and applications a variety of communication scenarios can be modeled by a set of parallel channels upper bounds on the achievable rates under maximum likelihood ml decoding and lower bounds on the decoding complexity per iteration of ensembles of low density parity check ldpc codes are presented the communication of these codes is assumed to take place over statistically independent parallel channels where the component channels are memoryless binary input and output symmetric the bounds are applied to ensembles of punctured ldpc codes where the puncturing patterns are either random or possess some structure our discussion is concluded by a diagram showing interconnections between the new theorems and some previously reported results

on ensembles of low density parity check codes asymptotic distance distributions we derive expressions for the average distance distributions in several ensembles of regular low density parity check codes ldpc among these ensembles are the standard one defined by matrices having given column and row sums ensembles defined by matrices with given column sums or given row sums and an ensemble defined by bipartite graphs

reliable channel regions for good binary codes transmitted over parallel channels we study the average error probability performance of binary linear code ensembles when each codeword is divided into j subcodewords with each being transmitted over one of j parallel channels this model is widely accepted for a number of important practical channels and signaling schemes including block fading channels incremental redundancy retransmission schemes and multicarrier communication techniques for frequency selective channels our focus is on ensembles of good codes whose performance in a single channel model is characterized by a threshold behavior e g turbo and low density parity check ldpc codes for a given good code ensemble we investigate reliable channel regions which ensure reliable communications over parallel channels under maximum likelihood ml decoding to construct reliable regions we study a modifed 1961 gallager bound for parallel channels by allowing codeword bits to be randomly assigned to each component channel the average parallel channel gallager bound is simplified to be a function of code weight enumerators and channel assignment rates special cases of this bound average union bhattacharyya ub shulman feder sf simplified sphere ss and modified shulman feder msf parallel channel bounds allow for describing reliable channel regions using simple functions of channel and code spectrum parameters parameters describing the channel are the average parallel channel bhattacharyya noise parameter the average channel mutual information and parallel gaussian channel signal to noise ratios snrs code parameters include the union bhattacharyya noise threshold and the weight spectrum distance to the random binary code ensemble reliable channel regions of repeat accumulate ra codes for parallel binary erasure channels becs and of turbo codes for parallel additive white gaussian noise awgn channels are numerically computed and compared with simulation results based on iterative decoding in addition an examp

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

degree distribution design for ldpc codes a derivative matching approach a deterministic method to design degree distributions for low density parity check codes over the binary erasure channel is proposed this method consists of matching the first and high order derivatives of the extrinsic information transfer exit function of the variable node set to the corresponding derivatives of the inverse exit function of the check node set in order to reduce the gap between the two curves in the exit chart a sufficient condition for a check concentrated distribution to achieve derivative matching up to some order is first obtained and then a deterministic design algorithm enabled by the fourier budan theorem is developed exploiting this sufficient condition a comparison with other deterministic design techniques is also provided revealing the potential of the proposed algorithm

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

existence of equilibrium solution for a non cooperative game with fuzzy goals and parameters in this paper we introduce a concept of equilibrium for a non cooperative game with fuzzy goals involving fuzzy parameters this equilibrium is based on zimmermann s approach for solving linear multiobjective problems with fuzzy goals and the concept of n s equilibrium introduced by zhukovskii for a non cooperative game with payoffs involving unknown parameters in the case of complete ignorance of their behavior we also provide a theorem of existence of this equilibrium based on the fan inequality

tree expectation propagation for ml decoding of ldpc codes over the bec we propose a decoding algorithm for ldpc codes that achieves the maximum likelihood ml solution over the binary erasure channel bec in this channel the tree structured expectation propagation tep decoder improves the peeling decoder pd by processing check nodes of degree one and two however it does not achieve the ml solution as the tree structure of the tep allows only for approximate inference in this paper we provide the procedure to construct the structure needed for exact inference this algorithm denoted as generalized tree structured expectation propagation gtep modifies the code graph by recursively eliminating any check node and merging this information in the remaining graph the gtep decoder upon completion either provides the unique ml solution or a tree graph in which the number of parent nodes indicates the multiplicity of the ml solution we also explain the algorithm as a gaussian elimination method relating the gtep to other ml solutions compared to previous approaches it presents an equivalent complexity it exhibits a simpler graphical message passing procedure and most interesting the algorithm can be generalized to other channels

the design of rate compatible protograph ldpc codes this paper presents a simple yet effective method for designing nested families of ldpc codes rate compatible codes are essential for many communication applications e g hybrid automatic repeat request harq systems and their design is nontrivial due to the difficulty of simultaneously guaranteeing the quality of several related codes puncturing can be used to generate rate compatible ldpc codes but it produces a gap to capacity that in practice often significantly exceeds the gap of the mother code we propose an alternative method based on successively extending a high rate protograph the resulting codes not only inherit the advantages of protograph codes namely low encoding complexity and efficient decoding algorithms but also cover a wide range of rates and have very good performance with iterative decoding thresholds that are within 0 2 db of their capacity limits

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

capacity achieving ldpc codes through puncturing the performance of punctured low definition parity check ldpc codes under maximum likelihood ml decoding is studied in this correspondence via deriving and analyzing their average weight distributions awds and the corresponding asymptotic growth rate of the awds in particular it is proved that capacity achieving codes of any rate and for any memoryless binary input output symmetric mbios channel under ml decoding can be constructed by puncturing some original ldpc code with small enough rate moreover it is shown that the gap to capacity of all the punctured codes can be the same as the original code with a small enough rate conditions under which puncturing results in no rate loss with asymptotically high probability are also given in the process these results show high potential for puncturing to be used in designing capacity achieving codes and in rate compatible coding under any mbios channel

nash equilibrium strategy for fuzzy non cooperative games in this article two person zero sum games are investigated in the fuzzy environment several models constructed by maeda in the symmetrical fuzzy environment are extended to the models in the asymmetric fuzzy environment the existence of equilibrium strategies for these extended models is proposed in the asymmetric fuzzy environment however in some cases nash equilibrium strategies may not exist therefore two special cases are presented for which nash equilibrium strategies do exist in order to investigate the existence of weak pareto nash equilibrium strategies for fuzzy matrix games we introduce the concept of crisp bi matrix games with parameters by solving the parametric bi matrix games we obtain the weak pareto nash equilibrium strategies for the fuzzy matrix games

hausdorff volume in non equiregular sub riemannian manifolds in this paper we study the hausdorff volume in a non equiregular sub riemannian manifold and we compare it with a smooth volume we first give the lebesgue decomposition of the hausdorff volume then we study the regular part show that it is not commensurable with the smooth volume and give conditions under which it is a radon measure we finally give a complete characterization of the singular part we illustrate our results and techniques on numerous examples and cases e g to generic sub riemannian structures

on the renyi divergence joint range of relative entropies and a channel coding theorem this paper starts by considering the minimization of the renyi divergence subject to a constraint on the total variation distance based on the solution of this optimization problem the exact locus of the points bigl d q p 1 d q p 2 bigr is determined when p 1 p 2 and q are arbitrary probability measures which are mutually absolutely continuous and the total variation distance between p 1 and p 2 is not below a given value it is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet this characterization yields a geometric interpretation of the minimal chernoff information subject to a constraint on the variational distance this paper also derives an exponential upper bound on the performance of binary linear block codes or code ensembles under maximum likelihood decoding its derivation relies on the gallager bounding technique and it reproduces the shulman feder bound as a special case the bound is expressed in terms of the renyi divergence from the normalized distance spectrum of the code or the average distance spectrum of the ensemble to the binomially distributed distance spectrum of the capacity achieving ensemble of random block codes this exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes or code ensembles as a function of the deviation of their distance spectra from the binomial distribution an efficient use of this bound is considered

accumulate repeat accumulate codes capacity achieving ensembles of systematic codes for the erasure channel with bounded complexity this paper introduces ensembles of systematic accumulate repeat accumulate ara codes which asymptotically achieve capacity on the binary erasure channel bec with bounded complexity per information bit of encoding and decoding it also introduces symmetry properties which play a central role in the construction of new capacity achieving ensembles for the bec the results here improve on the tradeoff between performance and complexity provided by previous constructions of capacity achieving code ensembles defined on graphs the superiority of ara codes with moderate to large block length is exemplified by computer simulations which compare their performance with those of previously reported capacity achieving ensembles of low density parity check ldpc and irregular repeat accumulate ira codes ara codes also have the advantage of being systematic

parity check density versus performance of binary linear block codes over memoryless symmetric channels we derive lower bounds on the density of parity check matrices of binary linear codes which are used over memoryless binary input output symmetric mbios channels the bounds are expressed in terms of the gap between the rate of these codes for which reliable communications is achievable and the channel capacity they are valid for every sequence of binary linear block codes if there exists a decoding algorithm under which the average bit error probability vanishes for every mbios channel we construct a sequence of ensembles of regular low density parity check ldpc codes so that an upper bound on the asymptotic density of their parity check matrices scales similarly to the lower bound the tightness of the lower bound is demonstrated for the binary erasure channel by analyzing a sequence of ensembles of right regular ldpc codes which was introduced by shokrollahi and which is known to achieve the capacity of this channel under iterative message passing decoding we show that this sequence of ensembles is asymptotically optimal in a sense to be defined in this paper strengthening a result of shokrollahi finally we derive lower bounds on the bit error probability and on the gap to capacity for binary linear block codes which are represented by bipartite graphs and study their performance limitations over mbios channels the latter bounds provide a quantitative measure for the number of cycles of bipartite graphs which represent good error correction codes

a fast approach to compute fuzzy values of matrix games with payoffs of triangular fuzzy numbers the aim of this paper is to develop an effective method for solving matrix games with payoffs of triangular fuzzy numbers tfns which are arbitrary in this method it always assures that players gain floor and loss ceiling have a common tfn type fuzzy value and hereby any matrix game with payoffs of tfns has a tfn type fuzzy value based on duality theorem of linear programming lp and the representation theorem for fuzzy sets the mean and the lower and upper limits of the tfn type fuzzy value are easily computed through solving the derived lp models with data taken from 1 cut set and 0 cut set of fuzzy payoffs hereby the tfn type fuzzy value of any matrix game with payoffs of tfns can be explicitly obtained moreover we can easily compute the upper and lower bounds of any alfa cut set of the tfn type fuzzy value for any matrix game with payoffs of tfns and players optimal mixed strategies through solving the derived lp models at any specified confidence level alfa the proposed method in this paper is demonstrated with a numerical example and compared with other methods to show the validity applicability and superiority

fuzzy linear programming models to solve fuzzy matrix games abstract in this paper a zero sum two person game with imprecise values in its matrix of payoffs is considered we propose a method for its solution based on the establishment of a fuzzy linear programming f l p problem for each player the method is shown as a generalization of that conventionally used in the solution of a classical game to solve such f l p problems we propose the auxiliary models resulting from the application of some of the methods for ranking fuzzy numbers hence according to the kind of ranking method that players want to use different solutions to the former game can be obtained we show that these solutions are of the same nature as the parameters defining the game and corresponding to a fuzzy predicate that can be established as the value of the game is around

new sequences of capacity achieving ldpc code ensembles over the binary erasure channel in this paper new sequences n n of capacity achieving low density parity check ldpc code ensembles over the binary erasure channel bec is introduced these sequences include the existing sequences by shokrollahi et al as a special case for a fixed code rate r in the set of proposed sequences shokrollahi s sequences are superior to the rest of the set in that for any given value of n their threshold is closer to the capacity upper bound 1 r for any given 0 n0 the sequence n n requires a smaller maximum variable node degree as well as a smaller number of constituent variable node degrees to achieve a threshold within neighborhood of the capacity upper bound 1 r moreover it is proven that the check regular subset of the proposed sequences are asymptotically quasi optimal i e their decoding complexity increases only logarithmically with the relative increase of the threshold a stronger result on asymptotic optimality of some of the proposed sequences is also established

comparison of constructions of irregular gallager codes the low density parity check codes whose performance is closest to the shannon limit are gallager codes based on irregular graphs we compare alternative methods for constructing these graphs and present two results first we find a super poisson construction which gives a small improvement in empirical performance over a random construction second whereas gallager codes normally take n sup 2 time to encode we investigate constructions of regular and irregular gallager codes that allow more rapid encoding and have smaller memory requirements in the encoder we find that these fast encoding gallager codes have equally good performance

distance verification for classical and quantum ldpc codes the techniques of distance verification known for general linear codes are first applied to the quantum stabilizer codes then these techniques are considered for classical and quantum stabilizer low density parity check ldpc codes new complexity bounds for distance verification with provable performance are derived using the average weight spectra of the ensembles of ldpc codes these bounds are expressed in terms of the erasure correcting capacity of the corresponding ensemble we also present a new irreducible cluster technique that can be applied to any ldpc code and takes advantage of parity checks sparsity for both the classical and quantum ldpc codes this technique reduces complexity exponents of all existing deterministic techniques designed for generic stabilizer codes with small relative distances which also include all known families of the quantum stabilizer ldpc codes

sub riemannian vs euclidean dimension comparison and fractal geometry on carnot groups we solve gromov s dimension comparison problem for hausdorff and box counting dimension on carnot groups equipped with a carnot caratheodory metric and an adapted euclidean metric the proofs use sharp covering theorems relating optimal mutual coverings of euclidean and carnot caratheodory balls and elements of sub riemannian fractal geometry associated to horizontal self similar iterated function systems on carnot groups inspired by falconer s work on almost sure dimensions of euclidean self affine fractals we show that carnot caratheodory self similar fractals are almost surely horizontal as a consequence we obtain explicit dimension formulae for invariant sets of euclidean iterated function systems of polynomial type jet space carnot groups provide a rich source of examples

characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff in this paper we consider fuzzy bimatrix games which mean bimatrix games with fuzzy payoff matrices we define two types of concepts of equilibrium strategy and investigate their relationships moreover we prove that these nash equilibrium strategies exist in any fuzzy bimatrix games

capacity achieving codes with bounded graphical complexity and maximum likelihood decoding in this paper the existence of capacity achieving codes for memoryless binary input output symmetric mbios channels under maximum likelihood ml decoding with bounded graphical complexity is investigated graphical complexity of a code is defined as the number of edges in the graphical representation of the code per information bit and is proportional to the decoding complexity per information bit per iteration under iterative decoding irregular repeat accumulate ira codes are studied first utilizing the asymptotic average weight distribution aawd of these codes and invoking divsalar s bound on the binary input additive white gaussian noise biawgn channel it is shown that simple nonsystematic ira ensembles outperform systematic ira and regular low density parity check ldpc ensembles with the same graphical complexity and are at most 0 124 db away from the shannon limit however a conclusive result as to whether these nonsystematic ira codes can really achieve capacity cannot be reached motivated by this inconclusive result a new family of codes is proposed called low density parity check and generator matrix ldpc gm codes which are serially concatenated codes with an outer ldpc code and an inner low density generator matrix ldgm code it is shown that these codes can achieve capacity on any mbios channel using ml decoding and also achieve capacity on any bec using belief propagation bp decoding both with bounded graphical complexity moreover it is shown that under certain conditions these capacity achieving codes have linearly increasing minimum distances and achieve the asymptotic gilbert varshamov bound for all rates

on designing good ldpc codes for markov channels this paper presents a reduced complexity approximate density evolution de scheme for low density parity check ldpc codes in channels with memory in the form of a hidden markov chain this approximation is used to design degree sequences representing some of the best known ldpc code ensembles for the gilbert elliott channel and example optimizations are also given for other markov channels the problem of approximating the channel estimation is addressed by obtaining a specially constructed message passing schedule in which the channel messages all approach their stable densities it is shown that this new schedule is much easier to approximate than the standard schedule but has the same ultimate performance in the limits of long block length and many decoding iterations this result is extended to show that all message passing schedules that satisfy mild conditions will have the same threshold under density evolution

on universal properties of capacity approaching ldpc code ensembles this paper is focused on the derivation of some universal properties of capacity approaching low density parity check ldpc code ensembles whose transmission takes place over memoryless binary input output symmetric mbios channels properties of the degree distributions graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information theoretic bounds these bounds are expressed in terms of the target block bit error probability and the gap in rate to capacity most of the bounds are general for any decoding algorithm and some others are proved under belief propagation bp decoding proving these bounds under a certain decoding algorithm validates them automatically also under any suboptimal decoding algorithm a proper modification of these bounds makes them universal for the set of all mbios channels which exhibit a given capacity bounds on the degree distributions and graphical complexity apply to finite length ldpc codes and to the asymptotic case of an infinite block length the bounds are compared with capacity approaching ldpc code ensembles under bp decoding and they are shown to be informative and are easy to calculate finally some interesting open problems are considered

design of rate compatible irregular repeat accumulate codes we consider the design of efficient rate compatible rc irregular repeat accumulate ira codes over a wide code rate range the goal is to provide a family of rc codes to achieve high throughput in hybrid automatic repeat request arq scheme for high speed data packet wireless systems as a subclass of low density parity check codes ira codes have an extremely simple encoder and a low complexity decoder while providing capacity approaching performance we focus on a hybrid design method which employs both puncturing and extending we propose a simple puncturing method based on minimizing the maximal recoverable step of the punctured nodes we also propose a new extending scheme for ira codes by introducing the degree 1 parity bits for the lower rate codes and obtaining the optimal proportions of extended nodes through density evolution analysis the throughput performance of the designed rc ira codes in hybrid arq is evaluated for both awgn and block fading channels simulation results demonstrate that our designed rc codes offer good error correction performance over a wide rate range and provide high throughput especially in the high and low signal to noise ratio regions

inequality relation between fuzzy numbers and its use in fuzzy optimization the inequality relation between two fuzzy numbers is investigated a certain type of such a relation motivated by practical interpretation is proposed and its correspondence with the usual lattice type relation generated by the extended maximum and minimum operators and its possible interpretation are discussed the concept of r l fuzzy number is introduced the class of all r l fuzzy numbers covering practically the whole set of normal convex fuzzy numbers comparing two r l fuzzy numbers of the same type the relation introduced in the paper may be replaced by four ordinary inequalities this fact may be taken advantage of in optimization problems with linear fuzzy constraints

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

bounds on the number of iterations for turbo like ensembles over the binary erasure channel this paper provides simple lower bounds on the number of iterations which is required for successful message passing decoding of some important families of graph based code ensembles including low density parity check ldpc codes and variations of repeat accumulate codes the transmission of the code ensembles is assumed to take place over a binary erasure channel and the bounds refer to the asymptotic case where we let the block length tend to infinity the simplicity of the bounds derived in this paper stems from the fact that they are easily evaluated and are expressed in terms of some basic parameters of the ensemble which include the fraction of degree 2 variable nodes the target bit erasure probability and the gap between the channel capacity and the design rate of the ensemble this paper demonstrates that the number of iterations which is required for successful message passing decoding scales at least like the inverse of the gap in rate to capacity provided that the fraction of degree 2 variable nodes of these turbo like ensembles does not vanish hence the number of iterations becomes unbounded as the gap to capacity vanishes

coding for parallel channels gallager bounds and applications to turbo like codes the transmission of coded communication systems is widely modeled to take place over a set of parallel channels this model is used for transmission over block fading channels rate compatible puncturing of turbo like codes multicarrier signaling multilevel coding etc new upper bounds on the maximum likelihood ml decoding error probability are derived in the parallel channel setting we focus on the generalization of the gallager type bounds and discuss the connections between some versions of these bounds the tightness of these bounds for parallel channels is exemplified for structured ensembles of turbo codes repeat accumulate ra codes and some of their recent variations e g punctured accumulate repeat accumulate codes the bounds on the decoding error probability of an ml decoder are compared to computer simulations of iterative decoding the new bounds show a remarkable improvement over the union bound and some other previously reported bounds for independent parallel channels this improvement is exemplified for relatively short block lengths and it is pronounced when the block length is increased in the asymptotic case where we let the block length tend to infinity inner bounds on the attainable channel regions of modern coding techniques under ml decoding are obtained based solely on the asymptotic growth rates of the average distance spectra of these code ensembles

on characterization of equilibrium strategy of two person zero sum games with fuzzy payoffs in this paper we consider fuzzy matrix games namely two person zero sum games with fuzzy payoffs based on fuzzy max order for such games we define three kinds of concepts of minimax equilibrium strategies and investigate their properties first we shall show that these equilibrium strategies are characterized as nash equilibrium strategies of a family of parametric bi matrix games with crisp payoffs second we investigate properties of values of fuzzy matrix games by means of possibility and necessity measures in addition we give a numerical example to illustrate utility of our approaches

systematic design of low density parity check code ensembles for binary erasure channels we propose a systematic method to design irregular low density parity check ldpc codes for binary erasure channels bec compared to the existing methods which are based on the application of asymptotic analysis tools such as density evolution or extrinsic information transfer exit charts in an optimization process the proposed method is much simpler and faster through a number of examples we demonstrate that the codes designed by the proposed method perform very closely to the best codes designed by optimization an important property of the proposed designs is the flexibility to select the number of constituent variable node degrees p the proposed designs include existing systematic designs as a special case with p n 1 where n is the maximum variable node degree compared to the existing systematic designs for a given rate and a given 0 the designed ensembles can have a threshold in neighborhood of the capacity upper bound with smaller values of p and n they can also achieve the capacity of the bec as n and correspondingly p and the maximum check node degree tend to infinity

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

matrix games with payoffs of belief structures imprecise matrix games such as interval valued matrix games and fuzzy matrix games have attracted much interest for a long time most of the previous studies on imprecise matrix games mainly focus on the fuzzy uncertainty of payoffs however the uncertainties of nonspecificity and discord involved in payoffs are not well addressed so far the purpose of this paper is to study the matrix game with such types of uncertainties in order to achieve that purpose we present a matrix game model with payoffs of belief structures so as to integrate discord and nonspecificity the proposed model can be used to express more imprecise interactions between players in the reality besides as another main contribution of the study an effective method for solving matrix games with belief structures payoffs is developed to help us find the equilibrium points of the kind of games at last an example is given to illustrate the proposed model and method

on convergence speed of capacity achieving sequences for erasure channel oswald and shokrollahi carried out a systematic study of capacity achieving sequences of low density parity check ldpc codes for the binary erasure channel two quantities mu and delta were suggested to measure the convergence speed of these sequences they gave an upper bound of mu for a certain type of capacity achieving sequences in this correspondence a lower bound of mu for this type of sequences is provided it follows that under a certain condition the exact value of mu can be determined which answers an open question posed by oswald and shokrollahi in their paper if the condition is not satisfied the upper bound may not be tight a counterexample is given

a more accurate one dimensional analysis and design of irregular ldpc codes we introduce a new one dimensional 1 d analysis of low density parity check ldpc codes on additive white gaussian noise channels which is significantly more accurate than similar 1 d methods our method assumes a gaussian distribution in message passing decoding only for messages from variable nodes to check nodes compared to existing work which makes a gaussian assumption both for messages from check nodes and from variable nodes our method offers a significantly more accurate estimate of convergence behavior and threshold of convergence similar to previous work the problem of designing irregular ldpc codes reduces to a linear programming problem however our method allows irregular code design in a wider range of rates without any limit on the maximum variable node degree we use our method to design irregular ldpc codes with rates greater than 1 4 that perform within a few hundredths of a decibel from the shannon limit the designed codes perform almost as well as codes designed by density evolution

on the design of fast convergent ldpc codes for the bec an optimization approach the complexity performance trade off is a fundamental aspect of the design of low density parity check ldpc codes in this paper we consider ldpc codes for the binary erasure channel bec use code rate for performance metric and number of decoding iterations to achieve a certain residual erasure probability for complexity metric we first propose a quite accurate approximation of the number of iterations for the bec moreover a simple but efficient utility function corresponding to the number of iterations is developed using the aforementioned approximation and the utility function two optimization problems w r t complexity are formulated to find the code degree distributions we show that both optimization problems are convex in particular the problem with the proposed approximation belongs to the class of semi infinite problems which are computationally challenging to be solved however the problem with the proposed utility function falls into the class of semi definite programming sdp and thus the global solution can be found efficiently using available sdp solvers numerical results reveal the superiority of the proposed code design compared to existing code designs from literature

pauls rectifiable and purely pauls unrectifiable smooth hypersurfaces abstract this paper is related to the problem of finding a good notion of rectifiability in sub riemannian geometry in particular we study which kind of results can be expected for smooth hypersurfaces in carnot groups our main contribution will be a consequence of the following result there exists a c hypersurface s without characteristic points that has uncountably many pairwise non isomorphic tangent groups on every positive measure subset the example is found in a carnot group of topological dimension 8 it has hausdorff dimension 12 and so we use on it the hausdorff measure h 12 as a consequence we show that any lipschitz map defined on a subset of a carnot group of hausdorff dimension 12 with values in s has negligible image with respect to the hausdorff measure h 12 in particular we deduce that s cannot be lipschitz parametrizable by countably many maps each defined on some subset of some carnot group of hausdorff dimension 12 as main consequence we have that a notion of rectifiability proposed by s pauls is not equivalent to one proposed by b franchi r serapioni and f serra cassano at least for arbitrary carnot groups in addition we show that given a subset u of a homogeneous subgroup of hausdorff dimension 12 of a carnot group every bi lipschitz map f u s satisfies h 12 f u 0 finally we prove that such an example does not exist in heisenberg groups we prove that all c hypersurfaces in h n with n 2 are countably h n 1 r rectifiable according to pauls definition even with bi lipschitz maps

performance of low density parity check codes with linear minimum distance this correspondence studies the performance of the iterative decoding of low density parity check ldpc code ensembles that have linear typical minimum distance and stopping set size we first obtain a lower bound on the achievable rates of these ensembles over memoryless binary input output symmetric channels we improve this bound for the binary erasure channel we also introduce a method to construct the codes meeting the lower bound for the binary erasure channel then we give upper bounds on the rate of ldpc codes with linear minimum distance when their right degree distribution is fixed we compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble

successive maximization for systematic design of universally capacity approaching rate compatible sequences of ldpc code ensembles over binary input output symmetric memoryless channels a systematic construction of capacity achieving low density parity check ldpc code ensemble sequences over the binary erasure channel bec has been proposed by saeedi et al based on a method here referred to as successive maximization sm in sm the fraction of degree i nodes are successively maximized starting from i 2 with the constraint that the ensemble remains convergent over the channel in this paper we propose sm to design universally capacity approaching rate compatible ldpc code ensemble sequences over the general class of binary input output symmetric memoryless biosm channels this is achieved by first generalizing the sm method to other biosm channels to design a sequence of capacity approaching ensembles called the parent sequence the sm principle is then applied to each ensemble within the parent sequence this time to design rate compatible puncturing schemes as part of our results we extend the stability condition which was previously derived for degree 2 variable nodes to other variable node degrees as well as to the case of rate compatible codes consequently we prove that using the sm principle one is able to design universally capacity achieving rate compatible ldpc code ensemble sequences over the bec unlike the previous results in the literature the proposed sm approach is naturally extendable to other biosm channels the performance of the rate compatible schemes designed based on our method is comparable to those designed by optimization

non cooperative fuzzy games in normal form a survey in this paper we present a review of most of the approaches for solving non cooperative fuzzy games in normal form applications of these games are reviewed as well we also present a new class of fuzzy games in normal form the class of non cooperative games involving fuzzy parameters finally some potential directions of research are provided throughout the paper and in the conclusion

cooperative fuzzy games we derive versions of known results dealing with core equilibria and shapley values of cooperative games in the case of cooperative fuzzy games i e games defined on fuzzy subsets of the set of n players a fuzzy coalition is an n vector i associating with each player i his rate of participation i 0 1 in the fuzzy coalition and the real number v is the coalition worth assumed to be positively homogeneous if also v is superadditive or equivalently concave the fuzzy game with side payments has a nonempty core associated with the coalition worth function for any ordinary game with side payments is a fuzzy extension thereof viz the fuzzy game whose coalition worth function is the least positively homogeneous superadditive function majorizing the coalition worth function of the original game the fuzzy extension always has a nonempty core moreover if the original game has a nonempty core it coincides with that of its fuzzy extension analogous results are established for games without side payments an axiomatization of values of fuzzy games with side payments is also given the results are applied to show that the set of walras equilibria coincides with the fuzzy core of an economy

robust rate compatible punctured ldpc convolutional codes a family of robust rate compatible rc punctured low density parity check convolutional codes ldpc ccs is derived from a time invariant ldpc cc mother code by periodically puncturing encoded bits variable nodes with respect to several criteria 1 ensuring the recoverability of punctured variable nodes 2 minimizing the number of completely punctured cycle trapping sets cpctss and 3 minimizing the number of punctured variable nodes involved in short cycles the influence of 1 and 3 on iterative decoding performance is felt most strongly in the waterfall region of the bit error rate ber curve while 2 has a larger effect in the error floor or high signal to noise ratio snr region we show that the length of the puncturing period is an important parameter when designing high rate punctured codes and moreover that extending the puncturing period can improve the decoding performance and extend the range of compatible rates as examples we obtain families of rc ldpc ccs from several time invariant ldpc cc mother codes with monomial and binomial entries in their polynomial syndrome former matrices

fuzzy nash equilibriums in crisp and fuzzy games in this paper we introduce fuzzy nash equilibrium to determine a graded representation of nash equilibriums in crisp and fuzzy games this interpretation shows the distribution of equilibriums in the matrix form of a game and handles uncertainties in payoffs in addition a new method to rank fuzzy values with the user s viewpoint is investigated by this means the definition of satisfaction function which provides the result of comparison in the form of real value is developed when users have preferences regarding the payoffs

pareto optimal security strategies in matrix games with fuzzy payoffs we present a new methodology for the analysis of fuzzy payoff matrix games the main difficulty that appears in the study of these games is the comparison between the payoff values associated to the strategies of the players because these payoffs are fuzzy quantities our approach does not transform the fuzzy payoffs to crisp numbers via standard defuzzyfication but we use standard fuzzy orders which allows us to find solutions within the same space of fuzzy numbers moreover we provide a method to solve these games finding equivalent fuzzy linear programs whose maximal solutions give the solutions of the games

nonuniform error correction using low density parity check codes this correspondence introduces a framework to design and analyze low density parity check ldpc codes over nonuniform channels we study ldpc codes for channels with nonuniform noise distributions rate adaptive coding and unequal error protection first we propose a technique to design ldpc codes for volume holographic memory vhm systems for which the noise distribution is nonuniform we show that the proposed coding scheme has an easy design procedure and results in efficient codes for holographic memories an important property of the proposed technique is the design of the codes that have a low error floor and low variable node degrees while maintaining performance close to the shannon limit we then show that punctured ldpc codes can be studied as a special case of our design methodology for nonuniform channels finally we propose a method to generate ldpc codes that can provide unequal error protection in addition to having a good overall performance moreover the highly protected bits can be decoded without requiring the entire word to be decoded

rate compatible punctured low density parity check codes with short block lengths we consider the problem of rate compatible puncturing of low density parity check ldpc codes over additive white gaussian noise awgn channels in previous work it was shown that good puncturing distributions exist for ldpc codes but the code length was large in this correspondence we give a procedure for determining the puncturing distributions for ldpc codes with short block lengths a few thousand bits and show that careful puncturing can produce good performance we compare the performance of the rate compatible punctured ldpc codes with dedicated ldpc codes across a range of rates and see that the rate compatible codes have a favorable complexity performance tradeoff

uep constructions of quasi cyclic low density parity check codes via masking in this paper the algebraic constructions of quasi cyclic low density parity check qc ldpc codes with the unequal error protection uep property are considered a criterion for constructing such codes via the masking technique is proposed based on which explicit conditions on the base parity check matrices and masking matrices to achieve uep are provided we also give three specific constructions of uep qc ldpc codes furthermore a sufficient condition to ensure strict uep is presented simulation results demonstrate the superiority of our constructed codes over time sharing schemes the constructed codes also have competitive error performance against randomly constructed equal error protection ldpc codes and irregular uep ldpc codes designed based on the degree distribution

finite length rate compatible ldpc codes a novel puncturing scheme transactions letters in this paper we study rate compatible puncturing of finite length low density parity check ldpc codes we present a novel rate compatible puncturing scheme that is easy to implement our scheme uses the idea that the degradation in performance is reduced by selecting a puncturing pattern wherein the punctured bits are far apart from each other in the tanner graph of the code although the puncturing scheme presented is tailored to regular codes it is also directly applicable to irregular parent ensembles by simulations the proposed rate compatible puncturing scheme is shown to be superior to the existing puncturing methods for both regular and irregular ldpc codes over the binary erasure channel bec and the additive white gaussian noise awgn channel

feasibility and pareto optimality for multiobjective nonlinear programming problems with fuzzy parameters abstract in this paper we focus on multiobjective nonlinear programming problems with fuzzy parameters and extend the ordinary feasibility and pareto optimality concepts based on the concepts of possibility and necessity for fuzzy numbers using the four indices for ranking two fuzzy numbers four types of feasibility and pareto optimality are defined and the relationships among them are examined in detail these concepts can be viewed as quite generalized versions of the well known feasibility and pareto optimality concepts and the generalized pareto optimal solutions for the multiobjective nonlinear programming problems with fuzzy parameters may be obtained on the basis of the method of nonlinear programming

two person zero sum game approach for fuzzy multiple attribute decision making problems the multiple attribute decision making madm problem deals with candidate priority alternatives with respect to various attributes madm techniques are popularly used in diverse fields such as engineering economics management science transportation planning etc several approaches have been developed for assessing the weights of madm problems e g the eigenvector method electre topsis etc in this paper we obtained weights of a madm problem with a fuzzy decision matrix by formulating it as a two person zero sum game with an uncertain payoff matrix moreover the equilibrium solution and the resolution method for the madm game have also been developed these results are validated by a product development example of nano materials our study is an innovative approach for applying the game theory on madm problems with management of technology mot

design of efficiently encodable moderate length high rate irregular ldpc codes this paper presents a new class of irregular low density parity check ldpc codes of moderate length 10 sup 3 spl les n spl les 10 sup 4 and high rate r spl ges 3 4 codes in this class admit low complexity encoding and have lower error rate floors than other irregular ldpc code design approaches it is also shown that this class of ldpc codes is equivalent to a class of systematic serial turbo codes and is an extension of irregular repeat accumulate codes a code design algorithm based on the combination of density evolution and differential evolution optimization with a modified cost function is presented moderate length high rate codes with no error rate floors down to a bit error rate of 10 sup 9 are presented although our focus is on moderate length high rate codes the proposed coding scheme is applicable to irregular ldpc codes with other lengths and rates

rate compatible ldpc convolutional codes achieving the capacity of the bec in this paper we propose a new family of rate compatible regular low density parity check ldpc convolutional codes the construction is based on graph extension i e the codes of lower rates are generated by successively extending the graph of the base code with the highest rate theoretically the proposed rate compatible family can cover all the rational rates from 0 to 1 in addition the regularity of degree distributions simplifies the code optimization we prove analytically that all the ldpc convolutional codes of different rates in the family are capable of achieving the capacity of the binary erasure channel bec the analysis is extended to the general binary memoryless symmetric channel for which a capacity approaching performance can be achieved analytical thresholds and simulation results for finite check and variable node degrees are provided for both becs and binary input additive white gaussian noise channels the results confirm that the decoding thresholds of the rate compatible codes approach the corresponding shannon limits over both channels

ranking fuzzy numbers in the setting of possibility theory the arithmetic manipulation of fuzzy numbers or fuzzy intervals is now well understood equally important for application purposes is the problem of ranking fuzzy numbers or fuzzy intervals which is addressed in this paper a complete set of comparison indices is proposed in the framework of zadeh s possibility theory it is shown that generally four indices enable one to completely describe the respective locations of two fuzzy numbers moreover this approach is related to previous ones and its possible extension to the ranking of n fuzzy numbers is discussed at length finally it is shown that all the information necessary and sufficient to characterize the respective locations of two fuzzy numbers can be recovered from the knowledge of their mutual compatibilities

random coding techniques for nonrandom codes this work provides techniques to apply the channel coding theorem and the resulting error exponent which was originally derived for totally random block code ensembles to ensembles of codes with less restrictive randomness demands as an example the random coding technique can even be applied for an ensemble that contains a single code for a specific linear code we get an upper bound for the error probability which equals gallager s 1968 random coding bound up to a factor determined by the maximum ratio between the weight distribution of the code and the expected random weight distribution

capacity achieving ensembles for the binary erasure channel with bounded complexity we present two sequences of ensembles of nonsystematic irregular repeat accumulate ira codes which asymptotically as their block length tends to infinity achieve capacity on the binary erasure channel bec with bounded complexity per information bit this is in contrast to all previous constructions of capacity achieving sequences of ensembles whose complexity grows at least like the log of the inverse of the gap in rate to capacity the new bounded complexity result is achieved by puncturing bits and allowing in this way a sufficient number of state nodes in the tanner graph representing the codes we derive an information theoretic lower bound on the decoding complexity of randomly punctured codes on graphs the bound holds for every memoryless binary input output symmetric mbios channel and is refined for the binary erasure channel

results on punctured low density parity check codes and improved iterative decoding techniques this paper first introduces an improved decoding algorithm for low density parity check ldpc codes over binary input output symmetric memoryless channels then some fundamental properties of punctured ldpc codes are presented it is proved that for any ensemble of ldpc codes there exists a puncturing threshold it is then proved that for any rates r1 and r2 satisfying 0 r1 r 2 1 there exists an ensemble of ldpc codes with the following property the ensemble can be punctured from rate r1 to r2 resulting in asymptotically good codes for all rates r1lesrlesr2 specifically this implies that rates arbitrarily close to one are achievable via puncturing bounds on the performance of punctured ldpc codes are also presented it is also shown that punctured ldpc codes are as good as ordinary ldpc codes for bec and arbitrary positive numbers r1 r2 1 the existence of the sequences of punctured ldpc codes that are capacity achieving for all rates r1 lesrlesr2 is shown based on the above observations a method is proposed to design good punctured ldpc codes over a broad range of rates finally it is shown that the results of this paper may be used for the proof of the existence of the capacity achieving ldpc codes over binary input output symmetric memoryless channels