codes for iterative decoding from partial geometries this paper develops codes suitable for iterative decoding using the sum product algorithm by considering a large class of combinatorial structures known as partial geometries we are able to define classes of low density parity check ldpc codes which include several previously known families of codes as special cases the existing range of algebraic ldpc codes is limited so the new families of codes obtained by generalizing to partial geometries significantly increase the range of choice of available code lengths and rates we derive bounds on minimum distance rank and girth for all the codes from partial geometries and present constructions and performance results for the classes of partial geometries which have not previously been proposed for use with iterative decoding we show that these new codes can achieve improved error correction performance over randomly constructed ldpc codes and in some cases achieve this with a significant decrease in decoding complexity

codes and designs related to lifted mrd codes lifted maximum rank distance mrd codes which are constant dimension codes are considered it is shown that a lifted mrd code can be represented in such a way that it forms a block design known as a transversal design a slightly different representation of this design makes it similar to a q analog of a transversal design the structure of these designs is used to obtain upper bounds on the sizes of constant dimension codes which contain a lifted mrd code codes that attain these bounds are constructed these codes are the largest known constant dimension codes for the given parameters these transversal designs can also be used to derive a new family of linear codes in the hamming space bounds on the minimum distance and the dimension of such codes are given

reversible low density parity check codes low density parity check ldpc codes may be decoded using a circuit implementation of the sum product algorithm which maps the factor graph of the code by reusing the decoder for encoding both tasks can be performed using the same circuit thus reducing area and verification requirements motivated by this iterative encoding techniques based upon the graphical representation of the code are proposed code design constraints which ensure encoder convergence are presented and then used to design iteratively encodable codes while also preventing 4 cycle creation we show how the jacobi method for iterative matrix inversion can be applied to finite field matrices viewed as message passing and employed as the core of an iterative encoder we present an algebraic construction of 4 cycle free iteratively encodable codes using circulant matrices analysis of these codes identifies a weakness in their structure due to a repetitive pattern in the factor graph the graph supports pseudo codewords of low pseudo weight in order to remove the repetitive pattern in the graph we propose a recursive technique for generating iteratively encodable codes the new codes offer flexibility in the choice of code length and rate and performance that compares well to randomly generated quasi cyclic and extended euclidean geometry codes

a lattice boltzmann equation for diffusion the formulation of lattice gas automata lga for given partial differential equations is not straightforward and still requires some sort of magic lattice boltzmann equation lbe models are much more flexible than lga because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements here a lbe is presented for diffusion in an arbitrary number of dimensions the model is probably the simplest lbe which can be formulated it is shown that the resulting algorithm with relaxation parameter 1 is identical to an explicit finite difference efd formulation at its stability limit underrelaxation 0 1 allows stable integration beyond the stability limit of efd the time step of the explicit lbe integration is limited by accuracy and not by stability requirements

codes on finite geometries new algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented the new construction methods result in a class of multistep majority logic decodable codes and three classes of low density parity check ldpc codes decoding methods for the class of majority logic decodable codes and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their tanner graphs are presented most of the codes constructed can be either put in cyclic or quasi cyclic form and hence their encoding can be implemented with linear shift registers

expander codes using expander graphs we construct a new family of asymptotically good linear error correcting codes these codes have linear time sequential decoding algorithms and logarithmic time parallel decoding algorithms that use a linear number of processors we present both randomized and explicit constructions of these codes experimental results demonstrate the good performance of the randomly chosen codes

theory of the lattice boltzmann method two fluid model for binary mixtures a two fluid lattice boltzmann model for binary mixtures is developed the model is derived formally from kinetic theory by discretizing two fluid boltzmann equations in which mutual collisions and self collisions are treated independently in the resulting lattice boltzmann model viscosity and diffusion coefficients can be varied independently by a suitable choice of mutual and self collision relaxation time scales further the proposed model can simulate miscible and immiscible fluids by changing the sign of the mutual collision term this is in contrast to most existing single fluid lattice boltzmann models that employ a single relaxation time scale and hence are restricted to unity prandtl and schmidt numbers the extension of binary mixing model to multiscalar mixing is quite straightforward

a matrix theoretic approach for analyzing quasi cyclic low density parity check codes a matrix theoretic approach for studying quasi cyclic codes based on matrix transformations via fourier transforms and row and column permutations is developed these transformations put a parity check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field the approach is amicable to the analysis and construction of quasi cyclic low density parity check codes since it takes into account the specific parity check matrix used for decoding with iterative message passing algorithms based on this approach the dimension of the codes and parity check matrices for the dual codes can be determined several algebraic and geometric constructions of quasi cyclic codes are presented as applications along with simulation results showing their performance over additive white gaussian noise channels decoded with iterative message passing algorithms

asymptotic stability for tracking control of nonlinear uncertain dynamical systems described by differential inclusions in this paper we will study the tracking control problem for nonlinear uncertain dynamical systems two generalized feedback control inputs have been proposed such that the feedback controlled systems satisfy the complete tracking control property with exponential asymptotic stability and the trajectories of the systems are steered to the pre specified observation map with an exponential convergence rate moreover an estimate of the tracking time of the trajectories attaining the observation map has also been given an example inspired from a guided missile problem illustrates the use of our main results

new classes of partial geometries and their associated ldpc codes the use of partial geometries to construct parity check matrices for binary low density parity check ldpc codes has resulted in the design of successful codes with a probability of error on the awgn channel close to the shannon capacity at bit error rate down to 10 15 such considerations have motivated this further investigation a new and simple construction of a type of partial geometries with a quasi cyclic qc structure is given and their properties are investigated two new classes of this type of partial geometries one based on prime fields and the other based on cyclic subgroups of prime orders of finite fields are constructed qc ldpc codes with good error performances are constructed based on these two new classes of partial geometries the trapping sets of the partial geometry codes were previously considered using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets this topic is further considered here finally there is a natural relationship between partial geometries and strongly regular graphs the eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the tanner graphs derived from the partial geometries are good expanders for certain parameter sets since it can be argued that codes with good geometric and expansion properties might perform well on the awgn channel under message passing decoding

ldpc block and convolutional codes based on circulant matrices a class of algebraically structured quasi cyclic qc low density parity check ldpc codes and their convolutional counterparts is presented the qc codes are described by sparse parity check matrices comprised of blocks of circulant matrices the sparse parity check representation allows for practical graph based iterative message passing decoding based on the algebraic structure bounds on the girth and minimum distance of the codes are found and several possible encoding techniques are described the performance of the qc ldpc block codes compares favorably with that of randomly constructed ldpc codes for short to moderate block lengths the performance of the ldpc convolutional codes is superior to that of the qc codes on which they are based this performance is the limiting performance obtained by increasing the circulant size of the base qc code finally a continuous decoding procedure for the ldpc convolutional codes is described

enhancing iterative decoding of cyclic ldpc codes using their automorphism groups for cyclic ldpc codes we propose to use their automorphism groups to improve the iterative decoding performance the basic idea is to construct nonequivalent parity check matrices via column permutations three types of iterative decoders are devised to take advantage of the code s automorphism group in this paper we focus on cyclic ldpc codes defined by a circulant parity check matrix and consider two known subgroups of the automorphism group of a cyclic code for the large class of idempotent based cyclic ldpc codes in the literature we show that the two subgroups only provide equivalent parity check matrices and thus cannot be harnessed for iterative decoding towards exploiting the automorphism group of a code we propose a new class of cyclic ldpc codes based on pseudo cyclic mds codes with two information symbols for which nonequivalent parity check matrices are obtained simulation results show that for our constructed codes of short lengths the automorphism group can significantly enhance the iterative decoding performance

momentum transfer correction for macroscopic gradient boundary conditions in lattice boltzmann methods the boundary conditions used to represent macroscopic gradient related effects in arbitrary geometries with the lattice boltzmann methods need a trade off between the complexity of the scheme due to the loss of localness and the difficulties for directly applying link based approaches and the accuracy obtained a generalization of the momentum transfer boundary condition is presented in which the arbitrary location of the boundary is addressed with link wise interpolation used for dirichlet conditions and the macroscopic gradient is taken into account with a finite difference scheme this leads to a stable approach for arbitrary geometries that can be used to impose neumann and robin boundary conditions the proposal is validated for stress boundary conditions at walls two dimensional steady and unsteady configurations are used as test case partial slip flow between two infinite plates and the slip flow past a circular cylinder

pseudo codeword analysis of tanner graphs from projective and euclidean planes we consider coded data transmission over a binary input output symmetric memoryless channel using a binary linear code in order to understand the performance of maximum likelihood ml decoding one studies the codewords in particular the minimal codewords and their hamming weights in the context of linear programming lp decoding one s attention needs to be shifted to the pseudo codewords in particular to the minimal pseudo codewords and their pseudo weights in this paper we investigate some families of codes that have good properties under lp decoding namely certain families of low density parity check ldpc codes that are derived from projective and euclidean planes we study the structure of their minimal pseudo codewords and give lower bounds on their pseudo weight besides this main focus we also present some results that hold for pseudo codewords and minimal pseudo codewords of any tanner graph and we highlight how the importance of minimal pseudo codewords under lp decoding varies depending on which binary input output symmetric memoryless channel is used

asymptotic analysis of lattice boltzmann boundary conditions in this article we use a general method for the analysis of finite difference schemes to investigate lattice boltzmann algorithms for navier stokes problems with dirichlet boundary conditions several link based boundary conditions for commonly used lattice boltzmann bgk models are considered with our method the accuracy of the algorithms can be exactly predicted moreover the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity the analysis is applicable to general geometries and instationary flows

feedback controlled differential inclusions and stabilization of uncertain dynamical systems deterministic design of stabilizing feedback controls for a class of uncertain dynamical systems is considered adopting a problem formulation based on differential inclusions to encompass all possible realizations of uncertainty the authors describe a class of generalized feedback controls with practical analogues in the form of discontinuous feedbacks that guarantees global uniform ultimate boundedness moreover under a matched uncertainty hypothesis this class of controls guarantees global uniform asymptotic stability of the zero state and ultimate attainment of prescribed model behaviour

stability radius optimization a geometric approach abstract the subject of this paper is a geometric approach to the robustness optimization problem for uncertain finite dimensional linear systems for the structurally perturbed stabilizable pair a b we show that optimally robust feedback stabilization is possible via the class of feedbacks parameterizing a b feedback invariant subspaces of codimension rk b a direct consequence of this geometric approach is a reduction in dimension of the optimization problem

minimum distance bounds by graph analysis the parity check matrix of a linear code is used to define a bipartite code constraint tanner graph in which bit nodes are connected to parity check nodes the connectivity properties of this graph are analyzed using both local connectivity and the eigenvalues of the associated adjacency matrix a simple lower bound on the minimum distance of the code is expressed in terms of the two largest eigenvalues for a more powerful bound local properties of the subgraph corresponding to a minimum weight word in the code are used to create an optimization problem whose solution is a lower bound on the code s minimum distance linear programming gives one bound the technique is illustrated by applying it to sparse block codes with parameters 7 3 4 and 42 23 6

ldpc codes from generalized polygons we use the theory of finite classical generalized polygons to derive and study low density parity check ldpc codes the tanner graph of a generalized polygon ldpc code is highly symmetric inherits the diameter size of the parent generalized polygon and has minimum one half diameter to girth ratio we show formally that when the diameter is four or six or eight all codewords have even hamming weight when the generalized polygon has in addition an equal number of points and lines we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length a new minimum distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines finally we prove that all codes derived from finite classical generalized quadrangles are quasi cyclic and we give the explicit size of the circulant blocks in the parity check matrix our simulation studies of several generalized polygon ldpc codes demonstrate powerful bit error rate ber performance when decoding is carried out via low complexity variants of belief propagation

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

output feedback control of uncertain coupled systems the stabilization of uncertain dynamical systems on n that admit a decomposition into two coupled subsystems of dimension nc and nr respectively is studied we refer to the nc dimensional sybsystem as the reduced order system and to the nr dimensional subsystem as the residual system the overall n dimensional system is termed the full system the prototype for which is a mechanical model arising in the study of the active control of structures each subsystem is modelled by a differential equation with a linear nominal part and a nonlinear perturbation of a specified class the output available for feedback purposes is an nc valued linear combination of the state components of the full system and is subject to bounded measurement noise with known bound an output feedback strategy is described and sufficient conditions are obtained for the existence of a calculable global uniform compact attractor containing the state origin for controlled systems of this class

multiple relaxation time lattice boltzmann model for the convection and anisotropic diffusion equation a lattice boltzmann model with a multiple relaxation time mrt collision operator for the convection diffusion equation is presented the model uses seven discrete velocities in three dimensions d3q7 model the off diagonal components of the relaxation time matrix which originate from the rotation of the principal axes enable us to take into account full anisotropy of diffusion an asymptotic analysis of the model equation with boundary rules for the dirichlet and neumann type specified flux conditions is carried out to show that the model is first and second order accurate in time and space respectively the results of the analysis are verified by several numerical examples it is also shown numerically that the error of the mrt model is less sensitive to the variation of the relaxation time coefficients than that of the classical bgk model in addition an alternative treatment for the neumann type boundary condition that improves the accuracy on a curved boundary is presented along with a numerical example of a spherical boundary

regular and irregular progressive edge growth tanner graphs we propose a general method for constructing tanner graphs having a large girth by establishing edges or connections between symbol and check nodes in an edge by edge manner called progressive edge growth peg algorithm lower bounds on the girth of peg tanner graphs and on the minimum distance of the resulting low density parity check ldpc codes are derived in terms of parameters of the graphs simple variations of the peg algorithm can also be applied to generate linear time encodeable ldpc codes regular and irregular ldpc codes using peg tanner graphs and allowing symbol nodes to take values over gf q q 2 are investigated simulation results show that the peg algorithm is a powerful algorithm to generate good short block length ldpc codes

fixed initialization decoding of ldpc codes over a binary symmetric channel we introduce in this paper the concept of a correctable error set and a fixed initialization decoding by noticing that the sum product decoder with a given iteration number only depends on the initialized probability of error for a bsc although this value has been conventionally selected as the bsc crossover probability we show that other selections can provide better performance or faster convergence we also prove that for any fixed initialization i e any given correctable error set the word error rate can be represented as a polynomial of the bsc crossover probability this suggests that the word error rate can be analytically derived from the knowledge of the correctable error set

several properties of short ldpc codes in this paper we present several properties on minimum distance d sub min and girth g sub min in tanner graphs for low density parity check ldpc codes with small left degrees we show that the distance growth of 2 4 ldpc codes is too slow to achieve the desired performance we further give a tight upper bound on the maximum possible girth the numerical results show that codes with large g sub min could outperform the average performance of regular ensembles of the ldpc codes over binary symmetric channels the same codes perform about 1 5 db away from the sphere packing bound on additive white gaussian noise channels

lattice boltzmann model for nonlinear convection diffusion equations a lattice boltzmann model for convection diffusion equation with nonlinear convection and isotropic diffusion terms is proposed through selecting equilibrium distribution function properly the model can be applied to the common real and complex valued nonlinear evolutionary equations such as the nonlinear schrodinger equation complex ginzburg landau equation burgers fisher equation nonlinear heat conduction equation and sine gordon equation by using a real and complex valued distribution function and relaxation time detailed simulations of these equations are performed and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies

advection diffusion lattice boltzmann scheme for hierarchical grids in this paper we describe an extension of a recently developed lattice boltzmann method for solving the advection diffusion equation our proposed approach allows to couple grids of different grid resolutions and includes a staggered timestepping scheme interpolations in space and time and finally a scaling step ensuring the continuity of the desired macroscopic quantities across the grid interface after validating the basic lattice boltzmann method on a uniform grid by a convergence study of analytic problems we demonstrate the consistency of our approach by solving benchmark problems and comparing results on uniform grids and multiply locally refined grids

linear time encodable and decodable error correcting codes we present a new class of asymptotically good linear error correcting codes these codes can be both encoded and decoded in linear time they can also be encoded by logarithmic depth circuits of linear size and decoded by logarithmic depth circuits of size o nlogn we present both randomized and explicit constructions of these codes

viable control for uncertain nonlinear dynamical systems described by differential inclusions in this paper we will study the viable control problem for a class of uncertain nonlinear dynamical systems described by a differential inclusion the goal is to construct a feedback control such that all trajectories of the system are viable in a map moreover for any initial states no viable in the map under the feedback control all solutions of the system are steered to the map with an exponential convergence rate and viable in the map after a finite time t in this case an estimate of the time t of all trajectories attaining the map is given in the nanomedicine system an example inspired from cerebral embolism and cerebral thrombosis problems illustrates the use of our main results

bounds on the expansion properties of tanner graphs this work focuses on the expansion properties of a tanner graph because they are known to be related to the performance of associated iterative message passing algorithms over various channels by analyzing the eigenvalues and corresponding eigenvectors of the normalized incidence matrix representing a tanner graph lower bounds on these expansion properties are derived specifically for the binary erasure channel these results lead to two lower bounds on stopping distance for any given binary linear code and an upper bound on stopping redundancy for the family of difference set codes type i 2 d projective geometry low density parity check ldpc codes

woven graph codes asymptotic performances and examples constructions of woven graph codes based on constituent block and convolutional codes are studied it is shown that within the random ensembles of such codes based on s partite s uniform hypergraphs where s depends only on the code rate there exist codes satisfying the gilbert varshamov gv and the costello lower bound on the minimum distance and the free distance respectively a connection between regular bipartite graphs and tailbiting tb codes is shown some examples of woven graph codes are presented among them an example of a rate rwg 1 3 woven graph code with dfree 32 based on heawood s bipartite graph containing n 7 constituent rate rc 2 3 convolutional codes with overall constraint lengths c 5 is given

resolvable 2 designs for regular low density parity check codes this paper extends the class of low density parity check ldpc codes that can be algebraically constructed we present regular ldpc codes based on resolvable steiner 2 designs which have tanner graphs free of four cycles the resulting codes are 3 spl rho regular or 4 spl rho regular for any value of spl rho and for a flexible choice of code lengths

on the minimum distance of array codes as ldpc codes for a prime q and an integer j spl les q the code c q j is a class of low density parity check ldpc codes from array codes which has a nice algebraic structure in this correspondence we investigate the minimum distance d q j of the code in an algebraic way we first prove that the code is invariant under a doubly transitive group of affine permutations then we show that d 5 4 8 d 7 4 8 and d q 4 spl ges 10 for any prime q 7 in addition we also analyze the codewords of weight 6 in the case of j 3 and the codewords of weight 8 in c 5 4 and c 7 4

an analysis of the orthogonality structures of convolutional codes for iterative decoding the structures of convolutional self orthogonal codes and convolutional self doubly orthogonal codes for both belief propagation and threshold iterative decoding algorithms are analyzed on the basis of difference sets and computation tree it is shown that the double orthogonality property of convolutional self doubly orthogonal codes improves the code structure by maximizing the number of independent observations over two successive decoding iterations while minimizing the number of cycles of lengths 6 and 8 on the code graphs thus the double orthogonality may improve the iterative decoding in both convergence speed and error performance in addition the double orthogonality makes the computation tree rigorously balanced this allows the determination of the best weighing technique so that the error performance of the iterative threshold decoding algorithm approaches that of the iterative belief propagation decoding algorithm but at a substantial reduction of the implementation complexity