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

weight distributions of non binary multi edge type ldpc code ensembles analysis and efficient evaluation non binary multi edge type ensembles of low density parity check codes are analyzed in terms of non binary codeword weight distribution and its growth rate in particular an exact expression of the growth rate for small weights is developed as a side result the stopping set distributions of these ensembles are developed examples of weight distributions are provided showing that the derived closed form expressions can be easily evaluated the obtained results can thus be exploited to analyze and design non binary low density parity check codes that fall within the multi edge type framework such as but not limited to protograph based codes

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

computing in the jacobian of a hyperelliptic curve in this paper we present algorithms suitable for computer use for computation in the jacobian of a hyperelliptic curve we present a reduction algorithm which is asymptotically faster than that of gauss when the genus g is very large

voronoi s algorithm in purely cubic congruence function fields of unit rank 1 the first part of this paper classifies all purely cubic function fields over a finite field of characteristic not equal to 3 in the remainder we describe a method for computing the fundamental unit and regulator of a purely cubic congruence function field of unit rank 1 and characteristic at least 5 the technique is based on voronoi s algorithm for generating a chain of successive minima in a multiplicative cubic lattice which is used for calculating the fundamental unit and regulator of a purely cubic number field

linear algebra algorithms for divisors on an algebraic curve we use an embedding of the symmetric dth power of any algebraic curve c of genus g into a grassmannian space to give algorithms for working with divisors on c using only linear algebra in vector spaces of dimension o g and matrices of size o g2 o g when the base field k is finite or if c has a rational point over k these give algorithms for working on the jacobian of c that require o g4 field operations arising from the gaussian elimination our point of view is strongly geometric and our representation of points on the jacobian is fairly simple to deal with in particular none of our algorithms involves arithmetic with polynomials we note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in florian hess 1999 ph d thesis which works with function fields as extensions of k x however for special classes of curves hess algorithms are asymptotically more efficient than ours generalizing other known efficient algorithms for special classes of curves such as hyperelliptic curves cantor 1987 superelliptic curves galbraith paulus and smart 2002 and cab curves harasawa and suzuki 2000 in all those cases one can attain a complexity of o g2

weight distribution of low density parity check codes we derive the average weight distribution function and its asymptotic growth rate for low density parity check ldpc code ensembles we show that the growth rate of the minimum distance of ldpc codes depends only on the degree distribution pair it turns out that capacity achieving sequences of standard unstructured ldpc codes under iterative decoding over the binary erasure channel bec known to date have sublinearly growing minimum distance in the block length

stationary measures and invariant subsets of homogeneous spaces iii let g be a real lie group be a lattice in g and g be a compactly generated closed subgroup of g if the zariski closure of the group ad g is semisimple with no compact factor we prove that every g orbit closure in g is a finite volume homogeneous space we also establish related equidistribution properties

non binary protograph based ldpc codes enumerators analysis and designs this paper provides a comprehensive analysis of nonbinary low density parity check ldpc codes built out of protographs we consider both random and constrained edge weight labeling and refer to the former as the unconstrained nonbinary protograph based ldpc codes u nbpb codes and to the latter as the constrained nonbinary protograph based ldpc codes c nbpb codes equipped with combinatorial definitions extended to the nonbinary domain ensemble enumerators of codewords trapping sets stopping sets and pseudocodewords are calculated the exact enumerators are presented in the finite length regime and the corresponding growth rates are calculated in the asymptotic regime an exit chart tool for computing the iterative decoding thresholds of protograph based ldpc codes is presented followed by several examples of finite length u nbpb and c nbpb codes with high performance throughout this paper we provide accompanying examples which demonstrate the advantage of nonbinary protograph based ldpc codes over their binary counterparts and over random constructions the results presented in this paper advance the analytical toolbox of nonbinary graph based codes

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

enumerators for protograph based ensembles of ldpc and generalized ldpc codes protograph based ldpc and generalized ldpc g ldpc codes have the advantages of a simple design procedure and highly structured encoders and decoders the design of such protograph based codes relies on what is effectively a computer based search as such following gallager it is prudent to restrict the search to a good ensemble for example an ensemble whose minimum distance grows linearly with codeword length a good ensemble can also mean one with good stopping set trapping set or pseudocodeword properties in this paper ensemble codeword weight enumerators for finite length ldpc and g ldpc codes based on protographs were derived and then the asymptotic case was considered the asymptotic results allow us to determine whether or not the typical relative minimum distance in the ensemble grows linearly with codeword length then the codeword weight enumerator technique is adapted to yield ensemble stopping set trapping set and pseudocodeword enumerators for protograph ldpc and g ldpc codes in this case the asymptotic results allow us to determine whether or not the typical relative smallest stopping set size trapping set size and pseudoweight grows linearly with codeword length trapping set enumerators for g ldpc code ensembles represent a more complex problem which we do not consider here

computing discrete logarithms in real quadratic congruence function fields of large genus the discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems recently real quadratic congruence function fields were used to construct a public key distribution system the security of this public key system is based on the difficulty of a discrete logarithm problem in these fields in this paper we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus this algorithm is a generalization of similar algorithms for real quadratic number fields

design of regular 2 d sub c ldpc codes over gf q using their binary images in this paper a method to design regular 2 dc ldpc codes over gf q with both good waterfall and error floor properties is presented based on the algebraic properties of their binary image first the algebraic properties of rows of the parity check matrix h associated with a code are characterized and optimized to improve the waterfall then the algebraic properties of cycles and stopping sets associated with the underlying tanner graph are studied and linked to the global binary minimum distance of the code finally simulations are presented to illustrate the excellent performance of the designed codes

construction of cm picard curves in this article we generalize the cm method for elliptic and hyperelliptic curves to picard curves we describe the algorithm in detail and discuss the results of our implementation

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

on decoding of low density parity check codes over the binary erasure channel this paper investigates decoding of low density parity check ldpc codes over the binary erasure channel bec we study the iterative and maximum likelihood ml decoding of ldpc codes on this channel we derive bounds on the ml decoding of ldpc codes on the bec we then present an improved decoding algorithm the proposed algorithm has almost the same complexity as the standard iterative decoding however it has better performance simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm we also provide some graph theoretic properties of different decoding algorithms of ldpc codes over the bec which we think are useful to better understand the ldpc decoding methods in particular for finite length codes

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

uniformly distributed orbits of certain flows on homogeneous spaces is non compact for a co compact lattice this result was strengthened by f irstenberg 1 lj proving that every orbit is uniformly distributed with respect to a g invariant measure for non uniform lattices in sl 2 r using a classification of invariant measures obtained by dani in 21 dani and smillie 31 proved that every non periodic orbit is uniformly distributed there are also various results obtained on orbit closures and invariant measures etc of larger subgroups consisting of unipotent elements especially the horospherical subgroups recently there was a spurt in the area initiated by margulis proof cf 151 see also 7 of oppenheim conjecture on values of quadratic forms at integral points using the study of unipotent flows the reader is referred to the survey articles by dani 41 and margulis 14j for various related developments we now note some conjectures expected to hold for orbits of a unipotent flow namely the u action on

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

computing the minimum distance of nonbinary ldpc codes finding the minimum distance of low density parity check ldpc codes is an np hard problem different from all existing works that focus on binary ldpc codes we in this paper aim to compute the minimum distance of nonbinary ldpc codes motivated by the fact that operating in a large galois field provides one important degree of freedom to achieve both good waterfall and error floor performance our method is based on the existing nearest nonzero codeword search nncs method but several modifications are incorporated for nonbinary ldpc codes including the modified error impulse pattern the dithering method and the nonbinary decoder numerical results on the estimated minimum distances show that a code s minimum distance can be increased by careful selection of nonzero elements of the parity check matrix or by increasing the mean column weight or by increasing the size of the galois field these results support observations that have been made based on simulated performance in the literature finally we provide an upper bound on the minimum distance for nonbinary quasi cyclic ldpc codes

trellis based extended min sum algorithm for non binary ldpc codes and its hardware structure in this paper we present an improvement and a new implementation of a simplified decoding algorithm for non binary low density parity check codes nb ldpc in galois fields gf q the base algorithm that we use is the extended min sum ems algorithm which has been widely studied in the recent literature and has been shown to approach the performance of the belief propagation bp algorithm with limited complexity in our work we propose a new way to compute modified configuration sets using a trellis representation of incoming messages to check nodes we call our modification of the ems algorithm trellis ems t ems in the t ems the algorithm operates directly on the deviation space by considering a trellis built from differential messages which serves as a new reliability measure to sort the configurations we show that this new trellis representation reduces the computational complexity without any performance degradation in addition we show that our modifications of the algorithm allows to greatly reduce the decoding latency by using a larger degree of hardware parallelization

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

quasi cyclic ldpc codes an algebraic construction this paper presents two new large classes of qc ldpc codes one binary and one non binary codes in these two classes are constructed by array dispersions of row distance constrained matrices formed based on additive subgroups of finite fields experimental results show that codes constructed perform very well over the awgn channel with iterative decoding based on belief propagation codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry ldpc codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low complexity reliability based iterative decoding algorithms such as binary message passing decoding algorithms non binary codes decoded with a fast fourier transform based sum product algorithm achieve significantly large coding gains over reed solomon codes of the same lengths and rates decoded with either the hard decision berlekamp massey algorithm or the algebraic soft decision kotter vardy algorithm they have potential to replace reed solomon codes in some communication or storage systems where combinations of random and bursts of errors or erasures occur

efficient maximum likelihood decoding of ldpc codes over the binary erasure channel we propose an efficient maximum likelihood ml decoding algorithm for decoding low density parity check ldpc codes over the binary erasure channel bec we also analyze the computational complexity of the proposed algorithm

convergence of spherical averages for actions of free groups let x v be a probability space and suppose a free group fm with m generators acts on x v by measure preserving transformations let ai am be a set of free generators for fm and let t1 tm x x be transformations corresponding to the generators write t i t 1 for i 1 m and set a m 1 1 m we also have the action fm on l1 x v defined by tg p o tg 1 g e fm consider the set wa of all finite words over the alphabet a

arithmetic on superelliptic curves this paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form yn c x which has only one point at infinity divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given we obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time an algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

the arithmetic of certain cubic function fields in this paper we discuss the properties of curves of the form y3 f x over a given field k of characteristic different from 3 if f x satisfies certain properties then the jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field we seek to make this connection concrete and then use it to develop an explicit arithmetic for the jacobian of such curves from a purely mathematical perspective this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory at the same time it provides another source of groups which are suitable for diffie hellman type protocols in cryptographic applications

spectral shape of doubly generalized ldpc codes efficient and exact evaluation this paper analyzes the asymptotic exponent of the weight spectrum for irregular doubly generalized ldpc d gldpc codes in the process an efficient numerical technique for its evaluation is presented involving the solution of a 4 4 system of polynomial equations the expression is consistent with previous results including the case where the normalized weight or stopping set size tends to zero the spectral shape is shown to admit a particularly simple form in the special case where all variable nodes are repetition codes of the same degree a case which includes tanner codes for this case it is also shown how certain symmetry properties of the local weight distribution at the cns induce a symmetry in the overall weight spectral shape function finally using these new results weight and stopping set size spectral shapes are evaluated for some example generalized and doubly generalized ldpc code ensembles

non binary ldpc erasure codes with separated low degree variable nodes the code design of non binary low density parity check codes for the erasure channel under maximum a posteriori decoding is addressed in particular a partially structured ensemble of codes characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees is proposed the identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade off between waterfall performance error floor and decoding complexity as an example the performance curve of a short 256 128 code on the memoryless 16 ary erasure channel tightly approaches the singleton bound at least down to a codeword error rate of 10 9 at low decoding complexity

random walks on random coset spaces with applications to furstenberg entropy we determine the range of furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces

the arithmetic of jacobian groups of superelliptic cubics we present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity the first algorithm inspired by cantor s reduction for hyperelliptic curves is easily implemented with a few lines of code making use of a polynomial arithmetic package we prove explicit reducedness criteria for superelliptic curves of genus 3 and 4 which show the correctness of the algorithm the second approach quite general in nature and applicable to further classes of curves uses the fglm algorithm for switching between grobner bases for different orderings carrying out the computations symbolically we obtain explicit reduction formulae in terms of the input data

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

low complexity decoding for non binary ldpc codes in high order fields in this paper we propose a new implementation of the extended min sum ems decoder for non binary ldpc codes a particularity of the new algorithm is that it takes into accounts the memory problem of the non binary ldpc decoders together with a significant complexity reduction per decoding iteration the key feature of our decoder is to truncate the vector messages of the decoder to a limited number nm of values in order to reduce the memory requirements using the truncated messages we propose an efficient implementation of the ems decoder which reduces the order of complexity to nm log2 nm this complexity starts to be reasonable enough to compete with binary decoders the performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction which is shown both with a simulated density evolution approach and actual simulations

rigidity of group actions on homogeneous spaces iii consider homogeneous g h and g f for an s algebraic group g a lattice acts on the left strictly conservatively the following rigidity results are obtained morphisms factors and joinings defined apriori only in the measurable category are in fact algebraically constrained arguing in an elementary fashion we manage to classify all the measurable commuting with the action assuming ergodicity we find they are algebraically defined

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16