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

linear systems with locally integrable trajectories abstract the locally integrable function space l 1 loc is made a module over the ring of proper rational functions using this module structure the notion of relative dimension is defined naturally for every linear subspace in l 1 loc q it is shown that the property of having finite relative dimension together with the property of having sufficiently many smooth trajectories and the evident property of differentiation invariance characterize the weak solutions sets of linear constant coefficient differential systems among all linear subspaces

algebraic construction of sparse matrices with large girth in this correspondence we present a method for constructing sparse matrices that have a compact description and whose associated bipartite graphs have large girth based on an arbitrary seed matrix of nonnegative integers a new matrix is constructed which replaces each entry of the seed matrix with a sum of permutation matrices algebraic conditions that lead to short cycles in the associated bipartite graph are analyzed and methods to achieve large girth in two special cases are presented in one all the permutation matrices are circulants in the other they are all affine permutation matrices when used to define a low density parity check ldpc code the compact description should lead to efficient implementation and the large girth to good error correction performance the method is adaptable to a variety of rates and a variety of row and column degrees

thue equations and the method of chabauty coleman let ok be any domain with field of fractions k let f x y ok x y be a homogeneous polynomial of degree n coprime to y and assumed to have unit content i e the coefficients of f generate the unit ideal in ok assume that gcd n char k 1 let h ok and assume that the polynomial hzn f x y is irreducible in k x y z we denote by x f h k the nonsingular complete model of the projective plane curve cf h k defined by the equation hzn f x y 0 we shall assume in this article that g x f h 2 when k is a number field mordell s conjecture now faltings theorem implies that x f h k caporaso harris and mazur chm 1 1 have shown that if lang s conjecture for varieties of general type is true then for any number field k the size x k of the set of k rational points of any curve x k of genus g x 2 can be bounded by a constant depending only on g x prior to the paper chm mazur and others had asked whether x k can be bounded by a constant depending only on g x and the mordell weil rank of x k over k that is the rank of the group j k of k rational points of the jacobian j k of x k these farreaching questions are totally open as we shall recall in sect 1 the method of chabauty coleman sometimes yields a bound for x f h k depending only on g x f h when it is known in advance that the mordell weil rank of x f h k is small unfortunately the chabauty coleman method does not yield a bound for x f h k independent of the coefficients of hzn f x y for all curves of the form x f h it does however produce such a nice bound for the number of primitive integral solutions of f x y h as we now explain let k q and ok z a classical thue equation is an equation f x y h where f x 1 does not have repeated roots thue showed in 1909 that such an equation has finitely many solutions x y z2 if n 3

on the representation of unity by binary cubic forms if f x y is a binary cubic form with integer coefficients such that f x 1 has at least two distinct complex roots then the equation f x y 1 possesses at most ten solutions in integers x and y nine if f has a nontrivial automorphism group if further f x y is reducible over z x y then this equation has at most 2 solutions unless f x y is equivalent under gl2 z action to either x x2 xy y2 or x x2 2y2 the proofs of these results rely upon the method of thue siegel as refined by evertse together with lower bounds for linear forms in logarithms of algebraic numbers and techniques from computational diophantine approximation along the way we completely solve all thue equations f x y 1 for f cubic and irreducible of positive discriminant df 106 as corollaries we obtain bounds for the number of solutions to more general cubic thue equations of the form f x y m and to mordell s equation y2 x3 k where m and k are nonzero integers

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

one parameter families of elliptic curves over ℚ with maximal galois representations let e be an elliptic curve over q and let q e n be its nth division field in 1972 serre showed that if e is without complex multiplication then the galois group of q e n q is as large as possible that is gl2 z nz for all integers n coprime to a constant integer m e q depending at most on e q serre also showed that the best one can hope for is to have gl2 z nz gal q e n q 2 for all positive integers n we study the frequency of this optimal situation in a one parameter family of elliptic curves over q and show that in essence for almost all one parameter families almost all elliptic curves have this optimal behavior

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

binary forms hypergeometric functions and the schwarz christoffel mapping formula in a previous paper it was shown that if f is a binary form with complex coefficients having degree n 3 and discriminant df 0 and if af is the area of the region if x y l 1 in the real affine plane then idlil n n i af 3b l i where b 1 1 denotes the beta function with arguments of 1 3 this inequality was derived by demonstrating that the sequence mn defined by mn maxidfi n n af where the maximum is taken over all forms of degree n with df 0 is decreasing and then by showing that m3 3b q 1 the resulting estimate af 3b 1 for such forms with integer coefficients has had significant consequences for the enumeration of solutions of thue inequalities this paper examines the related problem of determining precise values for the sequence mn by appealing to the theory of hypergeometric functions it is shown that m4 27 6b 2 and that m4 is attained for the form xy x2 y2 it is also shown that there is a correspondence arising from the schwarz christowfel mapping formula between a particular collection of binary forms and the set of equiangular polygons with the property that af is the perimeter of the polygon corresponding to f based on this correspondence and a representation theorem for idfii n n1 af it is conjectured that mn d lln n 1 f n where fn x y x sin kn y cos k i and that the limiting value of the sequence m is 27r

on the structure of behaviors a behavior is a closed shift invariant subspace of the space of sequences with entries in a field k we work out an explicit duality for k modules this duality is then used to derive properties of behaviors and their noncommutative generalizations

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

construction of nonbinary cyclic quasi cyclic and regular ldpc codes a finite geometry approach this paper presents five methods for constructing nonbinary ldpc codes based on finite geometries these methods result in five classes of nonbinary ldpc codes one class of cyclic ldpc codes three classes of quasi cyclic ldpc codes and one class of structured regular ldpc codes experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the awgn channel and they achieve significant coding gains over reed solomon codes of the same lengths and rates with either algebraic hard decision decoding or kotter vardy algebraic soft decision decoding at the expense of a larger decoding computational complexity

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

iterative decoding of one step majority logic deductible codes based on belief propagation previously the belief propagation bp algorithm has received a lot of attention in the coding community mostly due to its near optimum decoding for low density parity check ldpc codes and its connection to turbo decoding in this paper we investigate the performance achieved by the bp algorithm for decoding one step majority logic decodable osmld codes the bp algorithm is expressed in terms of likelihood ratios rather than probabilities as conventionally presented the proposed algorithm fits better the decoding of osmld codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding ldpc codes although it has been believed that osmld codes are far inferior to ldpc codes we show that for medium code lengths say between 200 1000 bits the bp decoding of osmld codes can significantly outperform bp decoding of their equivalent ldpc codes the reasons for this behavior are elaborated

sums of euler products and statistics of elliptic curves we present several results related to statistics for elliptic curves over a finite field mathbb f p as corollaries of a general theorem about averages of euler products that we demonstrate in this general framework we can reprove known results such as the average lang trotter conjecture the average koblitz conjecture and the vertical sato tate conjecture even for very short intervals not accessible by previous methods we also compute statistics for new questions such as the problem of amicable pairs and aliquot cycles first introduced by silverman and stange our technique is rather flexible and should be easily applicable to a wide range of similar problems the starting point of our results is a theorem of gekeler which gives a reinterpretation of deuring s theorem in terms of an euler product involving random matrices thus making a direct connection between the conjectural horizontal distributions and the vertical distributions our main technical result then shows that under certain conditions a weighted average of euler products is asymptotic to the euler product of the average factors

high rate quasi cyclic low density parity check codes derived from finite affine planes this paper shows that several attractive classes of quasi cyclic qc low density parity check ldpc codes can be obtained from affine planes over finite fields one class of these consists of duals of one generator qc codes presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum weight codewords further it is shown that the minimum hamming distance of a code in this class is equal to its minimum additive white gaussian noise awgn pseudoweight also discussed is a class consisting of codes from circulant permutation matrices and an explicit formula for the rank of the parity check matrix is presented for these codes additionally it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2 the construction is similar to the derivation of reed solomon rs based ldpc codes presented by chen and djurdjevic experimental results show that a number of high rate qc ldpc codes with excellent error performance are contained in these classes

on the diophantine equation ax n by n if a b and n are positive integers with b a and n 3 then the equation of the title possesses at most one solution in positive integers x and y with the possible exceptions of a b n satisfying b a 1 2 a min 0 3n 83 and 17 n 347 the proof of this result relies on a variety of diophantine approximation techniques including those of rational approximation to hypergeometric functions the theory of linear forms in logarithms and recent computational methods related to lattice basis reduction additionally we compare and contrast a number of these last mentioned techniques

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

on the method of thue siegel the methods of thue and siegel based on explicit pade approximations to algebraic functions are used to examine diophantine approximations to roots of

power free values of polynomials on symmetric varieties given a symmetric variety y defined over q and a non zero polynomial with integer coefficients we use techniques from homogeneous dynamics to establish conditions under which the polynomial can be made r free for a zariski dense set of integral points on y we also establish an asymptotic counting formula for this set in the special case that y is a quadric hypersurface we give explicit bounds on the size of r by combining the argument with a uniform upper bound for the density of integral points on general affine quadrics defined over q

algebraic construction of quasi cyclic ldpc codes for the awgn and erasure channels this paper is concerned with construction of quasi cyclic qc low density parity check ldpc codes for three different types of channels the additive white gaussian noise the binary random erasure and the binary burst erasure channels two algebraic methods for systematic construction of qc ldpc codes are presented codes constructed perform well over all three types of channels

reed solomon based quasi cyclic ldpc codes designs girth cycle structure and reduction of short cycles designs and constructions of quasi cyclic qc ldpc codes for the awgn channel are presented the codes are constructed based on the conventional parity check matrices of reed solomon rs codes and are referred to as rs qc ldpc codes several classes of rs qc ldpc codes are given cycle structural properties of the tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented the designed codes perform well in both waterfall and low error rate regions

on thue s equation and more precisely for the number of primitive solutions to 1 1 that is solutions in coprime integers x y the first important results on the number of solutions of thue s equation were obtained by c l siegel in the case r 3 and in the binomial case f x y a x r b y r in view of his results and comments we should attribute to siegel the problem to decide whether the number of primitive solutions to 1 1 for irreducible f of degree r 3 could be bounded by a function depending only on r and h but otherwise independent of f in 1983 siegel s question was answered in the affirmative by j h evertse 2 as a special case of his results he also treats equations in number fields evertse obtains the bound

on the representation of integers by binary forms let f be a binary form with integer coefficients non zero discriminant and degree d with d at least 3 let r f z denote the number of integers of absolute value at most z which are represented by f we prove that there is a positive number c f such that r f z is asymptotic to c f z frac 2 d

on s unit equations in two unknowns j h evertse 1 k gy6ry 2 c l stewart 3 and r tijdeman 4 department of pure mathematics centre for mathematics and computer science kruislaan 413 nl 1098 sj amsterdam the netherlands 2 mathematical institute kossuth lajos university h 4010 debrecen hungary 3 department of pure mathematics university of waterloo waterloo ontario canada n2l 3gi 4 mathematical institute university of leiden nl 2333 ca leiden the netherlands

pliable fractional repetition codes for distributed storage systems design and analysis a distributed storage system dss is one of the most vital components of a cloud computing system used for storing and sharing big data among authorized users a typical dss consists of n storage nodes each with a storage capacity of alpha units of data such that the entire file stored on the dss can be recovered by accessing any k nodes the objective of this paper is to construct fractional repetition fr codes with applications in dsss fr codes are the key to constructing a class of distributed storage codes with exact repair by transfer i e upon failure a failed storage node is exactly regenerated through simple downloading a major drawback of existing fr codes is that they are insufficiently flexible to adapt adequately to system changes in dsss to address this problem this paper introduces a new type of fr codes called pliable fr codes in which both the per node storage alpha and repetition degree rho can easily and simultaneously be adjusted in addition this paper presents several constructions of pliable fr codes and provides a relatively comprehensive analysis of the constructed pliable fr codes which reveals the following versatile properties of the codes 1 the transposed codes of the constructed codes are also pliable fr codes 2 the constructed codes attain a tight upper bound on the file size with equality at least for 1 le k where k is the smallest value of k such that k alpha k 1 frac n 1 rho 1 3 the constructed codes also meet a singleton like bound on the minimum distance at least for 1 le k which demonstrates their optimality 4 the computational complexity necessary for determining the file size or the minimum distance of the constructed codes can be greatly reduced when it is hard to exactly determine them and 5 the constructed codes can be used as fractional repetition batch codes to provide load balancing in dsss for which the batch size i e the number of symbols that can be read in parallel can be exactly determined

dyson s lemma with moving parts in its original form d dyson s lemma bounds the order of vanishing of a polynomial in two variables at a finite set of points and was used in studying rational approximations to a fixed algebraic irrational number bombieri b revived interest in dyson s lemma and subsequently esnault and viehweg ev and vojta v1 vastly extended the scope of the result to polynomials in several variables and to a section of a line bundle on a product of two algebraic curves respectively the theorems of vojta and of esnault and viehweg are highly refined geometric results which are used to prove the mordell conjecture and roth s theorem respectively the main theorems of both ev and up to a factor of two v1 were recovered in n as special cases of a more general result before n had been written vojta v3 formulated a result which would vastly improve the main result of n the goal of this note is to prove vojta s statement on a product of an arbitrary number of projective lines and to give a counterexample on a product of three or more curves of genus 1 in order to state our main result we need to fix some notation let

dyson s lemma and a theorem of esnault and viehweg in the classical theory of diophantine approximation as encountered in the work of thue siegel dyson roth and schmidt finiteness results are obtained by constructing an auxilliary polynomial one knows a priori that the polynomial vanishes to high order at certain approximating points and this contradicts an upper bound on the order of vanishing obtained by other techniques the difficult part of the argument is always bounding the order of vanishing from above this technique of diophantine approximation has recently been extended with great success v3 v4 v5 f1 f2 in order to obtain finiteness results for rational points on certain subvarieties of abelian varieties an intermediate stage in this development occurred when esnault and viehweg ev1 proved a generalized version of dyson s lemma strong enough to imply roth s theorem a few years later vojta v1 built upon this work to prove a variant of dyson s lemma valid on a product of curves of arbitrary genus this played an essential role in his proof v3 of the mordell conjecture faltings f1 further extending the work of vojta was able to prove finiteness results for certain higher dimensional subvarieties of abelian varieties dyson s lemma however plays no immediate role in faltings proof instead he uses a geometrical result called the product theorem though dyson s lemma is not used in f1 there is nonetheless a close relationship between the arguments used in section 4 of f1 and those occurring in section 5 of ev1 in particular in both instances the proof is by induction on the dimension of product subvarieties with lemma 2 9 of ev1 replacing corollary 4 3 of f1 and with the main lemma 5 3 in ev1 taking the place of the product theorem in f1 in this note we propose to generalize the main theorems of ev1 and v1 in order to state our results we need to fix some notation let

nonbinary cyclic ldpc codes derived from idempotents and modular golomb rulers based jointly on idempotents and modular golomb rulers we construct a class of nonbinary cyclic low density parity check ldpc codes the defining parity check matrix is a sparse circulant on which we put two constraints 1 the characteristic polynomial is an idempotent 2 the nonzero elements of the first row are located on a modular golomb ruler we show that the second constraint forms a necessary and sufficient condition for the tanner graph to have no cycles of length 4 the minimum distance of the code is proved equal to the column weight of the parity check matrix plus one a search algorithm is presented with which we obtain some high rate codes with large minimum distances the issue of code equivalence is also discussed simulation results show that the obtained codes perform well under iterative decoding

on duality for partial differential and difference equations abstract in this paper we provide a simple version of oberst s duality between finitely generated polynomial modules and the solution sets of partial differential and difference equations

on the number of solutions of polynomial congruences and thue equations has only finitely many solutions in integers x and yv in the first part of this paper we shall establish upper bounds for the number of solutions of 1 in coprime integers x and y under the assumption that the discriminant d f of f is nonzero for most integers h these bounds improve upon those obtained by bombieri and schmidt in 5 in the course of proving these bounds we shall establish a result on polynomial congruences that extends earlier work of nagell 30 ore 32 sandor 33 and huxley 19 in fact we shall establish an upper bound for the number of solutions of a polynomial congruence that is in general best possible in the second part we shall address the problem of finding forms f for which 1 has many solutions for arbitrarily large integers h finally we shall obtain upper bounds for the number of solutions of certain thue mahler and ramanujan nagell equations by appealing to estimates of evertse gy6ry stewart and tijdeman 17 for the number of solutions of s unit equations

quasicyclic low density parity check codes from circulant permutation matrices in this correspondence the construction of low density parity check ldpc codes from circulant permutation matrices is investigated it is shown that such codes cannot have a tanner graph representation with girth larger than 12 and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived these results suggest that families of ldpc codes with such girth values are relatively easy to obtain and consequently additional parameters such as the minimum distance or the number of redundant check sums should be considered to this end a necessary condition for the codes investigated to reach their maximum possible minimum hamming distance is proposed

the number of solutions of decomposable form equations has only finitely many solutions the diophantine approximation techniques of thue and mahler and improvements by siegel dyson roth and bombieri made it possible to derive good explicit upper bounds for the number of solutions of 1 1 the best such upper bound to date due to bombieri 1 is 2 12r bombieri assumed that f is irreducible and r 6 which was not essential in his proof for t 0 i e f x y 1 in x y z bombieri and schmidt 2 derived the upper bound constant r which is best possible in terms of r

brief paper axiomatic characterization of linear differential systems and operators in his ieee trans aut contr paper in 1991 willems posed the following question given a set of smooth trajectories when does there exist a linear constant coefficient differential operator whose kernel is precisely the given set we show that the properties that are necessary and sufficient for a set of smooth trajectories to admit a kernel representation are linearity time invariance jet closedness and jet determinedness it is interesting to note that the properties of jet closedness and jet determinedness were introduced by willems himself in his automatica paper in 1986

on ranks of twists of elliptic curves and power free values of binary forms let e be an elliptic curve over q by the rank of e we shall mean the rank of the group of rational points of e mestre 31 improving on the work of neron 34 cf 13 39 and 46 has shown that there is an infinite family of elliptic curves over q with rank at least 12 however computational work see 3 4 7 and 47 suggests that a typical elliptic curve will have much smaller rank with curves of rank 0 or 1 being predominant indeed brumer 6 has proved subject to the birch swinnerton dyer conjecture the shimura taniyama weil conjecture and the generalized riemann hypothesis that the average rank of an elliptic curve ordered according to its faltings height is at most 2 3 in this article we shall study the behaviour of the rank as we run over twists of a given elliptic curve over q that is we shall restrict our attention to families of elliptic curves defined over q which are isomorphic over c there are families of quadratic cubic quartic and sextic twists see for example proposition 5 4 of chapter x of 42 let e be an elliptic curve over q with weierstrass equation y2 x ax b and for any non zero integer d let ed 2 3 denote a quadratic twist of e given by the equation dy2 x ax b let r d denote the rank of ed note that if d1 and d2 are non zero integers then ed is isomorphic to ed over q if and only if d1 d2 is the square of a rational number subject to the conjectures of birch and swinnerton dyer and of shimura taniyama and weil goldfeld 14 conjectured in 1979 that

quasi cyclic representation and vector representation of rs ldpc codes rs ldpc codes constructed based on the codewords of reed solomon rs codes with two information symbols are an important class of ldpc codes in this paper we present two representations namely quasi cyclic qc representation and vector representation for rs ldpc codes under the first representation most part of the parity check matrix of a full length rs ldpc code consists of circulant permutation matrices and zero matrices as a result the class of codes can enjoy the advantages in hardware implementation as qc ldpc codes in addition the base matrix under the qc representation of an rs ldpc code can be explicitly given such that the rank of its parity check matrix can be analyzed combinatorially under the second representation each permutation matrix in the parity check matrix of an rs ldpc code is defined by a nonbinary vector whose entries are a permutation of entries in the field from which the rs code is constructed then the affine invariance property is proved for full length rs ldpc codes which can facilitate the structural analysis of the codes

representations of integers by systems of three quadratic forms it is classically known that the circle method produces an asymptotic for the number of representations of a tuple of integers n1 nr by a system of quadratic forms q1 qr in k variables as long as k is sufficiently large with respect to r reducing the required number of variables remains a significant open problem in this work we consider the case of three forms and improve on the classical result by reducing the number of required variables to k 10 for almost all tuples under a non singularity assumption on the forms q1 q2 q3 to accomplish this we develop a three dimensional analogue of kloosterman s circle method in particular capitalizing on geometric properties of appropriate systems of three quadratic forms

taylor approximations of multidimensional linear differential systems abstractthe taylor approximations of a multidimensional linear differential system are of importance as they contain a complete information about it it is shown that in order to construct them it is sufficient to truncate the exponential trajectories only a computation of the taylor approximations is provided using purely algebraic means without requiring explicit knowledge of the trajectories

the practical computation of areas associated with binary quartic forms we derive formulas for practically computing the area of the region f z y 1 defined by a binary quartic form f x y r x y these formulas which involve a particular hypergeometric function are useful when estimating the number of lattice points in certain regions of the type f x y h and will likely find application in many contexts we also show that for forms f of arbitrary degree the maximal size of the area of the region f x y 1 normalized with respect to the discriminant of f and taken with respect to the number of conjugate pairs of f x 1 increases as the number of conjugate pairs decreases and we give explicit numerical values for these normalized maxima when f is a quartic form

the number of solutions of norm form equations a norm form is a form f xi xn with rational coefficients which factors into linear forms over c but is irreducible or a power of an irreducible form over q it is known that a nondegenerate norm form equation f xl x m has only finitely many solutions xl x e zn we derive explicit bounds for the number of solutions when f has coefficients in z these bounds depend only on n m and the degree of f but are independent of the size of the coefficients of f

low density parity check codes based on finite geometries a rediscovery and new results this paper presents a geometric approach to the construction of low density parity check ldpc codes four classes of ldpc codes are constructed based on the lines and points of euclidean and projective geometries over finite fields codes of these four classes have good minimum distances and their tanner 1981 graphs have girth 6 finite geometry ldpc codes can be decoded in various ways ranging from low to high decoding complexity and from reasonably good to very good performance they perform very well with iterative decoding furthermore they can be put in either cyclic or quasi cyclic form consequently their encoding can be achieved in linear time and implemented with simple feedback shift registers this advantage is not shared by other ldpc codes in general and is important in practice finite geometry ldpc codes can be extended and shortened in various ways to obtain other good ldpc codes several techniques of extension and shortening are presented long extended finite geometry ldpc codes have been constructed and they achieve a performance only a few tenths of a decibel away from the shannon theoretical limit with iterative decoding

linear systems and arma and fliess models linear dynamical systems are central objects of study in linear system theory and arma and fliess models are two very important classes of models that are used to represent them this article is concerned with the question of what is a relation between them in case of higher dimensions it is shown that the category of linear systems the weak category of arma models and the category of fliess models are equivalent to each other

hierarchical and high girth qc ldpc codes we present an approach to designing capacity approaching high girth low density parity check ldpc codes that are friendly to hardware implementation and compatible with some desired input code structure defined using a protograph the approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi cyclic hqc ldpc codes whereas the parity check matrices of standard quasi cyclic qc ldpc codes are composed of circulant submatrices those of hqc ldpc codes are composed of a hierarchy of circulant submatrices that are in turn constructed from circulant submatrices and so on through some number of levels next we present a girth maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high girth hqc ldpc code subject to bounds imposed by the fact that hqc codes are still quasi cyclic finally we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a squashing procedure that results in a high girth qc ldpc code although not a hierarchical one we illustrate our approach with three design examples of qc ldpc codes two girth 10 codes of rates 1 3 and 0 45 and one girth 8 code of rate 0 7 all of which are obtained from protographs of one sided spatially coupled codes

improved low density parity check codes using irregular graphs we construct new families of error correcting codes based on gallager s 1973 low density parity check codes we improve on gallager s results by introducing irregular parity check matrices and a new rigorous analysis of hard decision decoding of these codes we also provide efficient methods for finding good irregular structures for such decoding algorithms our rigorous analysis based on martingales our methodology for constructing good irregular codes and the demonstration that irregular structure improves performance constitute key points of our contribution we also consider irregular codes under belief propagation we report the results of experiments testing the efficacy of irregular codes on both binary symmetric and gaussian channels for example using belief propagation for rate 1 4 codes on 16000 bits over a binary symmetric channel previous low density parity check codes can correct up to approximately 16 errors while our codes correct over 17 in some cases our results come very close to reported results for turbo codes suggesting that variations of irregular low density parity check codes may be able to match or beat turbo code performance