on mds codes via cauchy matrices a special form of cauchy matrix is used to obtain a tighter bound for the validity region of the maximum distance separable mds conjecture and a new compact characterization of generalized reed solomon codes the latter is further used to obtain constructions and some existence results for long 2k k double circulant mds codes

irreducibility criterion for germs of analytic functions of two complex variables let s x y be the germ of an analytic function of two complex variables near the origin to decide whether f is irreducible or not we can apply a succession of blowing ups to desingularize f and see if the proper transform off ever bifurcates and then use the fact that f is irreducible iff such a bifurcation never takes place alternatively we can use newton s theorem also called puiseux s theorem to completely factor s into linear factors in y by allowing fractional power series in x and then collate these factors into conjugacy classes and now use the fact that fis irreducible iff there is only one conjugacy class in a recent conversation t c kuo of the university of sydney in australia asked me the following question

systematic encoding via grobner bases for a class of algebraic geometric goppa codes any linear code with a nontrivial automorphism has the structure of a module over a polynomial ring the theory of grobner bases for modules gives a compact description and implementation of a systematic encoder we present examples of algebraic geometric goppa codes that can be encoded by these methods including the one point hermitian codes

maximum distance separable 2d convolutional codes maximum distance separable mds block codes and mds 1d convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1d convolutional codes of a certain rate and degree respectively in this paper we generalize this concept to the class of 2d convolutional codes for that we introduce a natural bound on the distance of a 2d convolutional code of rate k n and degree delta which generalizes the singleton bound for block codes and the generalized singleton bound for 1d convolutional codes then we prove the existence of 2d convolutional codes of rate k n and degree delta that reach such bound when n geq k lfloor delta k rfloor 2 lfloor delta k rfloor 3 2 if k nmid delta or n geq k delta k 1 delta k 2 2 if k mid delta by presenting a concrete constructive procedure

extension of the berlekamp massey algorithm to n dimensions abstract we present an algorithm for finding a minimal set of linear recurring relations which are valid for a given n dimensional array over any field where the minimality is defined with respect to the partial order over the n dimensional lattice the algorithm is an extension of our two dimensional version of the berlekamp massey algorithm to more than two dimensions the n dimensional theory is based on more general concepts which can be reduced into those of the two dimensional theory in the previous paper in a typical case the resulting set of polynomials characterizing the minimal linear recurring relations proves to be a groebner basis of the ideal defined by the array and consequently the structure of an n dimensional linear feedback shift register with the minimum number of storage devices which can generate the array is determined by it

decoding binary 2 d cyclic codes by the 2 d berlekamp massey algorithm a method of decoding two dimensional 2 d cyclic codes by applying the 2 d berlekamp massey algorithm is proposed to explain this decoding method the author introduces a subclass of 2 d cyclic codes which are called 2 d bch codes due to their similarity with bch codes it is shown that there are some short 2 d cyclic codes with a better cost parameter value the merit of the approach is verified by showing several simple examples of 2 d cyclic codes

algebraic function fields over finite fields with many rational places algebraic function fields or equivalently algebraic curves provide a useful tool for coding theory for instance algebraic geometric codes and trace codes but also for other branches of information theory in these applications the number of rational places of a function field plays a crucial role one is particularly interested in function fields having a large number of rational places after a short introduction into the mathematical theory of algebraic functions the paper gives a survey of old and new results on the number of rational places of function fields

on a decoding algorithm for codes on maximal curves a decoding algorithm for algebraic geometric codes that was given by a n skorobogatov and s g vladut preprint inst problems of information transmission 1988 is considered the author gives a modified algorithm with improved performance which he obtains by applying the above algorithm a number of times in parallel he proves the existence of the decoding algorithm on maximal curves by showing the existence of certain divisors however he has so far been unable to give an efficient procedure of finding these divisors

on determining deep holes of generalized reed solomon codes for a linear code deep holes are defined to be vectors that are further away from codewords than all other vectors the problem of deciding whether a received word is a deep hole for generalized reed solomon grs codes is proved to be co np complete by guruswami and vardy for the extended reed solomon codes rs q mathbb f q k a conjecture was made to classify deep holes by cheng and murray since then efforts have been made to prove the conjecture or its various forms in this paper we classify deep holes completely for grs codes rs p d k where p is a prime d k geqslant p 1 2 our techniques are built on the idea of deep hole trees and several results concerning the erdos heilbronn conjecture

remarks on codes from hermitian curves corresp parameters and generator matrices are given for the codes obtained by applying goppa s algebraic geometric construction method to hermitian curves in pg 2 q where q 2 2s for some s in bf n automorphisms of these codes are also discussed and some results on self duality and weak self duality are given

fast decoding of algebraic geometric codes up to the designed minimum distance we present a decoding algorithm for algebraic geometric codes from regular plane curves in particular the hermitian curve which corrects all error patterns of weight less than d 2 with low complexity the algorithm is based on the majority scheme of feng and rao 1993 and uses a modified version of sakata s 1988 generalization of the berlekamp massey algorithm

statistical resilience of random populations to random perturbations we consider populations represented by random collections of real valued points and explore their statistical resilience to random perturbations seeking populations whose statistics remain qualitatively unchanged by the action of arbitrary random perturbations of a certain type studying a general physical perturbation scheme we obtain an explicit characterization of statistically resilient populations show that these objects are fractal and comprehensively analyze their topological and statistical structures an application of statistical resilience attained is an alternative explanation of the ubiquity of power law statistics

a generalized forney formula for algebraic geometric codes this correspondence contains a straightforward generalization of decoding of bch codes to the decoding of algebraic geometric codes couched in terms of varieties ideals and grobner bases this consists of 1 a berlekamp massey type lattice shifting row reduction algorithm with majority voting similar to algorithms in the current literature 2 a realization that it produces a minimal grobner basis b for the error locator ideal i v relative to a particular weighted total degree monomial ordering 3 a factoring of that basis into several minimal plex bases that facilitates finding the variety v of error positions and 4 a direct generalization of forney s formula to calculate error magnitudes using functions spl sigma p which are by products of this factoring

on the covering radius of mds codes for a linear maximum distance separable mds code with redundancy r the covering radius is either r or r 1 however for r 3 few examples of q ary linear mds codes with radius r 1 are known including the reed solomon codes with length q 1 in this paper for redundancies r as large as sqrt 12 q infinite families of q ary mds codes with covering radius r 1 and length less than q 1 are constructed these codes are obtained from algebraic geometric codes arising from elliptic curves for most pairs r q with r le sqrt 12 q these are the shortest q ary mds codes with covering radius r 1

classification of pseudo cyclic mds codes linear codes are considered a code is characterized by the length n the dimension k and the minimum distance d an n k d code over the finite field gf q is said to be maximum distance separable mds if d n k 1 a krishna and d v sarwate 1990 investigated the existence of pseudo cyclic mds codes over gf q of length n where n divides q 1 or q 1 it is shown that the pseudo cyclic mds codes constructed by krishna and sarwate are generalised reed solomon codes pseudo cyclic codes are studied over gf q where n and q p sup h are not relatively prime it is proven that pseudo cyclic n k mds codes modulo x sup n a over gf q exist if and only if n p furthermore any pseudo cyclic p k code modulo x sup p a over gf q turns out to be mds and generalized reed solomon it is explicitly proven that some classes of pseudo cyclic n k mds codes over gf q are generalized reed solomon codes furthermore pseudo cyclic q 1 4 mds codes over gf q q even are completely classified

codes on the klein quartic ideals and decoding corresp a sequence of codes with particular symmetries and with large rates compared to their minimal distances is constructed over the field gf 2 3 in the sequence there is for instance a code of length 21 and dimension 10 with minimal distance 9 and a code of length 21 and dimension 16 with minimal distance 3 the codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by goppa the curve used in the present work is the klein quartic this curve has the maximal number of rational points over gf 2 3 allowed by serre s improvement of the hasse weil bound which together with the low genus accounts for the good code parameters the klein quartic has the frobenius group g of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes in fact the codes are given alternative descriptions as left ideals in the group algebra gf 2 3 g this description allows for easy decoding for instance in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3 decoding is obtained by multiplication with an idempotent in the group algebra

systematic encoders for generalized gabidulin codes and the q analogue of cauchy matrices abstract we characterize the generator matrix in standard form of generalized gabidulin codes the parametrization we get for the non systematic part of this matrix coincides with the q analogue of generalized cauchy matrices leading to the definition of q cauchy matrices these matrices can be represented very conveniently and their representation allows to define new interesting subfamilies of generalized gabidulin codes whose generator matrix is a structured matrix in particular as an application we construct gabidulin codes whose generator matrix is the concatenation of an identity block and a toeplitz hankel matrix in addition our results allow to give a new efficient criterion to verify whether a rank metric code of dimension k and length n is a generalized gabidulin code this criterion is only based on the computation of the rank of one matrix and on the verification of the linear independence of two sets of elements and it requires o m f k n field operations where f k n is the cost of computing the reduced row echelon form of a k n matrix moreover we also provide a characterization of the generator matrix in standard form of general mrd codes

construction and decoding of a class of algebraic geometry codes a class of codes derived from algebraic plane curves is constructed the concepts and results from algebraic geometry that were used are explained in detail no further knowledge of algebraic geometry is needed parameters generator and parity check matrices are given the main result is a decoding algorithm which turns out to be a generalization of the peterson algorithm for decoding bch decoder codes

on the generalized hamming weights of geometric goppa codes we study the generalized hamming weights of geometric goppa codes giving a translation of these weights into arithmetical terms concerning the arithmetic of the curve used to construct the code we derive some results and bounds that we apply to some known codes

decoding of codes defined by a single point on a curve a decoding algorithm for certain codes from algebraic curves is presented the dual code which is used for decoding is formed by evaluating rational functions having poles at a single point a theoretical foundation is developed from which an improved bound for the minimum distance and results on decoding up to that bound are derived the decoding is done by a computationally efficient berlekamp massey type algorithm which is intrinsic to the curve

the maximal process of nonlinear shot noise abstract in the nonlinear shot noise system model shots statistics are governed by general poisson processes and shots decay dynamics are governed by general nonlinear differential equations in this research we consider a nonlinear shot noise system and explore the process tracking along time the system s maximal shot magnitude this maximal process is a stationary markov process following a decay surge evolution it is highly robust and it is capable of displaying both a wide spectrum of statistical behaviors and a rich variety of random decay surge sample path trajectories a comprehensive analysis of the maximal process is conducted including its markovian structure its decay surge structure and its correlation structure all results are obtained analytically and in closed form

an explicit construction of a sequence of codes attaining the tsfasman vladut zink bound the first steps we present a sequence of codes attaining the tsfasman vladut zink bound the construction is based on the tower of artin schreier extensions described by garcia and stichtenoth 1995 we also determine the dual codes the first steps of the constructions are explicitly given as generator matrices

a construction of mds 2d convolutional codes of rate 1 n based on superregular matrices abstract in this paper two dimensional convolutional codes with finite support are considered i e convolutional codes whose codewords have compact support indexed in n 2 and take values in f n where f is a finite field the main goal of this work is to analyze the free distance properties of this type of codes of rate 1 n and degree we first establish an upper bound on the maximum possible distance for these codes we then present particular constructions of two dimensional convolutional codes with finite support of rate 1 n and degree that attain such a bound and therefore have the maximum distance among all two dimensional convolutional codes with finite support with the same rate and degree we call such codes maximum distance separable two dimensional convolutional codes

classical solution of the nonlinear boltzmann equation in all r 3 asymptotic behavior of solutions proof is given of the existence of a classical solution to the nonlinear boltzmann equation in allr3 the solution which is global in time exists if the initial data go to zero fast enough at infinity and the mean free path is sufficiently large the solution is smooth in the space variable if the initial value is smooth the asymptotic behavior of solutions is also given it is shown that ast the solution to the boltzmann equation can be approximated by the solution to the free motion problem

a convergence proof for nanbu s simulation method for the full boltzmann equation it is shown that the discrete measures given by the nanbu simulation method converge with respect to the weak topology of measures to solutions of the boltzmann equation the main conditions for this result are that the cauchy problem for the boltzmann equation has a sufficiently smooth solution and that the discretization parameters cell size timestep and test particle number satisfy suitable constraints

the minimum distance of codes in an array coming from telescopic semigroups the concept of an error correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound this gives a unified point of view which explains several improvements on the minimum distance of algebraic geometric codes moreover it is explained in terms of linear algebra and the theory of semigroups only

a determination of the parameters of a large class of goppa codes we determine the code parameters of a large class of goppa codes constructed from certain curves described by stichtenoth see arch math vol xxiv pp 615 631 1973 in terms of the genus and the weierstrass gap sequence of a given chosen point on each tune examples are codes from hermitian artin schreier and hyperelliptic curves

asymmetric inclusion process we introduce and explore the asymmetric inclusion process asip an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process asep in both processes random events cause particles to propagate unidirectionally along a one dimensional lattice of n sites in the asep particles are subject to exclusion interactions whereas in the asip particles are subject to inclusion interactions that coalesce them into inseparable clusters we study the dynamics of the asip derive evolution equations for the mean and probability generating function pgf of the sites occupancy vector obtain explicit results for the above mean at steady state and describe an iterative scheme for the computation of the pgf at steady state we further obtain explicit results for the load distribution in steady state with the load being the total number of particles present in all lattice sites finally we address the problem of load optimization and solve it under various criteria the asip model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with unlimited batch service and a tandem array of growth collapse processes

superregular matrices and applications to convolutional codes abstract the main results of this paper are twofold the first one is a matrix theoretical result we say that a matrix is superregular if all of its minors that are not trivially zero are nonzero given a a b a b superregular matrix over a field we show that if all of its rows are nonzero then any linear combination of its columns with nonzero coefficients has at least a b 1 nonzero entries secondly we make use of this result to construct convolutional codes that attain the maximum possible distance for some fixed parameters of the code namely the rate and the forney indices these results answer some open questions on distances and constructions of convolutional codes posted in the literature 6 9

majority coset decoding a majority coset decoding mcd procedure that can be applied to an arbitrary geometric code is discussed in general the basic algorithm for decoding of algebraic geometric codes does not correct up to the designed minimum distance in mcd a reduction step is added to the basic algorithm in case the basic algorithm fails a majority scheme is used to obtain an additional syndrome for the error vector thus a strictly smaller cost containing the error vector is obtained in this way the basic algorithm is applied to a decreasing chain of cosets and after finitely many steps the coset will be small enough for successful application of the basic algorithm

on the active periods of nonlinear shot noise abstract we study the temporal on and off structure of the recently introduced nonlinear shot noise model i eliazar j klafter pnas 102 2005 13779 i eliazar j klafter nonlinear shot noise levy noah joseph physica a 2006 to appear in which i shots of random magnitudes bombard a system stochastically in time ii the magnitudes of incoming shots decay to zero nonlinearly and iii the overall effect of the shots on the system is additive when the shot inflow and shot decay satisfy certain conditions the resulting shot noise alternates randomly between active periods in which the noise is on and silent periods in which noise is off the statistical properties of the active and silent periods are analyzed and the resulting on and off shot noise structure is explored explicit formulae for means variances laplace transforms and other statistics are derived in closed form based on this analysis shot noise systems are categorized into three classes transient null recurrent and recurrent and these classes are characterized both analytically and probabilistically within the null recurrent class systems whose active periods are governed by heavy tailed probability distributions are further characterized

contribution to munuera s problem on the main conjecture of geometric hyperelliptic mds codes in coding theory it is of great interest to know the maximal length of mds codes in fact the main conjecture says that the length of mds codes over f sub q is less than or equal to q 1 except for some special cases munuera see ibid vol 38 p 1573 7 1992 proposed a new way to attack the main conjecture on mds codes for geometric codes in particular he proved the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough he also asked whether a similar theorem can be proved for any hyperelliptic curve the purpose of this correspondence is to give an affirmative answer in fact our method also proves the main conjecture for geometric mds codes for q 2 if the genus of the hyperelliptic curve is either 1 2 or 3 and for q 3 if the genus of the curve is 1

generalized hamming weights of trace codes linear codes over f sub p often admit a natural representation as trace codes of codes that are defined over an extension field f sub p m in the paper the authors obtain estimates for the weights of subcodes of such trace codes their main result is a far reaching generalization of the carlitz uchiyama bound for the duals of binary bch codes in particular they prove sharp bounds for the generalized hamming weights of a large class of codes including duals of bch codes classical goppa codes melas codes and arbitrary cyclic codes of length n p sup m 1 the main tool is the theory of algebraic functions over finite fields in particular the hasse weil bound for the number of places of degree one

algebraic geometry codes the theory of error correcting codes derived from curves in an algebraic geometry was initiated by the work of goppa as generalizations of bose chaudhuri hocquenghem bch reed solomon rs and goppa codes the development of the theory has received intense consideration since that time and the purpose of the paper is to review this work elements of the theory of algebraic curves at a level sufficient to understand the code constructions and decoding algorithms are introduced code constructions from particular classes of curves including the klein quartic elliptic and hyperelliptic curves and hermitian curves are presented decoding algorithms for these classes of codes and others are considered the construction of classes of asymptotically good codes using modular curves is also discussed

shot noise displaying simultaneously the noah and joseph effects shot noise processes displaying the noah effect are of infinite variance hence the measurement of the correlation structure of such processes indicating whether the joseph effect is displayed as well cannot be conducted via their autocovariance functions which are undefined to circumvent this problem a poisson based analysis applicable to shot noise processes with infinite variance as well as to shot noise processes with divergent noise levels is developed a poissonian autocovariance function which characterizes the process distribution of general shot noise processes is introduced in particular this function governs and quantifies the processes stationary structure amplitudal behavior and correlation structure temporal behavior the poissonian methodology developed enables a precise quantitative analysis of shot noise processes displaying simultaneously the noah and joseph effects

modular curves and codes with a polynomial construction in this paper we study q ary codes arising from modular curves x 0 11l over gf p 2 and some binary codes attached to them all these codes have a polynomial construction and have good asymptotic parameters for certain values of the parameters the q ary codes with q p 2 geq 49 are better than the varshamov gilbert bound and the binary codes are better than the ziablov bounds

algebraic decoding using special divisors the basic algorithm for decoding of algebraic geometric codes corrects up to d sub c 1 2 g 2 errors where d sub c denotes the designed minimum distance of a code and g denotes the genus of a curve the modified algorithm improves on this but applies to a restricted class of codes an extended modified algorithm that applies to all codes is formulated it will correct up to d sub c 1 2 s errors s is called the clifford defect of a curve for curves with g or 1 this defect satisfies 0

algebraic geometric codes and multidimensional cyclic codes a unified theory and algorithms for decoding using grobner bases it is proved that any algebraic geometric ag code can be expressed as a cross section of an extended multidimensional cyclic code both ag codes and multidimensional cyclic codes are described by a unified theory of linear block codes defined over point sets ag codes are defined over the points of an algebraic curve and an m dimensional cyclic code is defined over the points in m dimensional space the power of the unified theory is in its description of decoding techniques using grobner bases in order to fit an ag code into this theory a change of coordinates must be applied to the curve over which the code is defined so that the curve is in special position for curves in special position all computations can be performed with polynomials and this also makes it possible to use the theory of grobner bases next a transform is defined for ag codes which generalizes the discrete fourier transform the transform is also related to a grobner basis and is useful in setting up the decoding problem in the decoding problem a key step is finding a grobner basis for an error locator ideal for ag codes multidimensional cyclic codes and indeed any cross section of an extended multidimensional cyclic code sakata s algorithm can be used to find linear recursion relations which hold on the syndrome array in this general context the authors give a self contained and simplified presentation of sakata s algorithm and present a general framework for decoding algorithms for this family of codes in which the use of sakata s algorithm is supplemented by a procedure for extending the syndrome array

a new class of superregular matrices and mdp convolutional codes abstract this paper deals with the problem of constructing superregular matrices that lead to mdp convolutional codes these matrices are a type of lower block triangular toeplitz matrices with the property that all the square submatrices that can possibly be nonsingular due to the lower block triangular structure are nonsingular we present a new class of matrices that are superregular over a sufficiently large finite field f such construction works for any given choice of characteristic of the field f and code parameters n k such that n k we also discuss the size of f needed so that the proposed matrices are superregular

on generator matrices of mds codes corresp it is shown that the family of q ary generalized reed solomon codes is identical to the family of q ary linear codes generated by matrices of the form i a where i is the identity matrix and a is a generalized cauchy matrix using cauchy matrices a construction is shown of maximal triangular arrays over gf q which are constant along diagonals in a hankel matrix fashion and with the property that every square subarray is a nonsingular matrix by taking rectangular subarrays of the described triangles it is possible to construct generator matrices i a of maximum distance separable codes where a is a hankel matrix the parameters of the codes are n k d for 1 leq n leq q 1 1 leq k leq n and d n k 1

on the main conjecture on geometric mds codes a new way to attack the main conjecture on mds codes for geometric codes is proposed in particular the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough is proven

a note on elliptic curves over finite fields let e be an elliptic curve over a finite field k and let e k be the group of k rational points on e we evaluate all the possible groups e k where e runs through all the elliptic curves over a given fixed finite field k

the weight hierarchy of higher dimensional hermitian codes studies a class of projective systems and linear codes corresponding to hermitian varieties over finite fields the weight hierarchy also known as the set of generalized hamming weights of the code is calculated the higher weight distribution is also found

on automorphism groups of the hermitian codes we determine the exact automorphism groups of the goppa geometric codes from the hermitian curves over gf q sup 2

simplified understanding and efficient decoding of a class of algebraic geometric codes an efficient decoding algorithm for algebraic geometric codes is presented for codes from a large class of irreducible plane curves including hermitian curves it can correct up to d 1 2 errors where d is the designed minimum distance with it we also obtain a proof of d sub min spl ges d without directly using the riemann roch theorem the algorithm consists of gaussian elimination on a specially arranged syndrome matrix followed by a novel majority voting scheme a fast implementation incorporating block hankel matrix techniques is obtained whose worst case running time is o mn sup 2 where m is the degree of the curve applications of our techniques to decoding other algebraic geometric codes to decoding bch codes to actual minimum distance and to two dimensional shift register synthesis are also presented

on the cauchy problem for the nonlinear boltzmann equation global existence uniqueness and asymptotic stability the analysis of the initial value problem for the nonlinear boltzmann equation is considered in this paper a theorem defining global existence and uniqueness for initial data which decay at infinity with an inverse power law is the main result of this work and is obtained by suitable application of fixed point theorems in banach spaces the theorem also defines the asymptotic stability of the solutions

strongly mds convolutional codes maximum distance separable mds convolutional codes have the property that their free distance is maximal among all codes of the same rate and the same degree in this paper a class of mds convolutional codes is introduced whose column distances reach the generalized singleton bound at the earliest possible instant such codes are called strongly mds convolutional codes they also have a maximum or near maximum distance profile the extended row distances of these codes will also be discussed briefly

stochastic ornstein uhlenbeck capacitors we introduce and study a class of random capacitor systems which are both charged and discharged stochastically a capacitor is fed by a random inflow with stationary and independent increments discharging occurs according to a markovian rate which is linear in the capacitor s level the resulting capacitor dynamics are markovian stochastically cyclic and regenerative we coin these systems levy charged ornstein uhlenbeck capacitors various random quantities associated with these systems are analyzed including the time to discharge the duration of the charging cycle the trajectory and the peak height of the capacitor level during a charging cycle and the capacitor s stationary equilibrium level furthermore we show that there are sharp distinctions between these capacitor systems and corresponding standard levy driven ornstein uhlenbeck systems

improved geometric goppa codes i basic theory in this paper we present a construction of improved geometric goppa codes which for the case of r 2g are often more efficient than the current geometric goppa codes derived from some varieties which include algebraic curves hyperplanes surfaces and other varieties for the special case of a plane in a three dimensional projective space the improved geometric goppa codes are reduced to linear multilevel codes for these improved geometric goppa codes a designed minimum distance can be easily determined and a decoding procedure which corrects up to half the designed minimum distance is also given

on cyclic mds codes of length q over gf q corresp it is shown that a cyclic code c of length q over gf q is the maximum distance separable if and only if either 1 q is a prime in which case c is equivalent up to a coordinate permutation to an extended reed solomon code or 2 c is a trivial code of dimension k in 1 q 1 q hence there exists a nontrivial cyclic extended reed solomon code of length q over gf q if and only if q is a prime

the boltzmann equation global existence for a rare gas in an infinite vacuum solutions of the boltzmann equation are proved to exist globally in time under conditions that include the case of a finite volume of gas in an infinite vacuum when the mean free path of the gas is large enough it is also proved as might be expected in this case that the density of the gas at each point in space goes to zero as time goes to infinity

a tower of artin schreier extensions of function fields attaining the drinfeld vladut bound for an algebraic function fieldf having a finite constant field letg f resp n f denote the genus off resp the number of places off of degree one we construct a tower of function fields f 1 subseteq f 2 subseteq f 3 subseteq ldots over mathbb f q 2 such that the ration fi g fi tends to the drinfeld vladut boundq 1

superregular lower triangular toeplitz matrices for low delay wireless streaming a matrix is termed superregular if all of its possible submatrices are non singular superregular lower triangular toeplitz matrices are useful for mds convolutional codes and sequential network codes in this paper we present the explicit matrix constructions for superregular lower triangular toeplitz matrices in mathrm gf 2 p k times k k leq 5 for k 5 we provide a greedy algorithm which over sufficiently large fields is guaranteed to find a superregular lower triangular toeplitz matrix we introduce product preserving joint superregularity and extend our explicit matrix constructions to these cases we provide methods for deriving the exact symbol loss probability and delay for any deterministic block code we derive the exact symbol loss probability and delay for codes using a superregular lower triangular matrix and for codes using two product preserving jointly superregular lower triangular matrices we then compare these results with those obtained from both simulations and our practical implementation and for each case we also compare with random based codes furthermore our experiments show a gain in coding throughput above 40 for superregular lower triangular toeplitz matrices over random matrices

decoding algebraic geometric codes up to the designed minimum distance a simple decoding procedure for algebraic geometric codes c sub omega d g is presented this decoding procedure is a generalization of peterson s decoding procedure for the bch codes it can be used to correct any d 1 2 or fewer errors with complexity o n sup 3 where d is the designed minimum distance of the algebraic geometric code and n is the codelength

projective reed muller codes a class of codes in the reed muller family the projective reed muller codes prm codes is studied the author defines the prm codes of all orders and discusses their relation to polynomial codes the exact parameters of prm codes are given the duals are characterized and in parallel to the classical works on generalized reed muller codes the cyclic properties are studied tables over parameters of the codes are given

non linear shot noise levy noah joseph abstract we introduce and study a generic non linear shot noise system model shots of random magnitudes arrive to the system stochastically following an arbitrary time homogeneous poisson point process after hitting the system the magnitude of an arriving shot decays to zero the decay is governed by an arbitrary differential equation dynamics shots are independent and their overall effect on the system is additive the system s noise level at time t equals the sum of the magnitudes at time t of all the shots arriving to the system prior to time t the resulting shot noise is i a levy process when the decay dynamics are degenerate ii a levy driven ornstein uhlenbeck process when the decay dynamics are linear and iii a stationary non markov process when the decay dynamics are non linear the resulting shot noise admits an underlying levy structure which we explicitly compute and can yield both the noah effect and the joseph effect closed form analytic formulae for various statistics are derived including the log laplace transform and cumulants of the stationary noise level the process auto covariance function and the process range of dependence

on the decoding of algebraic geometric codes this paper provides a survey of the existing literature on the decoding of algebraic geometric codes definitions theorems and cross references will be given we show what has been done discuss what still has to be done and pose some open problems

generalized reed solomon codes from algebraic geometry a few years ago tsfasman em et al using results from algebraic geometry showed that there is a sequence of codes which are generalizations of goppa codes and which exceed the gilbert varshamov bound we show that a similar sequence of codes in fact the duals of the previous codes can be found by generalizing the construction of reed solomon codes our approach has the advantage that it uses less complicated concepts from algebraic geometry

a revision on classical solutions to the cauchy boltzmann problem for soft potentials this short note complements the recent paper of the authors alonso gamba in j stat phys 137 5 6 1147 1165 2009 we revisit the results on propagation of regularity and stability using lp estimates for the gain and loss collision operators which had the exponent range misstated for the loss operator we show here the correct range of exponents we require a lebesgue s exponent 1 in the angular part of the collision kernel in order to obtain finiteness in some constants involved in the regularity and stability estimates as a consequence the lp regularity associated to the cauchy problem of the space inhomogeneous boltzmann equation holds for a finite range of p 1 explicitly determined

decoding geometric goppa codes using an extra place decoding geometric goppa codes can be reduced to solving the key congruence of a received word in an affine ring if the codelength is smaller than the number of rational points on the curve then this method can correct up to 1 2 d l 2 s errors where d is the designed minimum distance of the code and s is the clifford defect the affine ring with respect to a place p is the set of all rational functions which have no poles except at p and it is somehow similar to a polynomial ring for a special kind of geometric goppa code namely c sub omega d mp the decoding algorithm is reduced to solving the key equation in the affine ring which can be carried out by the subresultant sequence in the affine ring with complexity o n sup 3 where n is the length of codewords

a growth collapse model levy inflow geometric crashes and generalized ornstein uhlenbeck dynamics we introduce and study a stochastic growth collapse model the growth process is a steady random inflow with stationary independent and non negative increments crashes occur according to an arbitrary renewal process they are geometric and their magnitudes are random and are governed by an arbitrary distribution on the unit interval if the system s pre crash level is x 0 and the crash magnitude is 0 c 1 then an avalanche of size cx takes place after which the system collapses to the post crash level 1 c x this results in a stochastic growth collapse growth collapse system evolution governed by generalized non markovian ornstein uhlenbeck dynamics we explore the behavior of this growth collapse model and compute various system statistics including mean variance auto correlation laplace transform and in the case of heavy tailed inflows probability tails special emphasis is devoted to scale free systems where the crash magnitudes are governed by a power law distribution we show that i there is a hidden regular ornstein uhlenbeck structure underlying all scale free systems ii when crash rates tend to infinity these systems yield regular ornstein uhlenbeck limits and iii when the inflow is selfsimilar these systems yield linnik distributed equilibria

achieving the designed error capacity in decoding algebraic geometric codes a decoding algorithm for codes arising from algebraic curves explicitly constructable by goppa s construction is presented any configuration up to the greatest integer less than or equal to d 1 2 errors is corrected by the algorithm whenever d or 6g where d is the designed minimum distance of the code and g is the genus of the curve the algorithm s complexity is at most o d sup 2 n where n denotes the length of the code application to hermitian codes and connections with well known algorithms are explained

a note on hermitian codes over gf q sup 2 properties of codes defined by goppa s algebraic geometric method using hermitian curves are discussed generator and parity check matrices for such codes are derived explicitly results concerning the minimum distance and the weight distribution of the codes are provided

on the decoding of algebraic geometric codes over f sub q for q or 16 it is proved that for algebraic geometric codes on a curve over f sub q for q or 37 or on a curve of sufficiently large genus over f sub q for q or 16 there exists a polynomial decoding algorithm up to d 1 2 errors d being the designed minimum distance

on a time and space discretized approximation of the boltzmann equation in the whole space abstract in this paper convergence results on the solutions of a time and space discrete model approximation of the boltzmann equation for a gas of maxwellian particles in a bounded domain obtained by babovsky and illner 1989 are extended to approximate the solutions of the boltzmann equation in the whole physical space this is done for a class of particle interactions including maxwell and soft cut off potentials in the sense of grad the main result shows that the solutions of the discrete model converge in l 1 to the solutions of the boltzmann equation when the discretization parameters go simultaneously to zero the convergence is uniform with respect to the discretization parameters in addition a sufficient condition for the implementation of the main result is provided the techniques detailed in this paper may be also applied in other contexts

distributional and classical solutions to the cauchy boltzmann problem for soft potentials with integrable angular cross section this paper focuses on the study of existence and uniqueness of distributional and classical solutions to the cauchy boltzmann problem for the soft potential case assuming s n 1 integrability of the angular part of the collision kernel grad cut off assumption for this purpose we revisit the kaniel shinbrot iteration technique to present an elementary proof of existence and uniqueness results that includes the large data near local maxwellian regime with possibly infinite initial mass we study the propagation of regularity using a recent estimate for the positive collision operator given in alonso et al in convolution inequalities for the boltzmann collision operator arxiv 0902 0507 math ap by e carneiro and the authors that allows us to show such propagation without additional conditions on the collision kernel finally an l p stability result with 1 p is presented assuming the aforementioned condition

fast decoding of codes from algebraic plane curves improvement to an earlier decoding algorithm for codes from algebraic geometry is presented for codes from an arbitrary regular plane curve the authors correct up to d 2 m sup 2 8 m 4 9 8 errors where d is the designed distance of the code and m is the degree of the curve the complexity of finding the error locator is o n sup 7 3 where n is the length of the code for codes from hermitian curves the complexity of finding the error values given the error locator is o n sup 2 and the same complexity can be obtained in the general case if only d 2 m sup 2 2 errors are corrected