classification of convolutional codes convolutional codes have the structure of an f z module to study their properties it is desirable to classify them as the points of a certain algebraic variety by considering the correspondence of submodules and the points of certain quotient schemes and the inclusion of these as subvarieties of certain grassmannians one has a one to one correspondence of convolutional codes and the points of these subvarieties this classification of convolutional codes sheds light on their structure and proves to be helpful to give bounds on their free distance and to define convolutional codes with good parameters

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

applications of riordan matrix functions to bernoulli and euler polynomials abstract we define riordan matrix functions associated with riordan arrays and study their algebraic properties we also give their applications in the construction of new classes of bernoulli and euler polynomials and bernoulli and euler numbers referred to as the duals and conjugate bernoulli and euler polynomials and dual and conjugate bernoulli and euler numbers respectively

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

every convolutional code is a goppa code convolutional goppa codes cgc were defined in appl algebra eng comm comput vol 15 pp 51 61 2004 and ieee trans inf theory vol 52 340 344 2006 in this paper we prove that every convolutional code is a cgc defined over a smooth curve over bbf q z and we give an explicit construction of convolutional codes as cgc over the projective line bbp bbf q z 1 we characterize which convolutional codes are defined by a complete linear system over curves of genus 0 1 and over hyperelliptic curves we apply these results to provide detailed constructions of some linear block codes as goppa codes

on superregular matrices and mdp convolutional codes superregular matrices are a type of lower triangular toeplitz matrix that arises in the context of constructing convolutional codes having a maximum distance profile these matrices are characterized by the property that the only submatrices having a zero determinant are those whose determinants are trivially zero due to the lower triangular structure in this paper we discuss how superregular matrices may be used to construct codes having a maximum distance profile we also present an upper bound on the minimum size a finite field must have in order that a superregular matrix of a given size can exist over that field this in turn gives an upper bound on the smallest field size over which an mdp n k convolutional code can exist

computation of counting distributions arising from a single stage multiplicative process abstract the cumulative distribution of the number of secondary electrons in a single stage photomultiplier is calculated by numerically integrating the inversion integral for its probability generating function along a suitably chosen contour a residue series applicable in certain cases is also presented saddlepoint approximations to the contour integral are described which are the more accurate the greater the numbers of secondaries recurrent relations are developed for computing values of the distribution for purposes of comparison computation of the neyman type a distribution is treated as a limiting case

diagonally striped matrices and approximate inverse preconditioners the inverse of a banded matrix is in general dense if the structure of the original banded matrix is striped that is the non zero diagonals are separated by one or more zero diagonals the inverse may exhibit a similar striped structure the motivation for studying inverses of striped matrices is to obtain efficient preconditioners for systems arising from radiation transport equations whose matrices include dominant values along diagonal stripes linear systems whose system matrix has a striped inverse lend themselves to the use of a sparse approximate inverse spai preconditioner whose structure is derived from that of the actual inverse

quasi cyclic unit memory convolutional codes unit memory convolutional codes with generator matrices which are composed of circulant submatrices are introduced this structure facilitates the analysis of efficient search for good codes equivalences among such codes and some of the basic structural properties are discussed in particular catastrophic encoders and minimal encoders are characterized and dual codes treated further various distance measures are discussed and a number of good codes some of which result from efficient computer search and some of which result from known block codes are presented

the elements of finite order in the riordan group over the complex field abstract an element of finite order in the riordan group over the real field must have order 1 or 2 if we extend all the entries to be complex numbers then it may have any finite order in the present paper we investigate the elements of finite order of the riordan group over the complex field this notion leads us to define an element of pseudo order k 2 and k pseudo involution respectively it turns out that the inverse of a k pseudo involution only differs from it in signs we clarify some relationship between the elements of pseudo order k and k pseudo involutions in particular k pseudo involutions for k 0 mod 4 are characterized by a single sequence the subgroups of the riordan group formed by the elements having pseudo order of a prime power p r are also introduced

a method for the numerical evaluation of the oscillatory integrals associated with the cuspoid catastrophes application to pearcey s integral and its derivatives a numerical method for the evaluation of the cuspoid canonical integrals and their derivatives is described the method exploits cauchy s theorem and jordan s lemma to write the infinite integration path along different contours in the complex plane the method is straightforward to implement on a computer and in many cases results of high accuracy can be obtained using standard quadrature techniques application is made to pearcey s integral p x y and its two partial derivatives and the method is shown to have some significant advantages over other techniques that have been applied to this problem tables of p x y delta p x y delta x delta p x y delta y and the real zeros of p x y are presented for the grid 8 0 or x or 8 0 and 0 or y or 8 0

column distances of convolutional codes over mathbb z _ p r maximum distance profile codes over finite non binary fields have been introduced and thoroughly studied in the last decade these codes have the property that their column distances are maximal among all codes of the same rate and degree in this paper we aim at studying this fundamental concept in the context of convolutional codes over a finite ring we extensively use the concept of p encoder to establish the theoretical framework and derive several bounds on the column distances in particular a method for constructing not necessarily free maximum distance profile convolutional codes over mathbb z p r is presented

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

a priori sparsity patterns for parallel sparse approximate inverse preconditioners parallel algorithms for computing sparse approximations to the inverse of a sparse matrix either use a prescribed sparsity pattern for the approximate inverse or attempt to generate a good pattern as part of the algorithm this paper demonstrates that for pde problems the patterns of powers of sparsified matrices psms can be used a priori as effective approximate inverse patterns and that the additional effort of adaptive sparsity pattern calculations may not be required psm patterns are related to various other approximate inverse sparsity patterns through matrix graph theory and heuristics concerning the pde s green s function a parallel implementation shows that psm patterned approximate inverses are significantly faster to construct than approximate inverses constructed adaptively while often giving preconditioners of comparable quality

optimal and near optimal encoders for short and moderate length tail biting trellises the results of an extensive search for short and moderate length polynomial convolutional encoders for time invariant tail biting representations of block codes at rates r 1 4 1 3 1 2 and 2 3 are reported the tail biting representations found are typically as good as the best known block codes

a formula to construct all involutions in riordan matrix groups abstract we get a formula for all finite riordan involutions using the natural approximation induced by the inverse limit construction we also describe all infinite riordan matrices which are involutions in the riordan group finally we get some consequences

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

on the construction of 1 d mds convolutional goppa codes we show that the free distance as a function on a space parameterizing a family of convolutional codes is a lower semicontinuous function and that therefore the property of being maximum distance separable mds is an open condition for a class of convolutional codes an algorithm is offered to compute the free distance the behavior of the free distance by enlargements of the alphabet and by increasing the length is also studied as an application the algebraic equations characterizing the subfamily of mds codes are explicitly computed for families of 1 d convolutional goppa codes

riordan group involutions we study involutions in the riordan group especially those with combinatorial meaning we give a new determinantal criterion for a matrix to be a riordan involution and examine several classes of examples a complete characterization of involutions in the appell subgroup is developed in another direction we find several examples that generalize the rna matrix but are of independent interest

algorithmic construction of trellis codes an algorithmic approach is proposed whereby long convolutional codes of rate r sub c k n can easily be constructed for any chosen signal constellation in signal space these algorithms are iterative and in each step a number of candidate codes are found which locally maximize the distance hamming or euclidean between the codewords the result is not necessarily a free distance optimizing code however since the construction complexity can be chosen optimal codes are quite frequently found the codes ae constructed such that a rapid growth of the column distance is achieved a method of combining two codes into a single code of twice the constraint length is also presented

multidimensional quasi cyclic and convolutional codes we introduce multidimensional analogues of quasi cyclic qc codes and study their algebraic structure we demonstrate a concatenated structure for multidimensional qc codes and use this to prove that this class of codes is asymptotically good we also relate the new family of codes to convolutional codes it is known that the minimum distance of qc codes provides a natural lower bound on the free distance of convolutional codes we show that the same relation also holds between certain rank one 2 d convolutional codes and the related multidimensional qc codes we provide examples which show that our bound is sharp in some cases we also present some optimal 2 d qc codes along the way we provide a condition on the encoders of rank one convolutional codes which are equivalent to noncatastrophicity for 1 d convolutional codes in the n text d case n 1 our condition is sufficient for the noncatastrophicity of the encoder

algebraic aspects of some riordan arrays related to binary words avoiding a pattern we consider some riordan arrays related to binary words avoiding a pattern p which can be easily studied by means of an a matrix rather than their a sequence both concepts allow us to define every element as a linear combination of other elements in the array the a sequence is unique and corresponds to a linear dependence from the previous row the a matrix is not unique and corresponds to a linear dependence from several previous rows however for the problems considered in the present paper we show that the a matrix approach is more convenient we provide explicit algebraic generating functions for these riordan arrays and obtain many statistics on the corresponding languages we thus obtain a deeper insight of the languages l p of binary words avoiding p having a number of 0s less or equal to the number of 1s

an application of riordan arrays to the transient analysis of m m 1 queues abstract riordan arrays are used extensively in many contexts as a combinatorial tool for solving enumeration problems such as lattice path counting problems in this paper we first show how riordan array method can be applied to the transient analysis of m m 1 queue with zero customer at the initial point second we extend this method to the generalized riordan array with multiple support functions in order to deal with the transient analysis of m m 1 queue with non zero customers at the initial point numerical examples are also given to show how easy and quick the transient probability obtained from the riordan method can be computed

simple proofs of open problems about the structure of involutions in the riordan group abstract we prove that if d g x f x is an element of order 2 in the riordan group then g x exp x xf x for some antisymmetric function x z also we prove that every element of order 2 in the riordan group can be written as bmb 1 for some element b and m 1 1 in the riordan group these proofs provide solutions to two open problems presented by l shapiro l w shapiro some open questions about random walks involutions limiting distributions and generating functions adv appl math 27 2001 585 596

riordan graphs i structural properties abstract in this paper we use the theory of riordan matrices to introduce the notion of a riordan graph the riordan graphs are a far reaching generalization of the well known and well studied pascal graphs and toeplitz graphs and also some other families of graphs the riordan graphs are proved to have a number of interesting fractal properties which can be useful in creating computer networks with certain desirable features or in obtaining useful information when designing algorithms to compute values of graph invariants the main focus in this paper is the study of structural properties of families of riordan graphs obtained from infinite riordan graphs which includes a fundamental decomposition theorem and certain conditions on riordan graphs to have an eulerian trail cycle or a hamiltonian cycle we will study spectral properties of the riordan graphs in a follow up paper

construction of partial unit memory mds convolutional codes maximum distance separable mds convolutional codes form an optimal family of convolutional codes the study of which is of great importance there are very few general algebraic constructions of mds convolutional codes in this paper we construct a large family of partial unit memory mds convolutional codes over mathbb f q with flexible parameters compared with the previous work the field size q required to define these codes is much smaller the construction also leads to many new strongly mds convolutional codes an important subclass of mds convolutional codes some examples are presented at the end of this paper

a class of one dimensional mds convolutional codes a class of one dimensional convolutional codes will be presented they are all mds codes i e have the largest distance among all one dimensional codes of the same length n and overall constraint length furthermore their extended row distances are computed and they increase with slope at least n in certain cases of the algebraic parameters we will also derive parity check matrices of vandermonde type for these codes finally cyclicity in the convolutional sense of 8 will be discussed for our class of codes it will turn out that they are cyclic if and only if the field element used in the generator matrix has order n this can be regarded as a generalization of the block code case

a symbolic treatment of riordan arrays abstract we approach riordan arrays and their generalizations via umbral symbolic methods this new approach allows us to derive fundamental aspects of the theory of riordan arrays as immediate consequences of the umbral version of the classical abel s identity for polynomials in particular we obtain a novel non recursive formula for riordan arrays and derive from this new formula some known recurrences and a new recurrence relation for riordan arrays

convolutional codes with maximum distance profile abstract maximum distance profile codes are characterized by the property that two trajectories which start at the same state and proceed to a different state will have the maximum possible minimum distance from each other relative to any other convolutional code of the same rate and degree in this paper we use methods from systems theory to characterize maximum distance profile codes algebraically the main result shows that maximum distance profile codes form a generic set inside the variety which parametrizes the set of convolutional codes of a fixed rate and a fixed degree

half of a riordan array and restricted lattice paths abstract for an infinite lower triangular matrix g g n k n k 0 we define the half of g to be the infinite lower triangular matrix h h n k n k 0 such that h n k g 2 n k n for all n k 0 in this paper we will show that if g g n k n k 0 is a riordan array then its half h h n k n k 0 is also a riordan array and we obtain new combinatorial interpretations for some riordan arrays in terms of weighted lattice paths

elements of finite order in the riordan group and their eigenvectors abstract we consider elements of finite order in the riordan group r over a field f of characteristic 0 we solve for all n 2 two foundational questions posed by l shapiro 2001 for the case n 2 involutions given f x of finite compositional order n theorem 1 gives a formula for g x showing that those g x which make g x f x r a riordan element of order n are precisely those for which the matrix g x f x has an eigenvector h 0 h 1 t with h 0 0 theorem 2 classifies finite order riordan group elements up to conjugation in r and gives a normal form for finite order riordan arrays under similarity this leads to theorem 3 a formula for all eigenvectors of finite order riordan arrays and results in interesting combinatorial identities we relate our work to 2008 and 2013 papers of cheon and kim which motivated this paper and we solve their open question this circle of ideas closes with a new proof of c marshall s theorem which finds the unique f x given bi invertible g x such that g x f x is an involution

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

some inverse limit approaches to the riordan group abstract using the inverse limit tool we obtain the riordan group in its infinite and bi infinite representations from groups of finite riordan matrices we define two different reflections for bi infinite riordan matrices employing these definitions we give answers to the problems d d and d d in the riordan group so we describe the self complementary and self dual riordan matrices

convolutional codes with maximum column sum rank for network streaming the column hamming distance of a convolutional code determines the error correction capability when streaming over a class of packet erasure channels we introduce a metric known as the column sum rank that parallels the column hamming distance when streaming over a network with link failures we prove the rank analogues of several known column hamming distance properties and introduce a new family of convolutional codes that maximize the column sum rank up to the code memory our construction involves finding a class of super regular matrices that preserve this property after multiplication with non singular block diagonal matrices in the ground field

recurrence relations for polynomial sequences via riordan matrices we give recurrence relations for any family of generalized appell polynomials unifying so some known recurrences for many classical sequences of polynomials our main tool to get our goal is the riordan group we use the product of riordan matrices to interpret some relationships between different polynomial families moreover using the hadamard product of series we get a general recurrence relation for the polynomial sequences associated to the so called generalized umbral calculus

on negacyclic mds convolutional codes new families of classical and quantum optimal negacyclic convolutional codes are constructed in this paper this optimality is in the sense that they attain the classical quantum generalized singleton bound the constructions presented in this paper are performed algebraically and not by computational search

shift operators defined in the riordan group and their applications abstract in this paper we discuss a linear operator t defined in riordan group r by using the upper shift matrix u and lower shift matrix u t namely for each r r t r u r u t some isomorphic properties of the operator t and the structures of its range sets for different domains are studied by using the operator t and the properties of bell subgroup of r the riordan type chu vandermonde identities and the riordan equivalent identities of format last theorem and beal conjecture are shown the applications of the shift operators to the complementary riordan arrays and to the riordan involutions and riordan pseudo involutions are also presented

a unified approach for the catalan matrices by using riordan arrays abstract we study generalized catalan matrices based on the riordan array and fuss catalan numbers a unified combinatorial interpretation for the entries of the generalized catalan matrices is presented by means of m dyck paths some properties and examples of the generalized catalan matrices are given including a new convolution formula for the generalized catalan numbers finally we present applications of generalized catalan matrices to the problems in counting the hill free and lower peak free m dyck paths

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

on optimal constacyclic codes abstract in this paper we construct new families of maximum distance separable mds convolutional codes derived from classical constacyclic codes additionally we also construct new families of almost mds block and convolutional codes most of the new convolutional codes constructed here are unit memory and have degree 2 moreover based on a different method we construct families of asymmetric quantum codes known in the literature all these results are mainly derived from the investigation of suitable properties on cyclotomic cosets of such codes

a combinatorial equivalence relation for formal power series abstract in this paper we examine an equivalence relation on the set of formal power series with nonzero constant term this is done both in terms of functional equations and also by interlacing two concepts from riordan group theory the a sequence and the bell subgroup the best known example gives an equivalence class 1 z 1 1 z c z t z q z where c z t z and q z are generating functions of the catalan numbers ternary numbers and quaternary numbers respectively a power series for one member of an equivalence class can be transformed into power series for the rest of members in the equivalence class and interpretations in terms of weighted lattice paths can also be given

groups of generalized pascal matrices abstract we present a new approach to the study of generalized pascal matrices that yields general results about group structures and explicit formulas for powers and inverses of the pascal matrices shows how the groups of pascal matrices are related to groups of 2 2 matrices and clarifies the way in which symmetries and duality relate to the algebraic aspects of the theory we consider bilaterally infinite generalized pascal matrices obtained as the matrix representations of certain linear operators on spaces of formal laurent series their column generating functions are simple rational functions closely related to the linear fractional maps of the complex plane we obtain explicit expressions for the inverses and the powers of the pascal matrices that generalize most of the analogous results in the literature

identities induced by riordan arrays historically there exist two versions of the riordan array concept the older one better known as recursive matrix consists of bi infinite matrices d n k n k is an element of z k n implies d n k 0 deals with formal laurent series and has been mainly used to study algebraic properties of such matrices the more recent version consists of infinite lower triangular arrays d n k n k is an element of n k n implies d n k 0 deals with formal power series and has been used to study combinatorial problems here we show that every riordan array induces two characteristic combinatorial sums in three parameters n k m is an element of z these parameters can he specialized and generate an indefinite number of other combinatorial identities which are valid in the hi infinite realm of recursive matrices

some algebraic structure of the riordan group abstract in this paper we highlight some of the algebraic properties of the riordan group we present some new properties and provide proofs for some known properties for instance proofs of properties of the power bell and derivative subgroups are presented we also give a new property for generating stochastic riordan matrices which are elements of the stochastic riordan subgroup centralizers stabilizers and certain riordan group isomorphisms are also given and proved we conclude by proving some properties connecting similar riordan matrices and pseudo involutions in the riordan group

bounds on distances and error exponents of unit memory codes binary unit memory codes originally introduced by lee are investigated a few examples of constant unit memory codes are given and bounds on the distance profile and the free distances are discussed for time varying codes asymptotic lower bounds on the distance profile and the free distance are given the error probability for the codes used on a memoryless binary input output symmetric channel is asymptotically upper bounded the asymptotic results for the free distance and the error probability which are in some respects better than for conventional convolutional codes are interpreted by forney s inverse concatenation construction

the existence of strongly mds convolutional codes it is known that maximum distance separable and maximum distance profile convolutional codes exist over large enough finite fields of any characteristic for all parameters n k delta it has been conjectured that the same is true for convolutional codes that are strongly maximum distance separable using methods from linear systems theory we resolve this conjecture by showing that over a large enough finite field of any characteristic codes which are simultaneously maximum distance profile and strongly maximum distance separable exist for all parameters n k delta

riordan arrays and related polynomial sequences abstract in this paper we generalize the results due to luzon and moron and present a characterization of the generalized riordan arrays using this characterization we establish a recurrence for the generalized sheffer sequences and study some special types of polynomial sequences including the generalized lucas u sequence and v sequence and the generalized humbert polynomial sequence in particular by appealing to the factorizations of riordan arrays the umbral composition of polynomial sequences and the hadamard product of series we establish various relations among these polynomial sequences thus we give a unified approach to some well known polynomial sequences appeared in combinatorics and the theory of special functions

recursive matrices and umbral calculus two major objectives can be seen to guide much recent work in enumeration 1 to single out a limited variety of recurrences for numerical sequences which will encompass counting problems of wide enough type 2 to recover from empirical data an underlying set theoretic structure which would reveal the source of the given recursion we are here concerned with the first of these objectives though the eventual understanding of the second is tacitly present if only as a goal we noticed the coincidence of several computations which similar as they are in retrospect had failed to realize their kinship on leafing through the unique assembly of recursively solvable combinatorial problems in comtet s and sloane s invaluable collections one is struck by the repeated occurrence of one and the same kind of double recursion more strikingly the same recursion is seen to occur in the polynomial sequences of the umbra1 calculus of roman and rota see 25 everywhere the lagrange inversion formula for power series plays a pivotal role much work is nowadays going into the unraveling of the everdeeper layers of combinatorial significance of this formula both in the ordinary case and in its as yet partially worked out noncommutative and qanalogs andrews foata garsia gessel joni raney reiner schtitzenberger to name but a few whatever their origins the identities abutting lagrange inversion are expressed by integers alone this suggests not only a hidden set theoretic layer but a characteristic free generalization as well this generalization is the central theme of our work we define a monoid of infinite matrices recursive matrices for short the entries of these matrices give the sought out recursion for example that for coefficients of binomial and sheffer polynomials and factor sequences as well as that of the special sequences recently introduced by roman in 26 546 002 l 8693 82 040546 28 02 00 0

algebraic lower bounds on the free distance of convolutional codes a new module structure for convolutional codes is introduced and used to establish further links with quasi cyclic and cyclic codes the set of finite weight codewords of an n k convolutional code over fq is shown to be isomorphic to an fq x submodule of fq n x where fq n x is the ring of polynomials in indeterminate x over fq n an extension field of fq such a module can then be associated with a quasi cyclic code of index n and block length nl viewed as an fq x submodule of fq n x langxl 1rang for any positive integer l using this new module approach algebraic lower bounds on the free distance of a convolutional code are derived which can be read directly from the choice of polynomial generators links between convolutional codes and cyclic codes over the field extension fq n are also developed and bose chaudhuri hocquenghem bch type results are easily established in this setting techniques to find the optimal choice of the parameter l are outlined

notes on the total positivity of riordan arrays abstract let r d t h t be a riordan array we show that if both d t and h t are polya frequency formal power series then r is totally positive

constructions of mds convolutional codes maximum distance separable mds convolutional codes are characterized through the property that the free distance attains the generalized singleton bound the existence of mds convolutional codes was established by two of the authors by using methods from algebraic geometry this correspondence provides an elementary construction of mds convolutional codes for each rate k n and each degree spl delta the construction is based on a well known connection between quasi cyclic codes and convolutional codes

generalized column distances the notion of generalized hamming weights of block codes has been investigated since the nineties due to its significant role in coding theory and cryptography in this paper we extend this concept to the context of convolutional codes in particular we focus on column distances and introduce the novel notion of generalized column distances gcd we first show that the hierarchy of gcd is strictly increasing we then provide characterizations of such distances in terms of the truncated parity check matrix of the code that will allow us to determine their values finally the case in which the parity check matrix is in systematic form is treated

pascal eigenspaces and invariant sequences of the first or second kind abstract an infinite real sequence a n is called an invariant sequence of the first resp second kind if a n k 0 n n k 1 k a k resp a n k n k n 1 k a k we review and investigate invariant sequences of the first and second kind and study their relationships using similarities of pascal type matrices and their eigenspaces

finite and infinite dimensional lie group structures on riordan groups abstract we introduce a frechet lie group structure on the riordan group we give a description of the corresponding lie algebra as a vector space of infinite lower triangular matrices we describe a natural linear action induced on the frechet space k n by any element in the lie algebra we relate this to a certain family of bivariate linear partial differential equations we obtain the solutions of such equations using one parameter groups in the riordan group we show how a particular semidirect product decomposition in the riordan group is reflected in the lie algebra we study the stabilizer of a formal power series under the action induced by riordan matrices we get stabilizers in the fraction field k x using bi infinite representations we provide some examples the main tool to get our results is the paper 18 where the riordan group was described using inverse sequences of groups of finite matrices

on behaviors and convolutional codes it is well known that a convolutional code is essentially a linear system defined over a finite field in this paper we elaborate on this connection we define a convolutional code as the dual of a complete linear behavior in the sense of willems 1979 using ideas from systems theory we describe a set of generalized first order descriptions for convolutional codes as an application of these ideas we present a new algebraic construction for convolutional codes

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

a general umbral calculus in infinitely many variables on etudie des suites de fonctions factorielles satisfaisant des analogues des recurrences binomiales et de sheffer et on montre comment un calcul operatoriel ressemblant au calcul umbral peut etre construit dessus

on cyclotomic cosets and code constructions abstract new properties of q ary cyclotomic cosets modulo n q m 1 where q 3 is a prime power are investigated in this paper based on these properties the dimension as well as bounds for the designed distance of some families of classical cyclic codes can be computed as an application new families of nonbinary calderbank shor steane css quantum codes as well as new families of convolutional codes are constructed in this work these new css codes have parameters better than the ones available in the literature the convolutional codes constructed here have free distance greater than the ones available in the literature

the uniform asymptotic swallowtail approximation practical methods for oscillating integrals with four coalescing saddle points practical methods for the numerical implementation of the uniform swallowtail approximation have been developed this approximation arises in the uniform asymptotic evaluation of oscillating integrals with four coalescing saddle points a complex contour quadrature technique has been used to evaluate the swallowtail canonical integral s x y z and its partial derivatives delta s delta x delta s delta y delta s delta z this method has the advantage that it is straightforward to implement on a computer and results of high accuracy are readily obtained a comparison is made with other methods that have been reported in the literature for the evaluation of s x y z isometric plots of mod s x y z mod mod delta s delta x mod mod delta s delta y mod mod delta s delta z mod are presented and some properties of the zeros of s 0 y z that lie on the line y 0 are also discussed two methods for the evaluation of the mapping parameters x y z are described an iterative method that is valid when x y z is not close to the swallowtail caustic and an algebraic method valid for x y z on the caustic and for y 0 symbolic algebraic computer programs have been used to carry out the necessary algebraic manipulations in practice both methods for determining x y z are complementary an application of the uniform swallowtail approximation to the butterfly canonical integral has been made the uniform asymptotic swallowtail approximation can now be regarded as a practical tool for the evaluation of oscillating integrals with four coalescing saddle points

on the algebraic parameters of convolutional codes with cyclic structure in this paper convolutional codes with cyclic structure will be investigated these codes can be understood as left principal ideals in a suitable skew polynomial ring it has been shown in 4 that only certain combinations of the algebraic parameters field size length dimension and forney indices can occur for such cyclic codes we will investigate whether all these combinations can indeed be realized by a suitable cyclic code and if so how to construct such a code a complete characterization and construction will be given for minimal cyclic codes it is derived from a detailed investigation of the units in the skew polynomial ring

a determinantal approach to sheffer sequences abstract in this paper by using the theory of riordan arrays and the relations between sheffer sequences and riordan arrays we give a determinantal definition for sheffer sequences based on this new definition some general properties of sheffer sequences are reproved and the determinantal representations of some well known sheffer sequences are presented

bijections and the riordan group one of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space in this paper we examine combinatorial sequences in the context of the riordan group various subgroups of the riordan group each give us a different view of the original sequence in many cases this leads to both a combinatorial interpretation and to eco rewriting rules in this paper we will concentrate on just four of the subgroups of the riordan group to demonstrate some of the possibilities of this approach

structural properties of riordan matrices and extending the matrices abstract we consider an infinite lower triangular matrix l l n k n k n 0 and a sequence n n n 0 called the a b sequence such that every element l n 1 k 1 except lying in column 0 can be expressed as l n 1 k 1 i 0 n k m i l n ai k bi 0 0 where a and b are integers with a b m 0 and b 0 this concept generalizes the a sequence of a riordan matrix as a result we explore several structural properties of riordan matrices by means of a b sequences in particular if b

on the weight distribution of convolutional codes detailed information about the weight distribution of a convolutional code is given by the adjacency matrix of the state diagram associated with a minimal realization of the code we will show that this matrix is an invariant of the code moreover it will be proven that codes with the same adjacency matrix have the same dimension and the same forney indices and finally that for one dimensional binary convolutional codes the adjacency matrix determines the code uniquely up to monomial equivalence

convolutional goppa codes in this correspondence we define convolutional goppa codes over algebraic curves and construct their corresponding dual codes examples over the projective line and over elliptic curves are described obtaining in particular some maximum distance separable mds convolutional codes

riordan matrices in the reciprocation of quadratic polynomials we iterate contractive one degree polynomials with coefficients in the ring k x of formal power series to calculate the reciprocal in k x of a quadratic polynomial doing this we meet the structure of riordan array we interpret certain changes of variable as a riordan array we finish the paper by using our techniques to find new ways to get known formulas for the sum of powers of natural numbers involving stirling and eulerian numbers

on the parametrization of conditioned invariant subspaces and observer theory the present paper is an in depth analysis of the set of conditioned invariant subspaces of a given observable pair c a we do this analysis in two different ways one based on polynomial models starting with a characterization obtained in p a fuhrmann linear operators and systems in hilbert space 1981 ieee trans automat control ac 26 1981 284 the other being a state space approach toeplitz operators projections in polynomial and rational models wiener hopf factorizations and factorization indices all appear and are tools in the characterizations we single out an important subclass of conditioned invariant subspaces namely the tight ones which already made an appearance in p a fuhrmann u helmke systems control lett 30 1997 217 a precursor of the present paper of particular importance for the study of the parametrization of the set of conditioned invariant subspaces of an observable pair c a is the structural map that associates with any reachable pair with only the input dimension constrained a uniquely determined conditioned invariant subspace the construction of this map uses polynomial models and the shift realization new objects the partial observability and reachability matrices are introduced which are needed for the state space characterizations kernel and image representations for conditioned invariant subspaces are derived uniqueness of a kernel representation of a conditioned invariant subspace is shown to be equivalent to tightness we pass on to an analysis and derivation of the kronecker hermite canonical form for full column rank rectangular polynomial matrices this extends the work of a e eckberg a characterization of linear systems via polynomial matrices and module theory ph d thesis mit cambridge ma 1974 g d forney siam j control optim 13 1973 493 and d hinrichsen h f munzner and d pratzel wolters systems

on cyclic codes of composite length and the minimum distance in an interesting paper prof c ding provided three constructions of cyclic codes of length being a product of two primes numerical data shows that many codes from these constructions are best cyclic codes of the same length and dimension over the same finite field however not much is known about these codes in this paper we explain some of the numerical data by developing a general method on cyclic codes of composite length and on estimating the minimum distance we also provide a general construction of cyclic codes of composite length which are related to ding s constructions numerical data shows that it produces many best cyclic codes as well finally we point out how these cyclic codes can be used to construct convolutional codes with large free distance

dual operators and lagrange inversion in several variables the original lagrange inversion formula which gives explicitly the inverse under composition of a formal series was obtained by lagrange in 1770 through formal computations involving logarithms of inlmite products see lagrange 121 there exists an extensive literature on various versions of the inversion formula obtained by algebraic analytic and combinatorial methods comtet 3 gives a survey and an ample bibliography during the last ten years there have appeared several papers on generalizations and new proofs of the inversion formula in 1974 abhyankar obtained an inversion formula using algebraic methods see bass connell and wright 11 several authors have related lagrange inversion with rota s finite operator calculus 151 for example joni 8 garsia and joni 4 hofhauer 7 and roman and rota 14 joyal and labelle used a combinatorial theory of formal series to obtain inversion formulas see 9 l i and viskov 161 used ideas from lie algebra theory to prove formulas similar to abhyankar s in the present paper we use operator methods to obtain inversion formulas in several variables that generalize the formulas of abhyankar joni and viskov we get our results studying an algebra with involution of linear operators on the algebra of formal laurent series in several indeterminates we also show that by means of an anti isomorphism of operator algebras which we call the operator bore1 transform the study of finite operator calculus can be reduced to the study of certain groups of operators on the algebra of formal laurent series our approach may be considered as an algebraic analogue of the complex variable methods used by good 5 to obtain the lagrange inversion formula as a consequence of the change of variables theorem for the