linear block codes over cyclic groups the main building block for the construction of a geometrically uniform coded modulation scheme is a subgroup of g sup i where g is a group generating a low dimensional signal constellation and i is an index set in this paper we study the properties of these subgroups when g is cyclic we exploit the fact that any cyclic group of q elements is isomorphic to the additive group of z sub q the ring of integers modulo q so that we can make use of concepts related to linearity our attention is focused mainly on indecomposable cyclic groups i e of prime power order since they are the elementary building blocks of any abelian group in analogy with the usual construction of linear codes over fields we define a generator matrix and a parity check matrix trellis construction and bounds on the minimum euclidean distance are also investigated some examples of coded modulation schemes based on this theory are also exhibited and their performance evaluated

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

kernel estimates for nonautonomous kolmogorov equations abstract using time dependent lyapunov functions we prove pointwise upper bounds for the heat kernels of some nonautonomous kolmogorov operators with possibly unbounded drift and diffusion coefficients

some properties of traces for stochastic and deterministic parabolic weighted sobolev spaces abstract the results presented provide information on the behavior near x1 0 for fixed t 0 in the sense of lp spaces with weights for functions u t u t x1 x x1 0 x r d 1 t 0 satisfying stochastic partial differential equations of type du t u t f t dt gk t dwkt with zero boundary condition on x1 0

minimality and observability of group systems abstract group systems are a generalization of willems type linear systems that are useful in error control coding it is shown that the basic ideas of willems s treatment of linear systems are easily generalized to linear systems over arbitrary rings and to group systems the interplay between systems behaviors and trellises evolution laws is discussed with respect to completeness minimality controllability and observability it is pointed out that for trellises of group systems and willems type linear systems minimality is essentially the same as observability the development is universal algebraic in nature and holds unconditionally for linear systems over the real numbers

global properties of invariant measures abstract we study global regularity properties of invariant measures associated with second order differential operators in r n under suitable conditions we prove global boundedness of the density sobolev regularity a harnack inequality and pointwise upper and lower bounds

some structural properties of convolutional codes over rings convolutional codes over rings have been motivated by phase modulated signals some structural properties of the generator matrices of such codes are presented successively stronger notions of the invertibility of generator matrices are studied and a new condition for a convolutional code over a ring to be systematic is given and shown to be equivalent to a condition given by massey and mittelholzer 1990 it is shown that a generator matrix that can be decomposed into a direct sum is basic minimal and noncatastrophic if and only if all generator matrices for the constituent codes are basic minimal and noncatastrophic respectively it is also shown that if a systematic generator matrix can be decomposed into a direct sum then all generator matrices of the constituent codes are systematic but that the converse does not hold some results on convolutional codes over z p sup e are obtained

dynamical systems and convolutional codes over finite abelian groups polynomial algebraic techniques have always played a central role in linear systems theory and also in the theory of convolutional codes we show how such techniques can be generalized to study systems and codes defined over abelian groups the systems are considered from the behavioral point of view as developed by willems in the 1980s and some of our results can be seen as extensions of willems results to group systems we also address a certain number of coding oriented questions and we propose concrete methods based on these algebraic techniques for the synthesis of encoders inverters and syndrome formers for codes over finite abelian groups

monotonic solutions of a perturbed quadratic fractional integral equation abstract we present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in c 0 1 the concept of a measure of noncompactness and a fixed point theorem due to darbo are the main tools in carrying out our proof finally we give an example for indicating the natural realizations of our abstract result presented in the paper

on a perturbed quadratic fractional integral equation of abel type we present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in c 0 1 our equation contains the famous chandrasekhar integral equation as a special case the concept of the measure of noncompactness related to monotonicity introduced by j banas and l olszowy and a fixed point theorem due to darbo are the main tools in carrying out our proof moreover we give an example for indicating the natural realizations of our abstract result presented in this paper

assignable polynomials to linear systems over von neumann regular rings abstract given a system a b over a commutative von neumann regular ring r it is proved that there exist a matrix f and a vector u such that the single input system a b f b u and the original system a b have the same module of reachable states and the same set of polynomials assignable by state feedback moreover there is a bijection between reachable states and assignable polynomials in the form of a certain isomorphism of r modules and the existence of this isomorphism for all systems actually characterizes von neumann regular rings finally the set of assignable polynomials to a single input system is completely described for arbitrary commutative rings which in the case of von neumann regular rings completes the study of assignable polynomials to multi input systems

feedback decoding of fixed point arithmetic convolutional codes convolutional codes defined over the integers modulo a power of two an arithmetic structure used for fixed point arithmetic computations employ well known binary convolutional codes as their underlying generators a recursive decoding technique that exploits binary expansion components of the code symbols uses any binary decoding algorithm valid for the underlying code

nonconcurrent error detection and correction in fault tolerant linear finite state machines in this paper we construct fault tolerant linear finite state machines lfsms in which error detection and correction can be performed nonconcurrently e g periodically more specifically by jointly choosing the state encoding constraints and the redundant dynamics of the fault tolerant lfsm we enable an external checker to detect and identify errors due to past faults based on the current possibly corrupted state of the lfsm the paper presents systematic constructions of fault tolerant lfsms based on a characterization of nonconcurrent error detection correction in terms of state encoding constraints and redundant dynamics in particular we develop a scheme that uses bose chaudhuri hocquenghem bch coding and obtains fault tolerant lfsms that require 2d additional state variables and have the ability to correct up to d errors in any state variable at any time step in the time interval consisting of the latest n time steps of operation the construction uses the minimum possible number of additional state variables and requires an error detecting correcting mechanism with computational complexity that is only linear in n

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

on a perturbed functional integral equation of urysohn type abstract we study the existence of monotonic solutions for a perturbed functional integral equation of urysohn type in the space of lebesgue integrable functions on an unbounded interval the technique associated with measures of noncompactness in both the weak and the strong sense and the darbo fixed point are the main tool to prove our main result

block coded m psk modulation over gf m channel codes where the redundancy is obtained not from parity symbols but from expanding the channel signal set are addressed they were initially proposed by g ungerboeck 1982 using a convolutional code here a block coding approach is given rate m m 1 coded 2 sup m 1 ary phase shift keying psk is considered the expanded signal set is given the structure of a finite field the code is defined by a square nonsingular circulant generator matrix over the field binary data are mapped on a dataword of the same length as the codewords over an additive subgroup of the field the codes using trellises are described and then the viterbi algorithm for decoding is applied the asymptotic coding gain ranges from 1 8 to 6 0 db for qpsk going from blocklength 3 to 12 for 8 psk the gain is from 0 7 to 3 0 db with blocklength 4 to 8 with only four states in the trellis codes of any length for qpsk and 8 psk are constructed each having an asymptotic coding gain of 3 0 db simulation results are presented it is found that the bit error rate performance at moderate signal to noise ratios is sensitive to the number of nearest and next nearest neighbors

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

the predictable degree property and row reducedness for systems over a finite ring motivated by applications in communications we consider linear discrete time systems over the finite ring z p r we solve the open problem of deriving a theory of row reduced representations for these systems we introduce a less restrictive form of representation than the adapted form introduced by fagnani and zampieri we call this form the composed form we define the concept of p predictable degree property and p row reduced we demonstrate that these concepts coupled with the composed form provide a natural setting that extends several classical results from the field case to the ring case in particular the classical rank test on the leading row coefficient matrix is generalized we show that any annihilator of a behavior mathfrak b of a pre specified degree is uniquely parametrized with finitely many coefficients in terms of a kernel representation in p row reduced composed form the underlying theory is the theory of reduced p basis for submodules of z q p r xi that is developed in this paper we show how to construct a p row reduced kernel representation in composed form by constructing a reduced p basis for the module mathfrak b perp

nonautonomous kolmogorov parabolic equations with unbounded coefficients we study a class of elliptic operators a with unbounded coeffi cients defined in i r d for some unbounded interval ir we prove that for any s 2 i the cauchy problem u s f 2 cb r d for the parabolic equation dtu au admits a unique bounded classical solution u this allows to associate an evolution family g t s with a in a natural way we study the main properties of this evolution family and prove gradient estimates for the function g t s f under suitable assumptions we show that there exists an evolution system of measures for g t s and we study the first properties of the extension of g t s to the l p spaces with respect to such measures

construction of linear ring codes for 6 psk the algebraic construction of linear codes over zm for m psk is considered for the case that m is the product of two distinct primes by means of the chinese remainder theorem crt such codes can be constructed from linear codes over the corresponding prime fields it is shown that for given component codes the crt composition yields psk codes that are at least as good and often much better than the codes obtained from the standard coset coding technique a lower bound on the minimum euclidean distance of 6 psk codes is derived that is based on the hamming weight distribution of the component codes

minimal trellises for block codes basic concepts in the study of trellises of block codes are defined it is shown that minimal proper trellises exists for all block codes bounds on the size of such trellises are established these bounds are shown to be exact for maximum distance separable codes and nearly so for perfect codes

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

on a polynomial scalar perturbation of a schrodinger system in lp spaces abstract in the paper 10 the l p realization l p of the matrix schrodinger operator l u d i v q u v u was studied the generation of a semigroup in l p r d c m and characterization of the domain d l p has been established in this paper we perturb the operator l p by a scalar potential belonging to a class including all polynomials and show that still we have a strongly continuous semigroup on l p r d c m with domain embedded in w 2 p r d c m we also study the analyticity compactness positivity and ultracontractivity of the semigroup and prove gaussian kernel estimates further kernel estimates and asymptotic behaviour of eigenvalues of the matrix schrodinger operator are investigated

a matrix based list decoding algorithm for linear codes over integer residue rings abstract in this paper we address the problem of list decoding of linear codes over an integer residue ring z q where q is a power of a prime p the proposed procedure exploits a particular matrix representation of the linear code over z p r called the standard form and the p adic expansion of the to be decoded vector in particular we focus on the erasure channel in which the location of the errors is known this problem then boils down to solving a system of linear equations with coefficients in z p r from the parity check matrix representations of the code we recursively select certain equations that a codeword must satisfy and have coefficients only in the field p r 1 z p r this yields a step by step procedure obtaining a list of the closest codewords to a given received vector with some of its coordinates erased we show that such an algorithm actually computes all possible erased coordinates that is the provided list is minimal

convolutional codes over groups the basic algebraic structure theory of convolutional codes and their trellises is developed simultaneously for codes over groups rings and fields the first part which covers fundamental notions such as minimality and observability is semi tutorial in that most definitions are already standard within the modern behavioral theory as are some of the formally stated results however some of the pivotal results emphasizing the role of observability as the basic well behavedness condition for codes are new and several previous results are given simplified proofs the usefulness of the behavioral approach even for convolutional codes over fields is demonstrated by a new minimality test for encoders as well as by the straightforward derivation of some known minimality criteria for generator matrices from the basic minimality criteria for group trellises the second part of the paper deals with issues that are specific to codes over rings and groups the main result is a concise characterization the first such of those groups that can appear as the branch group of any group trellis it is further shown how such groups are presented by shift registers a new large class of noncommutative convolutional codes is also given

mds convolutional codes over a finite ring in this paper we present an analog of the singleton bound for convolutional codes over finite rings codes meeting that bound are called mds a constructive method for the mds codes restricted to free codes is presented via cyclic codes over finite rings examples of nonfree mds codes are provided

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

the z sub 4 linearity of kerdock preparata goethals and related codes certain notorious nonlinear binary codes contain more codewords than any known linear code these include the codes constructed by nordstrom robinson 1967 kerdock 1972 preparata 1968 goethals 1974 and delsarte goethals 1975 it is shown here that all these codes can be very simply constructed as binary images under the gray map of linear codes over z sub 4 the integers mod 4 although this requires a slight modification of the preparata and goethals codes the construction implies that all these binary codes are distance invariant duality in the z sub 4 domain implies that the binary images have dual weight distributions the kerdock and preparata codes are duals over z sub 4 and the nordstrom robinson code is self dual which explains why their weight distributions are dual to each other the kerdock and preparata codes are z sub 4 analogues of first order reed muller and extended hamming codes respectively all these codes are extended cyclic codes over z sub 4 which greatly simplifies encoding and decoding an algebraic hard decision decoding algorithm is given for the preparata code and a hadamard transform soft decision decoding algorithm for the i kerdock code binary first and second order reed muller codes are also linear over z sub 4 but extended hamming codes of length n spl ges 32 and the golay code are not using z sub 4 linearity a new family of distance regular graphs are constructed on the cosets of the preparata code

the weighted coordinates bound and trellis complexity of block codes and periodic packings weighted entropy profiles and a new bound the weighted coordinates bound on the state complexity profile of block codes are presented these profiles and bound generalize the notion of dimension length profile dlp and entropy length profile elf to block codes whose symbols are not drawn from a common alphabet set and in particular group codes likewise the new bound may improve upon the dlp and elf bounds fur linear and nonlinear block codes over fields however it seems that the major contribution of the proposed bound is to the study of trellis complexity of block codes whose different coordinates are drawn from different alphabet sets the label code of lattice and nonlattice periodic packings usually has this property the construction of a trellis diagram for a lattice and some related bounds are generalized to periodic packings by introducing the fundamental module of the packing and using the new bound on the state complexity profile this generalization is limited to a given coordinate system we show that any bounds on the trellis structure of block codes and in particular the bound presented in this work are applicable to periodic packings

multilevel construction of block and trellis group codes the theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes in this paper we study the problem of building multilevel group codes i e codes obtained combining separate coding at different levels in such a way that the resulting code is a group code a construction leading to multilevel group codes for semi direct and direct products is illustrated the codes that can be obtained in this way are identified new geometrically uniform euclidean space codes obtained from multilevel codes over abelian and nonabelian groups are presented

on the hamming distance of linear codes over a finite chain ring let r be a finite chain ring e g a galois ring k its residue field and c a linear code over r we prove that d c the hamming distance of c is d c spl alpha where c spl alpha is a submodule quotient spl alpha is a certain element of r and denotes the canonical projection to k these two codes also have the same set of minimal codeword supports we explicitly construct a generator matrix polynomial of c spl alpha from the generator matrix polynomials of c we show that in general d c spl les d c with equality for free codes i e for free r submodules of r sup n and in particular for hensel lifts of cyclic codes over k most of the codes over rings described in the literature fall into this class we characterize minimum distance separable mds codes over r and prove several analogs of properties of mds codes over finite fields we compute the hamming weight enumerator of a free mds code over r

on a quadratic fractional hammerstein volterra integral equation with linear modification of the argument abstract we study the existence of monotonic solutions of a quadratic fractional hammerstein volterra integral equation with linear modification of the argument the quadratic integral equation studied below contains as a special case numerous integral equations encountered in the theory of radiative transfer and in the kinetic theory of gases we show that the quadratic fractional hammerstein volterra integral equation with linear modification of the argument has a monotonic solution in the banach space of all real functions defined and continuous on a bounded and closed interval the concept of measure of noncompactness and a fixed point theorem due to darbo are the main tools in carrying out our proof

sharp upper bounds for the density of some invariant measures we prove sharp upper bounds for invariant measures of markov processes in n associated with second order elliptic differential operators with unbounded coefficients

dimension length profiles and trellis complexity of linear block codes this semi tutorial paper discusses the connections between the dimension length profile dlp of a linear code which is essentially the same as its generalized hamming weight hierarchy and the complexity of its minimal trellis diagram these connections are close and deep dlp duality is closely related to trellis duality the dlp of a code gives tight bounds on its state and branch complexity profiles under any coordinate ordering these bounds can often be met a maximum distance separable mds code is characterized by a certain extremal dlp from which the main properties of mds codes are easily derived the simplicity and generality of these interrelationships are emphasized

coset codes ii binary lattices and related codes for pt i see ibid vol 34 no 5 p 1123 51 1988 the family of barnes wall lattices including d sub 4 and e sub 8 of lengths n 2 sup n and their principal sublattices which are useful in constructing coset codes are generated by iteration of a simple construction called the squaring construction the closely related reed muller codes are generated by the same construction the principal properties of these codes and lattices are consequences of the general properties of iterated squaring constructions which also exhibit the interrelationships between codes and lattices of different lengths an extension called the cubing construction generates good codes and lattices of lengths n 3 2 sup n including the golay code and leech lattice with the use of special bases for 8 space another related construction generates the nordstrom robinson code and an analogous 16 dimensional nonlattice packing these constructions are represented by trellis diagrams that display their structure and interrelationships and that lead to efficient maximum likelihood decoding algorithms

quaternary convolutional codes from linear block codes over galois rings from a linear block code b over the galois ring gr 4 m with a k times n generator matrix and minimum hamming distance d a rate k n convolutional code over the ring z4 with squared euclidean free distance at least 2d and a nonrecursive encoder with memory at most m 1 is constructed when the generator matrix of b is systematic the convolutional encoder is systematic basic noncatastrophic and minimal long codes constructed in this manner are shown to satisfy a gilbert varshnmov bound

two 16 state rate r 2 4 trellis codes whose free distances meet the heller bound for rate r 1 2 convolutional codes with 16 states there exists a gap between heller s 1968 upper bound on the free distance and its optimal value this article reports on the construction of 16 state binary rate r 2 4 nonlinear trellis and convolutional codes having d sub free 8 a free distance that meets the heller upper bound the nonlinear trellis code is constructed from a 16 state rate r 1 2 convolutional code over z sub 4 using the gray map to obtain a binary code both convolutional codes are obtained by computer search systematic feedback encoders for both codes are potential candidates for use in combination with iterative decoding regarded as modulation codes for 4 psk these codes have free squared euclidean distance d sub e free sup 2 16

space time coded systems using continuous phase modulation we develop space time trellis coded stc schemes using continuous phase modulation cpm we employ the rimoldi model of cpm to create a decomposed model of stc cpm the decomposition separates the coding from the modulation the space time encoding and the inherent cpm encoding is combined into a single trellis encoder on the ring of integers modulo this is followed by a bank of memoryless modulators the model allows the search for good space time codes to take into account the inherent encoding of the modulation

the oseen navier stokes flow in the exterior of a rotating obstacle the non autonomous case consider the navier stokes flow past a rotating obstacle with a general time dependent angular velocity and a time dependent outflow condition at infinity sometimes called an oseen condition by a suitable change of coordinates the problem is transformed to an non autonomous problem with unbounded drift terms on a fixed exterior domain r d it is shown that the solution to the linearized problem is governed by a strongly continuous evolution system ft t s gts 0 on l p for 1 p 1 moreover l p l q smoothing properties and gradient estimates of t t s 0 st are obtained these results are the key ingredients to show local in time existence of mild solutions to the full nonlinear problem for pd and initial value in l p

lower bounds on trellis complexity of block codes the trellis state complexity s of a linear block code is defined as the logarithm of the maximum number of states in its minimal trellis we present a new lower bound on the state complexity of linear codes which includes most of the existing bounds as special cases the new bound is obtained by dividing the time axis for the code into several sections of varying lengths as opposed to the division into two sections the past and the future employed in the well known dlp bounds for a large number of codes this results in a considerable improvement upon the dlp bound moreover we generalize the new bound to nonlinear codes and introduce several alternative techniques for lower bounding the trellis complexity based on the distance spectrum and other combinatorial properties of the code we also show how the general ideas developed in this paper may be employed to lower bound the maximum and the total number of branches in the trellis leading to considerably tighter bounds on these quantities furthermore the asymptotic behavior of the new bounds is investigated and shown to improve upon the previously known asymptotic estimates of trellis state complexity

existence of nondecreasing solutions of a quadratic integral equation of volterra type nondecreasing solutions of the quadratic integral equations is an interesting subject in the area of mathematics sciences and engineering using the theory of measures of noncompactness and applying a new method we prove the existence of nondecreasing solutions of a quadratic integral equation of volterra type in c i

on minimality of convolutional ring encoders convolutional codes are considered with code sequences modeled as semi infinite laurent series it is well known that a convolutional code c over a finite group g has a minimal trellis representation that can be derived from code sequences it is also well known that for the case that g is a finite field any polynomial encoder of c can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis in this paper we seek to extend this result to the finite ring case g bbzpr by introducing a so called ldquo p encoderrdquo we show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over bbzpr to produce a particular type of p encoder ldquominimal p encoderrdquo whose controller canonical realization is a minimal trellis with nonlinear features the minimum number of trellis states is then expressed as p gamma where gamma is the sum of the row degrees of the minimal p encoder in particular we show that any convolutional code over bbzpr admits a delay free p encoder which implies the novel result that delay freeness is not a property of the code but of the encoder just as in the field case we conjecture that a similar result holds with respect to catastrophicity i e any catastrophic convolutional code over bbzpr admits a noncatastrophic p encoder

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

the dynamics of group codes state spaces trellis diagrams and canonical encoders a group code c over a group g is a set of sequences of group elements that itself forms a group under a component wise group operation a group code has a well defined state space sigma sub k at each time k each code sequence passes through a well defined state sequence the set of all state sequences is also a group code the state code of c the state code defines an essentially unique minimal realization of c the trellis diagram of c is defined by the state code of c and by labels associated with each state transition the set of all label sequences forms a group code the label code of c which is isomorphic to the state code of c if c is complete and strongly controllable then a minimal encoder in controller canonical feedbackfree form may be constructed from certain sets of shortest possible code sequences called granules the size of the state space sigma sub k is equal to the size of the state space of this canonical encoder which is given by a decomposition of the input groups of c at each time k if c is time invariant and nu controllable then mod sigma sub k mod pi sub 1

system theoretic properties of convolutional codes over rings convolutional codes over rings are particularly suitable for representing codes over phase modulation signals in order to develop a complete structural analysis of this class of codes it is necessary to study rational matrices over rings which constitutes the generator matrices encoders for such convolutional codes noncatastrophic minimal systematic and basic generator matrices are introduced and characterized by using a canonical form for polynomial matrices over rings finally some classes of convolutional codes defined according to the generator matrix they admit are introduced and analyzed from a system theoretic point of view

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

geometrically uniform codes a signal space code c is defined as geometrically uniform if for any two code sequences in c there exists an isometry that maps one sequence into the other while leaving the code c invariant geometrical uniformity a strong kind of symmetry implies such properties as a the distance profiles from code sequences in c to all other code sequences are all the same and b all voronoi regions of code sequences in c have the same shape it is stronger than ungerboeck zehavi wolf symmetry or calderbank sloane regularity nonetheless most known good classes of signal space codes are shown to be generalized coset codes and therefore geometrically uniform including a lattice type trellis codes based on lattice partitions lambda lambda such that z sup n lambda lambda 4z sup n is a lattice partition chain and b phase shift keying psk type trellis codes based on up to four way partitions of a 2 sup n psk signal set

design of ring convolutional trellis codes for map decoding of mpeg 4 images we propose a trellis coded modulation system using continuous phase frequency shift keying cpfsk and ring convolutional codes for transmitting the bits generated by an embedded zerotree wavelet encoder improved performance is achieved by using maximum a posteriori decoding of the zerotree symbols and ring convolutional trellis codes are determined for this decoding method the cpfsk transmitter is decomposed into a memoryless modulator and a continuous phase encoder over the ring of integers modulo 4 the latter is combined with a polynomial convolutional encoder over the same ring in the code design process a search is made of the combined trellis where the branch metrics are modified to include the source transition matrix simulation results of image transmission are provided using the optimized system including mismatched channel cases

minimal and systematic convolutional codes over finite abelian groups abstract convolutional codes over abelian groups provide an effective theoretical framework for the analysis of some classes of tcm codes the encoder synthesis for this class of codes is not as simple as in the binary case since minimal encoders in the group case might be necessarily nonlinear in this contribution an algorithmic method testing whether a convolutional code over an abelian group admits a systematic or a minimal homomorphic encoder is provided this test consists in verifying whether a subgroup splits in a group through this method the class of codes admitting systematic encoders and the class of codes admitting minimal encoders can be compared finally this test is applied to some examples of practical relevance

a linear algebra approach to minimal convolutional encoders the authors review the work of g d forney jr on the algebraic structure of convolutional encoders upon which some new results regarding minimal convolutional encoders rest an example is given of a basic convolutional encoding matrix whose number of abstract states is minimal over all equivalent encoding matrices however this encoding matrix can be realized with a minimal number of memory elements neither in controller canonical form nor in observer canonical form thus this encoding matrix is not minimal according to forney s definition of a minimal encoder to resolve this difficulty the following three minimality criteria are introduced minimal basic encoding matrix minimal encoding matrix and minimal encoder it is shown that all minimal basic encoding matrices are minimal and that there exist minimal encoding matrices that are not minimal basic several equivalent conditions are given for an encoding matrix to be minimal it is proven that the constraint lengths of two equivalent minimal basic encoding matrices are equal one by one up to a rearrangement all results are proven using only elementary linear algebra

on monotonic solutions of an integral equation of volterra type using a technique associated with measures of noncompactness we prove the existence of nondecreasing solutions to integral equations of volterra type in c 0 1

minimal grobner bases and the predictable leading monomial property abstract we focus on grobner bases for modules of univariate polynomial vectors over a ring we identify a useful property the predictable leading monomial plm property that is shared by minimal grobner bases of modules in f x q no matter what positional term order is used the plm property is useful in a range of applications and can be seen as a strengthening of the wellknown predictable degree property row reducedness a terminology introduced by forney in the 70 s because of the presence of zero divisors minimal grobner bases over a finite ring of the type z p r where p is a prime integer and r is an integer 1 do not necessarily have the plm property in this paper we show how to derive from an ordered minimal grobner basis a so called minimal grobner p basis that does have a plm property we demonstrate that minimal grobner p bases lend themselves particularly well to derive minimal realization parametrizations over z p r applications are in coding and sequences over z p r

concatenated linear systems over rings and their application to construction of concatenated families of convolutional codes abstract we present a generalization of the theory of concatenated linear systems to commutative rings with identity moreover we highlight sufficient conditions to obtain reachable and observable concatenated linear systems this approach provides us with minimal input state output representations by means of which we can construct observable concatenated families of convolutional codes with different parameters over some particular rings this work focuses on the characterization of models of serial systematic serial and parallel concatenation

serially concatenated continuous phase modulation with convolutional codes over rings in this paper we investigate serially concatenated continuous phase modulation sccpm with convolutional codes cc over rings the transmitted signals are disturbed by additive white gaussian noise the properties for systems with both infinite and finite block lengths are investigated for an infinite length system we check the convergence threshold using the extrinsic information transfer chart for a finite length system we use union bound techniques to estimate the error floors in the union bound analysis we consider both the order and the position of nonzero permuted symbols a simple method for determining a cpm error event through the sum of the input symbol sequence is shown thus we can determine if the output symbol sequence of an error event in the ring cc can form an error event in cpm two properties concerning the interleaver gain ig are investigated a recursive search algorithm for the maximal ig is shown compared with previous sccpm with a binary cc the proposed system shows an improvement concerning the convergence threshold or error floors

an efficient algorithm for constructing minimal trellises for codes over finite abelian groups we present an efficient algorithm for computing the minimal trellis for a group code over a finite abelian group given a generator matrix for the code we also show how to compute a succinct representation of the minimal trellis for such a code and present algorithms that use this information to compute efficiently local descriptions of the minimal trellis this extends the work of kschischang and sorokine see ibid vol 41 no 6 p 1926 37 1995 who treated the case of linear codes over fields an important application of our algorithms is to the construction of minimal trellises for lattices a key step in our work is handling codes over cyclic groups c sub p spl alpha where p is a prime such a code can be viewed as a module over the ring z sub p spl alpha because of the presence of zero divisors in the ring modules do not share the useful properties of vector spaces we get around this difficulty by restricting the notion of linear combination to a p linear combination and by introducing the notion of a p generator sequence which enjoys properties similar to those of a generator matrix for a vector space

the capacity of finite abelian group codes over symmetric memoryless channels the capacity of finite abelian group codes over symmetric memoryless channels is determined for certain important examples such as m psk constellations over additive white gaussian noise awgn channels with m a prime power it is shown that this capacity coincides with the shannon capacity i e there is no loss in capacity using group codes this had previously been known for binary linear codes used over binary input output symmetric memoryless channels on the other hand a counterexample involving a three dimensional geometrically uniform constellation is presented in which the use of abelian group codes leads to a loss in capacity the error exponent of the average group code is determined and it is shown to be bounded away from the random coding error exponent at low rates for finite abelian groups not admitting galois field structure

solvability of volterra stieltjes operator integral equations and their applications abstract we investigate a class of operator integral equations of volterra stieltjes type and we study the solvability of those equations in the space of continuous functions equations in question create a generalization of numerous integral equations considered in nonlinear analysis the main tool used in our considerations is the technique associated with measures of noncompactness we show the applicability of our existence result in the study of a few integral equations of volterra type