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

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

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

on the convergence rate of a quasi newton method for inverse eigenvalue problems in this paper we first note that the proof of the quadratic convergence of the quasi newton method as given in friedland nocedal and overton siam j numer anal 24 1987 pp 634 667 is incorrect then we give a correct proof of the convergence

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

strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems it is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable a classical result of ky fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions which implies that the eigenvalues are semismooth functions based on a recent result of the authors it is further proved in this paper that the eigenvalues of a symmetric matrix are strongly semismooth everywhere as an application it is demonstrated how this result can be used to analyze the quadratic convergence of newton s method for solving inverse eigenvalue problems ieps and generalized ieps with multiple eigenvalues

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

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

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

fuzzy and stochastic programming abstract fuzzy mathematical programming is considered on the basis of the concept of fuzzy variable 5 that is one of the most complete patterns of fuziness statements of problems under consideration are analogous to those in stochastic programming equivalent precise analogues are derived for these problems the relationship between fuzzy programming and stochastic programming is determined

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

numerical methods for inverse singular value problems3 two numerical methods one continuous and the other discrete are proposed for solving inverse singular value problems the first method consists of solving an ordinary differential equation obtained from an explicit calculation of the projected gradient of a certain objective function the second method generalizes an iterative process proposed originally by friedland nocedal and overton siam j numer anal 24 1987 pp 634 667 for solving inverse eigenvalue problems with the geometry understood from the first method it is shown that the second method also the method proposed by friedland nocedal and overton for inverse eigenvalue problems is a variation of the newton method while the continuous method is expected to converge globally at a slower rate in finding a stationary point of the objective function the discrete method is proved to converge locally at a quadratic rate if there is a solution some numerical examples are presented

a quadratically convergent algorithm for inverse generalized eigenvalue problems abstract in 2018 for inverse generalized symmetric eigenvalue problems a new quadratically convergent algorithm was discovered from simple matrix equations although this algorithm has some nice features compared with the standard newton s method it cannot be applied to multiple eigenvalues in this paper we propose a modified algorithm adapted to an arbitrary set of prescribed eigenvalues moreover we prove its quadratic convergence in a neighborhood of the solutions that satisfy a mild condition

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

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

a quadratically convergent algorithm for inverse eigenvalue problems with multiple eigenvalues abstract in 2017 for inverse symmetric eigenvalue problems a new quadratically convergent algorithm has been derived from simple matrix equations although this algorithm has some nice features compared with the other quadratically convergent methods it is not applied to multiple eigenvalues in this paper we improve this algorithm with the aid of an optimization problem for the eigenvectors associated with multiple eigenvalues the proposed algorithm is adapted to an arbitrary set of given eigenvalues the main contribution is our convergence theorem formulated in a different manner from previous work for the existing quadratically convergent methods our theorem ensures the quadratic convergence in a neighborhood of the solutions that satisfy a mild condition

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

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

an algorithm for symmetric generalized inverse eigenvalue problems abstract using qr like decomposition with column pivoting and least squares techniques we propose a new and efficient algorithm for solving symmetric generalized inverse eigenvalue problems and give its locally quadratic convergence analysis we also present some numerical experiments which illustrate the behaviour of our algorithm

approximate cayley transform methods for inverse eigenvalue problems and convergence analysis abstract based on an approximation to cayley transform we propose an approximate cayley transform method and its inexact version for solving inverse eigenvalue problems which has the advantage over other known methods in the sense that it avoids solving systems in obtaining the approximate eigenvectors under the nonsingular condition used in d sun and j sun 29 we show that the proposed methods converge at least superlinearly moreover numerical experiments are given which illustrate that comparing with the cayley transform methods our methods need much less inner iterations and cpu time

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

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

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

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

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

inverse eigenvalue problems a collection of inverse eigenvalue problems are identified and classified according to their characteristics current developments in both the theoretic and the algorithmic aspects are summarized and reviewed in this paper this exposition also reveals many open questions that deserve further study an extensive bibliography of pertinent literature is attached

a solution of the affine quadratic inverse eigenvalue problem abstract the quadratic inverse eigenvalue problem qiep is to find the three matrices m c and k given a set of numbers closed under complex conjugations such that these numbers become the eigenvalues of the quadratic pencil p 2 m c k the affine inverse quadratic eigenvalue problem aqiep is the qiep with an additional constraint that the coefficient matrices belong to an affine family that is these matrices are linear combinations of substructured matrices an affine family of matrices very often arise in vibration engineering modeling and analysis research on qiep and aqiep are still at developing stage in this paper we propose three methods and the associated mathematical theories for solving aqiep a newton method an alternating projections method and a hybrid method combining the two validity of these methods are illustrated with results on numerical experiments on a spring mass problem and comparisons are made with these three methods amongst themselves and with another newton method developed by elhay and ram 2002 12 the results of our experiments show that the hybrid method takes much smaller number of iterations and converges faster than any of these methods

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

a quadratically convergent algorithm based on matrix equations for inverse eigenvalue problems abstract we propose a quadratically convergent algorithm for inverse symmetric eigenvalue problems based on matrix equations the basic idea is seen in a recent study by ogita and aishima while they derive an efficient iterative refinement algorithm for symmetric eigenvalue problems using special matrix equations in other words this study is interpreted as a unified view on quadratically convergent algorithms for eigenvalue problems and inverse eigenvalue problems based on matrix equations to the best of our knowledge such a unified development of algorithms is provided for the first time since the proposed algorithm for the inverse eigenvalue problems can be regarded as the newton s method for the matrix equations the quadratic convergence is naturally proved our algorithm is interpreted as an improved version of the cayley transform method for the inverse eigenvalue problems although the cayley transform method is one of the effective iterative methods the cayley transform takes o n 3 arithmetic operations to produce an orthogonal matrix using a skew symmetric matrix in each iteration our algorithm can refine orthogonality without the cayley transform which reduces the operations in each iteration it is worth noting that our approach overcomes the limitation of the cayley transform method to the inverse standard eigenvalue problems resulting in an extension to inverse generalized eigenvalue problems