parallel concatenated convolutional codes from linear systems theory viewpoint abstract the aim of this work is to characterize two models of concatenated convolutional codes based on the theory of linear systems the problem we consider can be viewed as the study of composite linear system from the classical control theory or as the interconnection from the behavioral system viewpoint in this paper we provide an input state output representation of both models and introduce some conditions for such representations to be both controllable and observable we also introduce a lower bound on their free distances and the column distances

unequal error protection for convolutional codes in this correspondence unequal error correcting capabilities of convolutional codes are studied for errors in the information symbols and code symbols the free input and output distances respectively serve as unequal counterparts to the free distance when communication takes place close to or above the channel capacity the error bursts tend to be long and the free distance is not any longer useful as the measure of the error correcting capability thus the active burst distance for a given output and the active burst distance for a given input are introduced as unequal counterparts to the active burst distance and improved estimates of the unequal error correcting capabilities of convolutional codes are obtained and illustrated by examples finally it is shown how to obtain unequal error protection for both information and code symbols using woven convolutional codes

a lower bound on the cutoff rate for gaussian channels with peak limited inputs a lower bound on the cutoff rate r sub o of a dispersive gaussian channel with peak rather than average power limited input is evaluated and compared to the lower bound on capacity under the same input constraints it is assumed that the input symbols are independent and uniformly distributed both bounds exhibit the same behavior as a function of an appropriately defined signal to noise ratio snr with a discrepancy of 1 68 db to the disadvantage of r sub o at high snrs the lower bound on r sub o is evaluated for peak and slope limited signals that model magnetic recording systems and is compared to the lower bound on capacity

low complexity decoding of partial unit memory codes on precoded partial response channels cosets of convolutional codes can be used to obtain large free euclidean distances and short maximum zero run lengths at the output of the 1 d partial response channel prc we present a new soft decision decoding technique for cosets of convolutional codes on the preceded 1 d prc the decoding technique is especially well suited for cosets of partial unit memory pum convolutional codes a connection between the decoder trellises for cosets of block codes and pum codes on the 1 d prc is exploited to obtain a small number of operations per decoded information bit we prove that the new decoding technique needs fewer operations than the viterbi algorithm for any n n r r spl ges 1 pum code coset with n spl les 2 sup v r where v is the constraint length it is also indicated how to prove the same result for other classes of codes for many of the best known pum code cosets the new decoding technique requires both fewer operations per decoded information bit and smaller path memories than the viterbi algorithm needs for comparable cosets of punctured convolutional codes

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

distance spectra of convolutional codes over partial response channels this paper is concerned with the well known coding technique that uses cosets of convolutional codes along with precoding to increase the free squared euclidean distance at the output of a partial response channel the focus is on totally trellis matched codes for which the decoding trellis has the fewest number of states codes totally trellis matched to the extended partial response class 4 epr4 and the doubly extended partial response class 4 e sup 2 pr4 channels for a variety of code rates and overall constraint lengths are sought for with respect to the squared euclidean distance spectrum criterion furthermore some previously published codes are shown to contain flawed codewords and are replaced with new ones

capacity of the discrete time gaussian channel with intersymbol interference the discrete time gaussian channel with intersymbol interference isi where the inputs are subject to a per symbol average energy constraint is considered the capacity of this channel is derived by means of a hypothetical channel model called the n circular gaussian channel ncgc whose capacity is readily derived using the theory of the discrete fourier transform the results obtained for the ncgc are used further to prove that in the limit of increasing block length n the capacity of the discrete time gaussian channel dtgc with isi using a per block average energy input constraint n block dtgc is indeed also the capacity when using the per symbol average energy constraint

dc free coset codes a class of block coset codes with disparity and run length constraints are studied they are particularly well suited for high speed optical fiber links and similar channels where dc free pulse formats channel error control and low complexity encoder decoder implementations are required the codes are derived by partitioning linear block codes the encoder and decoder structures are the same as those of linear block codes with only slight modifications a special class of dc free coset block codes are derived from bch codes with specified bounds on minimum distance disparity and run length the codes have low disparity levels a small running digital sum and good error correcting capabilities

generalized concatenation of encoded tamed frequency modulation a novel construction for encoded tamed frequency modulation tfm is introduced which is based on the principles of generalized concatenation the inner tfm is partitioned into nested subsystems which increases the free euclidean distances in order to obtain a large distance among the nested tfm subsystems the scrambler matrices have to be computed which transfer the original tfm into the equivalent tfm with better partitioning properties then outer convolutional codes with different error correcting capabilities are used to protect the partitioning the new concatenated and generalized concatenated constructions were simulated in an additive white gaussian noise channel a multistep decoding algorithm based on soft output demodulation was used we present various simulation results which show a significant coding gain in comparison with the best known trellis codes having the same trellis state complexity

new precoding for intersymbol interference cancellation using nonmaximally decimated multirate filterbanks with ideal fir equalizers in this paper we propose a new precoding method for intersymbol interference isi cancellation by using non maximally decimated multirate filterbanks unlike the existing preceding methods such as the th and trellis precodings the new preceding i may be independent of the isi channel ii is linear and does not have to implement any modulo operation iii gives the ideal fir equalization at the receiver for any fir isi channel including spectral null channels iv expands the transmission bandwidth in a minimum amount the precoding is built on non maximally decimated multirate filterbanks based on multirate filterbank theory we present a necessary and sufficient condition on an fir isi transfer function in terms of its zero set such that there is a linear fir n spl times m precoder so that an ideal fir equalizer exists where the integers k and n are arbitrarily fixed the condition is easy to check as a consequence of the condition for any given fir isi transfer function not identically 0 there always exist such linear fir precoders moreover for almost all given fir isi transfer functions there exist linear fir precoders with size n spl times n 1 i e the bandwidth is expanded by 1 n in addition to the conditions on the isi transfer functions a method for the design of the linear fir precoders and the ideal fir equalizers is also given numerical examples are presented to illustrate the theory

on the time evolution of macroscopic systems autodiffusion a basse densite et theoreme de lanford correlations dans l etat initial auto diffusion a densite finie le gaz de lorentz et le theoreme de bunimovich et sinai

optimization of signal sets for partial response channels i numerical techniques given a linear time invariant discrete time channel the problem of constructing n input signals of finite length k that maximize minimum l sub 2 distance between pairs of outputs is considered two constraints on the input signals are considered a power constraint on each of the n inputs hard constraint and an average power constraint over the entire set of inputs soft constraint the hard constraint problem is equivalent to packing n points in an ellipsoid in min k n 1 dimensions to maximize the minimum euclidean distance between pairs of points gradient based numerical algorithms and a constructive technique based on dense lattices are used to find locally optimal solutions to the preceding signal design problems two numerical examples are shown for which the average spectrum of an optimized signal set resembles the water pouring spectrum that achieves shannon capacity assuming additive white gaussian noise

coset codes for partial response channels or coset codes with spectral nulls known coset codes are adapted for use on partial response channels or to generate signals with spectral nulls by using coset precoding and running digital sum feedback any desired tradeoff can be achieved between the power and spectra of the relevant sequences up to the optimum tradeoff possible a fundamental theorem specifying this optimum tradeoff is given a maximum likelihood sequence estimation mlse decoder for the original code may be used for the adapted code and such a decoder then attains the minimum squared distance of the original code these methods sometimes generate codes with greater minimum squared distance than that of the original code this distance can be attained by augmented decoders although such decoders inherently require long decoding delays and may be subjected to quasi catastrophic error propagation the authors conclude that at least for sequences supporting large numbers of bits per symbol coset codes can be adapted to achieve effectively the same performance and complexity on partial response channels or for sequences with spectral nulls as they do in the ordinary memoryless case

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

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

cosets of convolutional codes with least possible maximum zero and one run lengths a communication or storage system may use a coset of a binary convolutional code for both symbol synchronization and error control to facilitate symbol synchronization the coset must have a short maximum zero run length l sub max general upper and lower bounds on l sub max were given previously by hole in this correspondence we use these bounds to identify which convolutional codes have cosets with short l sub max for such a code we then show how to determine a coset with the least possible l sub max among all cosets of the code exact expressions for the least possible l sub max of convolutional code cosets are given and examples of such cosets with large free distances are tabulated bounds on l sub max for cosets of block codes are also provided it is indicated how to tighten the bounds for block codes satisfying the one way chain condition we show that the cosets obtained from traditional high rate block code constructions have larger l sub max than cosets of convolutional codes with approximately the same rates in some systems the convolutional code cosets must have short maximum one run lengths as well as short maximum zero run lengths to avoid loss of symbol synchronization it is shown how to determine convolutional codes whose cosets with least possible maximum zero run lengths also have least possible maximum one run lengths

a general error correcting code construction for run length limited binary channels a general method is presented for constructing a single error correcting run length limited code for the class of input restricted binary symmetric channels described by the parameters d k where d is the minimum and k the maximum number of consecutive zeros in any allowable channel input sequence upper and lower bounds on the rates of these codes are derived

new short constraint length convolutional code constructions for selected rational rates corresp new short constraint length convolutional code constructions are tabulated for rates r n k n k 1 2 cdots n 1 with n 2 3 cdots 8 and for constraint lengths k 3 4 cdots 8 these codes have been determined by iterative search based upon a criterion of optimizing the free distance profile specifically these codes maximize the free distance d f while minimizing the number of adversaries in the distance or weight spectrum in several instances we demonstrate the superiority of these codes over previously published code constructions at the same rate and constraint length these codes are expected to have a number of applications including combined source channel coding schemes as well as coding for burst or impulsive noise channels

error trellises for convolutional codes i construction an error trellis is a directed graph that represents all the sequences belonging to the coset which contains the symbol by symbol detected version of a given received sequence a modular construction of error trellises for an n k convolutional code over gf q is presented the trellis is designed on the basis of partitioning the scalar check matrix of the code into submatrices of l rows accompanied with a corresponding segmentation of the syndrome the value of the design parameter l is an essentially unconstrained multiple of n k for all the cosets of the code the sections of the error trellis are drawn from a collection of only q sup l modules the module for each section is determined by the value of the associated syndrome segment in case the construction is based on a basic polynomial check matrix either canonical or noncanonical then the error trellis is minimal in the sense that spl sigma spl les spl mu where spl sigma is the dimension of the state space of the trellis and spl mu is the constraint length of a canonical generator matrix for the code for basic check matrices with delay free columns the inequality reduces to spl sigma spl mu

transport in polygonal billiards we present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain this billiard is a system with no exponential instability our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion our conclusions are drawn from the numerical evaluation of the mean square displacement the velocity autocorrelation function and its spectral analysis furthermore we analyze the properties of the incoherent scattering function the super burnett coefficient seems to be ill defined in all systems the multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported finally we study the heat conduction in our polygonal chain

a modified trellis coding technique for partial response channels the problem of trellis coding for multilevel baseband transmission over partial response channels with transfer polynomials of the form 1 or d sup n is addressed the novel method presented here accounts for the channel memory by using multidimensional signal sets and partitioning the signal set present at the noiseless channel output it is shown that this coding technique can be viewed as a generalization of a well known procedure for binary signaling the concatenation of convolutional codes and inner block codes that are tuned to the channel polynomial it results in high coding gains with moderate complexity if some bandwidth expansion is accepted

convolutional codes using finite field wavelets time varying codes and more this paper introduces a procedure for constructing convolutional codes using finite field wavelets this provides novel insight into the study of convolutional codes and permits the design of the new convolutional codes that is not possible by conventional methods exploiting algebraic properties of the wavelet codes we show that a rate k l wavelet convolutional encoder is a basic encoder that is noncatastrophic in addition we prove that any rate 1 l wavelet convolutional encoder is minimal basic and that every l band orthogonal wavelet system generates a rate 1 l self orthogonal code as an application of wavelet convolutional codes we construct time varying convolutional codes these codes have unique trellis structures that result in fast and low computational complexity decoding algorithms as examples we present some time varying wavelet convolutional codes that can be decoded faster than comparable time invariant convolutional codes we construct 16and 32 state time varying wavelet convolutional codes with minimum free distances of seven and eight respectively these codes have the same minimum free distance as the best time invariant codes of the same rate and state complexity but they can be decoded almost twice as fast we show that a 32 state time varying wavelet convolutional code is superior to the lauer code in performance while having almost the same decoding complexity although the 32 state wavelet code is inferior to the 16 state golay convolutional code as far as computational complexity it outperforms this code in the performance we also prove that orthogonal filterbanks generate self dual time varying codes we give a design for doubly even self dual time varying convolutional codes by imposing some constraints on the filters that define the l band orthogonal wavelets as another application of wavelet convolutional codes we propose a new scheme for generating rate adaptive codes these codes have the property that multiple rates of the code can be decoded on one trellis and its subtrellises

an estimate for lipschitz constants of metric projections abstract the main purpose of the paper is to give a global estimate for lipschitz constants of metric projections in a p uniformly convex and q uniformly smooth banach space

block and trellis codes for the binary 1 d partial response channel with simple maximum likelihood decoders we consider block and convolutional codes for improving the reliability of data transmission over the binary precoded noisy 1 d partial response channel we concentrate on a class of codes for which the maximum likelihood decoder matched to the encoder precoder and the channel has the same trellis structure as the encoder thus doubling the number of states due to the channel memory is avoided we show that the necessary and sufficient condition to belong to this class is that all codewords be of the same parity the reed muller and golay codes belong to this class

improved coding techniques for preceded partial response channels a coset of a convolutional code may be used to generate a zero run length limited trellis code for a 1 d partial response channel the free squared euclidean distance d sub free sup 2 at the channel output is lower bounded by the free hamming distance of the convolutional code the lower bound suggests the use of a convolutional code with maximal free hamming distance d sub max r n for given rate r and number of decoder states n in this paper we present cosets of convolutional codes that generate trellis codes with d sub free sup 2 d sub max r n for rates 1 5 spl les r spl les 7 9 and d sub free sup 2 d sub max r n for r 13 16 29 32 61 64 the tabulated convolutional codes with r spl les 7 9 were not optimized for hamming distance instead a computer search was used to determine cosets of convolutional codes that exploit the memory of the 1 d channel to increase d sub free sup 2 at the channel output the search was limited by only considering cosets with certain structural properties the r spl ges 13 16 codes were obtained using a new construction technique for convolutional codes with free hamming distance 4 newly developed bounds on the maximum zero run lengths of cosets were used to ensure a short maximum run length at the 1 d channel output

proximal analysis in reflexive smooth banach spaces abstract in the present paper we introduce and study a new proximal normal cone in reflexive banach spaces in terms of a generalized projection operator two new variants of generalized proximal subdifferentials are also introduced in reflexive smooth banach spaces the density theorem for both proximal subdifferentials has been proved in p uniformly convex and q uniformly smooth banach spaces various important properties and applications of our concepts are also proved

minimal and canonical rational generator matrices for convolutional codes a full rank k spl times n matrix g d over the rational functions f d generates a rate r k n convolutional code c g d is minimal if it can be realized with as few memory elements as any encoder for c and g d is canonical if it has a minimal realization in controller canonical form we show that g d is minimal if and only if for all rational input sequences u d the span of u d g d covers the span of u d alternatively g d is minimal if and only if g d is globally zero free or globally invertible we show that g d is canonical if and only if g d is minimal and also globally orthogonal in the valuation theoretic sense of monna 1970

bounds on the symmetric binary cutoff rate for dispersive gaussian channels bounds on the symmetric binary cutoff rate for a pulse amplitude modulated pam signaling over dispersive gaussian channels are evaluated and discussed these easily calculable bounds can be used to estimate the reliable rate of information transmission and the error exponent behavior for binary two level pam schemes operating through a prefiltered additive white gaussian channel the memory of which is long enough to make the exact evaluation of the cutoff rate formidable the core of the bounding technique relies on a probabilistic interpretation of a fundamental theorem in matrix theory regarding the logarithm of the largest eigenvalue of a nonnegative primitive matrix commonly applied in large deviation problems these bounds are calculated for some examples and their respective tightness is considered further potential applications of the proposed bounding technique are pointed out

trellis coding for partial response channels we consider trellis coding techniques for improving the reliability of digital transmission over noisy partial response channels such channels are commonly encountered in digital communication systems and also play a role in devices for data recording concentrating on the channels with characteristics 1 mp d we study methods to obtain codes which increase free euclidean distance between permitted sequences of channel outputs and avoid the occurrence of unlimited runs of identical outputs at the expense of some loss in data rate one technique employs the concept of set partitioning the other is based on using convolutional codes designed for maximum free hamming distance in conjunction with a precoder both methods lead to essentially equivalent codes

concatenated codes for class iv partial response channels we present the design and analysis of parallel and serial concatenated codes with iterative decoding for class iv partial response pr4 signaling code rates of the form k sub 0 k sub 0 1 for k sub 0 4 8 16 32 and 64 are considered we show that the 1 1 spl oplus d sup 2 tomlinson precoder first proposed by wolf and ungerboeck 1991 for trellis coded pr4 channels is effective for both parallel and serial concatenations but the error rate floor is relatively high for the serial case this leads to new interleaver and precoder designs for serial concatenations the latter involving an examination of the conditional distance enumerator of each precoded pr4 channel the new precoders substantially reduce the floors of the serial concatenated codes but degrade the performance of the parallel concatenated codes for a bit error rate of 10 sup 5 the best parallel and serial concatenated codes come within 1 7 db of a lower bound on the capacity limit for pr4 signaling on an additive white gaussian noise channel

effects of the catastrophic behavior of tcm schemes with partially overlapped signal constellations the paper shows that the catastrophic behavior of tcm codes based on partially overlapped signal constellations significantly increases both the effective error coefficient and the decoding delay of such codes resulting in a non negligible performance degradation with respect to the asymptotic coding gain

generalised prox regularity in reflexive smooth banach spaces with smooth dual norm abstract in this paper we continue the study of the properties of the generalised projection on nonconvex sets in reflexive smooth banach spaces with smooth dual norm started by the first author in 9 we define a new class of nonconvex sets in banach spaces containing the convex closed sets and having various important properties whenever the space is assumed to be uniformly convex with smooth norm

further exploration of convolutional encoders for unequal error protection and new uep convolutional codes in this paper the unequal error protection uep capability of convolutional encoders in terms of the separation vector is studied from an algebraic viewpoint a simple procedure is presented for constructing a generator matrix which is basic and has the largest separation vector for every convolutional code such a generator matrix would be desirable since the corresponding encoder not only achieves uep optimality but also avoids undesired catastrophic error propagation in addition canonical generator matrices which are both basic and reduced are even more preferable for encoding since they attain the lowest complexity for viterbi decoding however the direct transformation from a uep optimal generator matrix to a canonical generator matrix may come with an unexpected loss of the separation vector we also propose a specific type of transformation matrix that reduces the external degrees of the generator matrices from which canonical generator matrices can be constructed that include the mitigated degradation of the separation vector finally beneficial uep convolutional codes that achieve the maximum free distances for the given code parameters are provided

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

cosets of convolutional codes with short maximum zero run lengths communication systems and storage systems derive symbol synchronization from the received symbol stream to facilitate symbol synchronization the channel sequences must have a short maximum zero run length one way to achieve this is to use a coset of an n k convolutional code to generate the channel inputs for k spl les n 2 it is shown that there exist cosets with short maximum zero run length for any constraint length any coset of an n n 1 code with high rate and or large constraint length is shown to have a large maximum zero run length a systematic procedure for obtaining cosets with short maximum zero run length from n k codes is presented and new cosets with short maximum zero run length and large minimum hamming distance are tabulated

optimization of a lattice based constellation for signaling over a partial response channel this article discusses the problem of optimizing a lattice based signal constellation for signaling over a partial response channel the objective is to minimize the probability of error which is determined by the combined effects of the additive gaussian noise and channel memory

error performance analysis of quadrature partial response trellis coded modulation qpr tcm in fading mobile satellite channels in order to improve the bandwidth efficiency and error performance partial response signaling prs and trellis coded modulation tcm are combined together this method is named qpr tcm whereas previous contributions have considered the performance of tcm transmitted over an additive white gaussian noise awgn channel the emphasis in this paper is on the performance of trellis coded m ary qam combined with prs in the fading environment we only consider the case where interleaving deinterleaving is employed to further combat the fading and assume that the ideal channel state information csi is available analytical upper bounds are derived using the well known transfer function bounding technique the bandwidth efficiencies are computed and the numerical results for several practical schemes are compared to each other one interesting result is that when the combined system is used on a rician fading channel the bit error probability upper bounds of the proposed 6qpr tcm and 9qpr tcm systems are better than their counterparts the 4qam tcm systems with 2 and 4 states respectively for snr values greater than a threshold which have the best error performances in the literature

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 codes satisfying mth order running digital sum constraints multi level sequences with a spectral null of order m at frequency f meaning that the power spectral density and its first 2m 1 derivatives vanish at f are characterized by finite state transition diagrams fstds whose edge labels satisfy bounds on the variation of the mth order running digital sum rds necessary and sufficient conditions for fstds with higher order null constraints at dc and at an arbitrary submultiple of the symbol frequency are derived analytical results are given concerning the performance of codes satisfying an mth order rds constraint on partial response channels specific code designs for quaternary channel inputs are presented the euclidean distance properties of this new class of codes aside from their spectral shaping properties are demonstrated

punctured convolutional codes for the 1 d partial response channel a coded 1 d system containing a punctured convolutional encoder is analyzed in general a punctured encoder with 2 sup v states results in a decoder trellis with 2 sup v 1 states and a periodically time varying branch structure a class c of punctured convolutional encoders for which the decoder trellis has a fixed branch structure with 2 sup v states is described an encoder in this class can be combined with a precoder to form a new punctured convolutional encoder that generates the output from the precoder the author determines the set of good variable rate trellis codes for selectable rate encoding and viterbi decoding a set of variable rate trellis codes is generated by a variable rate punctured convolutional encoder in c the author also determines punctured convolutional encoders in c that generate a single good high rate trellis code

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

binary convolutional codes with application to magnetic recording calderbank heegard and ozarow 1 have suggested a method of designing codes for channels with intersymbol interference such as the magnetic recording channel these codes are designed to exploit intersymbol interference the standard method is to minimize intersymbol interference by constraining the input to the channel using run length limited sequences calderbank heegard and ozarow considered an idealized model of an intersymbol interference channel that leads to the problem of designing codes for a partial response channel with transfer function 1 d n 2 where the channel inputs are constrained to be pm 1 this problem is considered here channel inputs are generated using a nontrivial coset of a binary convolutional code the coset is chosen to limit the zero run length of the output of the channel and so maintain clock synchronization the minimum squared euclidean distance between outputs corresponding to distinct inputs is bounded below by the free distance of a second convolutional code which we call the magnitude code an interesting feature of the analysis is that magnitude codes that are catastrophic may perform better than those that are noncatastrophic

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

energy transport and detailed verification of fourier heat law in a chain of colliding harmonic oscillators the authors study a simple nonlinear classical hamiltonian system with positive k entropy a model for heat conduction and they find that it obeys the fourier heat law numerical simulation of its dynamics can be performed very efficiently so they are able to explore it in detail they verify the fourier heat law and calculate the coefficient of thermal conductivity k by three independent methods the first is direct simulation i e simulating the dynamics of the chain between two heat reservoirs the second is the green kubo formalism which is derived in a self contained manner the third method the one sided heating of a semi infinite cold chain is new and gives the best results it yields the entire temperature dependence k t in a single numerical simulation and definitely demonstrates the validity of the fourier heat law at all temperatures for the given system they believe that this method can also be useful for other systems they derive analytically the asymptotic behaviour of the coefficient of thermal conductivity at low temperatures t to 0 and observe that it agrees with numerical results obtained by the green kubo formalism which gives by far the best results at very low temperatures

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

woven convolutional codes i encoder properties encoders for convolutional codes with large free distances can be constructed by combining several less powerful convolutional encoders this paper is devoted to constructions in which the constituent convolutional codes are woven together in a manner that resembles the structure of a fabric the general construction is called twill and it is described together with two special cases viz woven convolutional encoders with outer warp and with inner warp the woven convolutional encoders inherit many of their structural properties such as minimality and catastrophicity from their constituent encoders for all three types of woven convolutional codes upper and lower bounds on their free distances as well as lower bounds on the active distances of their encoders are derived

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

active distances for convolutional codes a family of active distance measures for general convolutional codes is defined these distances are generalizations of the extended distances introduced by thommesen and justesen 1983 for unit memory convolutional codes it is shown that the error correcting capability of a convolutional code is determined by the active distances the ensemble of periodically time varying convolutional codes is defined and lower bounds on the active distances are derived for this ensemble the active distances are very useful in the analysis of concatenated convolutional encoders

error pattern correcting cyclic codes tailored to a prescribed set of error cluster patterns a new class of cyclic codes is discussed which is highly tailored to a prescribed set of dominant error cluster patterns the cyclic code construction is based on a generator polynomial that produces a distinct syndrome set for each error pattern in the target set by tailoring the generator polynomial specifically to the set of dominant error patterns the code becomes highly effective in handling single and multiple occurrences of dominant error patterns at a very high code rate a list decoding strategy based on a set of test word error events is developed for the proposed codes which efficiently utilizes both the algebraic information from the captured syndrome and the reliability measures provided by the local correlators matched to the dominant error patterns by forcing a decoder to correct a single pattern event for each test input word multiple decoders running in parallel on the list of test words can effectively correct multiple error pattern occurrences within the channel detector output word

on unequal error protection of convolutional codes from an algebraic perspective in this paper convolutional codes are studied for unequal error protection uep from an algebraic theoretical viewpoint we first show that for every convolutional code there exists at least one optimal generator matrix with respect to uep the uep optimality of convolutional encoders is then combined with several algebraic properties e g systematic basic canonical and minimal to establish the fundamentals of convolutional codes for uep in addition a generic lower bound on the length of a uep convolutional code is proposed good uep codes with their lengths equal to the derived lower bound are obtained by computer search

matched information rate codes for partial response channels in this paper we design capacity approaching codes for partial response channels the codes are constructed as concatenations of inner trellis codes and outer low density parity check ldpc codes unlike previous constructions of trellis codes for partial response channels we disregard any algebraic properties e g the minimum distance or the run length limit in our design of the trellis code our design is purely probabilistic in that we construct the inner trellis code to mimic the transition probabilities of a markov process that achieves a high capacity approaching information rate hence we name it a matched information rate mir design we provide a set of five design rules for constructions of capacity approaching mir inner trellis codes we optimize the outer ldpc code using density evolution tools specially modified to fit the superchannel consisting of the inner mir trellis code concatenated with the partial response channel using this strategy we design degree sequences of irregular ldpc codes whose noise tolerance thresholds are only fractions of a decibel away from the capacity examples of code constructions are shown for channels both with and without spectral nulls

a first approximation of concatenated convolutional codes from linear systems theory viewpoint this article focuses on the characterization of two models of concatenated convolutional codes from the perspective of linear systems theory we present an input state output representation of these models and study the conditions for obtaining a minimal input state output representation and non catastrophic concatenated convolutional code we also establish conditions so that the concatenated codes are observable and give a lower bound for their free distances

convolutional codes in rank metric with application to random network coding random network coding recently attracts attention as a technique to disseminate information in a network this paper considers a noncoherent multishot network where the unknown and time variant network is used several times in order to create dependence between the different shots particular convolutional codes in rank metric are used these codes are so called partial unit memory p um codes i e convolutional codes with memory one first distance measures for convolutional codes in rank metric are shown and two constructions of p um codes in rank metric based on the generator matrices of maximum rank distance codes are presented second an efficient error erasure decoding algorithm for these codes is presented its guaranteed decoding radius is derived and its complexity is bounded finally it is shown how to apply these codes for error correction in random linear and affine network coding