bounds on the trellis size of linear block codes the size of minimal trellis representation of linear block codes is addressed two general upper bounds on the trellis size based on the zero concurring codewords and the contraction index of the subcodes are presented the related permutations for attaining the bounds are exhibited these bounds evidently improve the previously published general bound additional bounds based on certain code constructions are derived the focus is on the squaring construction and specific constructive bounds for reed muller and repeated root cyclic codes are obtained in particular the recursive squaring construction of reed muller codes is explored and the exact minimal trellis size of this design is obtained efficient permutations in the sense of the trellis size are also demonstrated by using shortening and puncturing methods the corresponding bounds are specified

on the complexity of multiplication in finite fields abstract in this paper we study the bilinear complexity of multiplying two arbitrary elements from an nth degree extension of a finite field f and the related problem of multiplying over f two polynomials of degree n 1 with indeterminate coefficients we derive a new linear lower bound and we describe an algorithm leading to a quasi linear upper bound

on the rectangularity of nonlinear block codes we give simple sufficient conditions for a code to be rectangular and show that large families of well known nonlinear codes are rectangular these include hadamard 1893 levenshtein 1964 delsarte goethals 1975 kerdock 1972 and nordstrom robinson 1967 codes being rectangular each of these codes has a unique minimal trellis that can be used for soft decision maximum likelihood decoding

convergence of the maximum a posteriori path estimator in hidden markov models in a hidden markov model hmm the underlying finite state markov chain cannot be observed directly but only by an additional process we are interested in estimating the unknown path of the markov chain the most widely used estimator is the maximum a posteriori path estimator map path estimator it can be calculated effectively by the viterbi 1967 algorithm as is e g frequently done in the field of coding theory correction of intersymbol interference and speech recognition we investigate component wise convergence of the map path estimator convergence is shown under the condition of unbounded likelihood ratios this condition is satisfied in the important case of hmms with additive white gaussian noise we also prove convergence if the markov chain has two states the so called viterbi paths are an important tool for obtaining these results

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

a comparison of trellis modules for binary convolutional codes a convolutional code can be represented by the conventional trellis module or the minimal trellis module based on the bcjr trellis for block codes for many convolutional codes the trellis complexity tc of the minimal module is significantly less than the tc of the conventional module an alternative representation consisting of an extended bcjr trellis module and a pruned version of the conventional module was previously introduced we prove that the overall tc of the two new modules is less than the tc of the conventional module for infinitely many codes furthermore we show that the overall tc of the new modules is smaller than the tc of the minimal module for many codes considered in the literature

a table of state complexity bounds for binary linear codes this article contains a table of bounds on the state complexity of binary linear codes with length smaller than 25 general results on the state complexity of binary linear codes with low dimension or low minimum distance are included

signal space characterization of iterative decoding by tracing the flow of computations in the iterative decoders for low density parity check codes we formulate a signal space view for a finite number of iterations in a finite length code on a gaussian channel maximum a posteriori map codeword decoding or maximum likelihood decoding decodes to the codeword signal that is closest to the channel output in euclidean distance in contrast we show that iterative decoding decodes to the pseudosignal that has highest correlation with the channel output the set of pseudosignals corresponds to pseudocodewords only a vanishingly small number of which correspond to codewords we show that some pseudocodewords cause decoding errors but that there are also pseudocodewords that frequently correct the deleterious effects of other pseudocodewords

an updated table of minimum distance bounds for binary linear codes in 1973 helgert and sfinaff published a table of upper and lower bounds on the maximum minimum distance for binary linear error correcting codes up to length 127 this article presents an updated table incorporating numerous improvements that have appeared since then to simplify the updating task the author has developed a computer program that systematically investigates the consequences of each improvement by applying several well known general code construction techniques this program also made it possible to check the original table furthermore it offers a quick and reliable update service for future improvements

linear tail biting trellises the square root bound and applications for reed muller codes linear tail biting trellises for block codes are considered by introducing the notions of subtrellis merging interval and sub tail biting trellis some structural properties of linear tail biting trellises are proved it is shown that a linear tail biting trellis always has a certain simple structure the parallel merged cosets structure a necessary condition required from a linear code in order to have a linear tail biting trellis representation that achieves the square root bound is presented finally the above condition is used to show that for r spl ges 2 and m spl ges 4r 1 or r spl ges 4 and r 3 spl les m spl les 4r 5 3 the reed muller code rm r m under any bit order cannot be represented by a linear tail biting trellis whose state complexity is half of that of the minimal conventional trellis for the code under the standard bit order

a new look at the generalized distributive law in this paper we develop a measure theoretic version of the junction tree algorithm to compute desired marginals of a product function we reformulate the problem in a measure theoretic framework where the desired marginals are viewed as corresponding conditional expectations of a product of random variables we generalize the notions of independence and junction trees to collections of spl sigma fields on a space with a signed measure we provide an algorithm to find such a junction tree when one exists we also give a general procedure to augment the spl sigma fields to create independencies which we call lifting this procedure is the counterpart of the moralization and triangulation procedure in the conventional generalized distributive law gdl framework in order to guarantee the existence of a junction tree our procedure includes the conventional gdl procedure as a special case however it can take advantage of structures at the atomic level of the sample space to produce junction tree based algorithms for computing the desired marginals that are less complex than those gdl can discover as we argue through examples our formalism gives a new way by which one can hope to find low complexity algorithms for marginalization problems

entropy length profiles bounds on the minimal covering of bipartite graphs and trellis complexity of nonlinear codes the trellis representation of nonlinear codes is studied from a new perspective we introduce the new concept of entropy length profile elp this profile can be considered as an extension of the dimension length profile dlp to nonlinear codes this elaboration of the dlp the entropy length profiles appears to be suitable to the analysis of nonlinear codes additionally and independently we use well known information theoretic measures to derive novel bounds on the minimal covering of a bipartite graph by complete subgraphs we use these bounds in conjunction with the elp notion to derive both lower and upper bounds on the state complexity and branch complexity profiles of nonlinear block codes represented by any trellis diagram we lay down no restrictions on the trellis structure and we do not confine the scope of our results to proper or one to one trellises only the basic lower bound on the state complexity profile implies that the state complexity at any given level cannot be smaller than the mutual information between the past and the future portions of the code at this level under a uniform distribution of the codewords we also devise a different probabilistic model to prove that the minimum achievable state complexity over all possible trellises is not larger than the maximum value of the above mutual information over all possible probability distributions of the codewords this approach is pursued further to derive similar bounds on the branch complexity profile to the best of our knowledge the proposed upper bounds are the only upper bounds that address nonlinear codes the novel lower bounds are tighter than the existing bounds the new quantities and bounds reduce to well known results when applied to linear codes

new constructions of high performance low complexity convolutional codes in this paper new constructions of low trellis complexity convolutional codes are presented new codes are found by searching into a specific class of time varying convolutional codes which is shaped by some basic properties and search restrictions an efficient technique for obtaining minimal trellis modules for the proposed codes is provided finally new low complexity convolutional codes of various code rates and memory sizes are tabulated

properties of the maximum a posteriori path estimator in hidden markov models a hidden markov model hmm consists of a nonobservable markov chain x x0 x1 and a measured process y y 0 y1 whose distribution is determined by x to estimate the hidden path of x up to time n x0 x1 xn by the observations y 0 y1 yn usually the maximum a posteriori probability path estimator map path estimator is applied an effective means for calculating this estimator is the viterbi algorithm which is widely employed in the fields of coding theory correction of intersymbol interference and text recognition here properties of the map estimator are derived under a certain condition c it is shown that the limiting process u u0 u1 is a regenerative process particularly this means that u has an asymptotic distribution satisfies the central limit theorem and possesses a mean error furthermore condition c is satisfied for a broad class of hmms including the most important case for applications the hmm with additive white gaussian noise

ordered binary decision diagrams and minimal trellises ordered binary decision diagrams obdds are graph based data structures for representing boolean functions they have found widespread use in computer aided design and in formal verification of digital circuits minimal trellises are graphical representations of error correcting codes that play a prominent role in coding theory this paper establishes a close connection between these two graphical models as follows let spl cscr be a binary code of length n and let f sub c x sub 1 x sub n be the boolean function that takes the value 0 at x sub 1 x sub n if and only if x sub 1 x sub n spl isin spl cscr given this natural one to one correspondence between boolean functions and binary codes we prove that the minimal proper trellis for a code spl cscr with minimum distance d 1 is isomorphic to the single terminal obdd for its boolean indicator function f sub c x sub 1 x sub n prior to this result the extensive research during the past decade on binary decision diagrams in computer engineering and on minimal trellises in coding theory has been carried out independently as outlined in this work the realization that binary decision diagrams and minimal trellises are essentially the same data structure opens up a range of promising possibilities for transfer of ideas between these disciplines

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

characteristic generators and dualization for tail biting trellises this paper focuses on dualizing tail biting trellises particularly kv trellises these trellises are based on characteristic generators as introduced by koetter vardy 2003 and may be regarded as a natural generalization of minimal conventional trellises even though they are not necessarily minimal two dualization techniques will be investigated the local dualization introduced by forney 2001 for general normal graphs and a linear algebra based dualization tailored to the specific class of tail biting bahl cocke jelinek raviv bcjr trellises introduced by nori shankar 2006 it turns out that in general the bcjr dual is a subtrellis of the local dual while for kv trellises these two coincide furthermore making use of both the bcjr construction and the local dualization it will be shown that for each complete set of characteristic generators of a code there exists a complete set of characteristic generators of the dual code such that their resulting kv trellises are dual to each other if paired suitably this proves a stronger version of a conjecture formulated by koetter vardy

tanner graphs for group block codes and lattices construction and complexity we develop a tanner graph tg construction for an abelian group block code l with arbitrary alphabets at different coordinates an important application of which is the representation of the label code of a lattice the construction is based on the modular linear constraints imposed on the code symbols by a set of generators for the dual code l as a necessary step toward the construction of a tg for l we devise an efficient algorithm for finding a generating set for l in the process we develop a construction for lattices based on an arbitrary abelian group block code called generalized construction a gca and explore relationships among a group code its gca lattice and their duals we also study the problem of finding low complexity tgs for abelian group block codes and lattices and derive tight lower bounds on the label code complexity of lattices it is shown that for many important lattices the minimal label codes which achieve the lower bounds cannot be supported by cycle free tanner graphs

nonlinear contraction theorems in probabilistic spaces abstract in this paper first we prove a common fixed point theorem in menger probabilistic metric spaces then we present a nonlinear contraction case of jungck s common fixed point theorem in menger probabilistic metric spaces finally we prove a common fixed point theorem for six self maps in a menger probabilistic metric space

on viewing block codes as finite automata block codes are viewed from a formal language theoretic perspective it is shown that properties of trellises for subclasses of block codes called rectangular codes follow naturally from the myhill nerode theorem a technique termed subtrellis overlaying is introduced with the object of reducing decoder complexity necessary and sufficient conditions for trellis overlaying are derived from the representation of the block code as a group partitioned into a subgroup and its cosets the conditions turn out to be simple constraints on coset leaders it is seen that overlayed trellises are tail biting trellises for which decoding is generally more efficient than that for conventional trellises finally a decoding algorithm for tail biting trellises is described and the results of some simulations are presented

quasi cyclic generalized ldpc codes with low error floors in this paper a novel methodology for designing structured generalized low density parity check g ldpc codes is presented the proposed design results in quasi cyclic g ldpc codes for which efficient encoding is feasible through shift register based circuits the structure imposed on the bipartite graphs together with the choice of simple component codes leads to a class of codes suitable for fast iterative decoding a pragmatic approach to the construction of g ldpc codes is proposed the approach is based on the substitution of check nodes in the protograph of a low density parity check code with stronger nodes based for instance on hamming codes such a design approach which we call low density parity check ldpc code doping leads to low rate quasi cyclic g ldpc codes with excellent performance in both the error floor and waterfall regions on the additive white gaussian noise channel

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

iterative decoding of linear block codes a parity check orthogonalization approach it is widely accepted that short cycles in tanner graphs deteriorate the performance of message passing algorithms this discourages the use of these algorithms on tanner graphs tgs of well known algebraic codes such as hamming codes bose chaudhuri hocquenghem codes and reed solomon codes yedidia et al presented a method to generate code representations suitable for message passing algorithms this method does not guarantee a representation free of four cycles in this correspondence we present an algorithm to convert an arbitrary linear block into a code with orthogonal parity check equations a combinatorial argument is used to prove that the algorithm guarantees a four cycle free representation for any linear code the effects of removing four cycles on the performance of a belief propagation decoder for the binary erasure channel are studied in detail by analyzing the structures in different representations finally we present bit error rate ber and block error rate bler performance curves of linear block codes under belief propagation algorithms for the binary erasure channel and the additive white gaussian noise awgn channel in order to demonstrate the improvement in performance achieved with the help of the proposed algorithm

on the algebraic structure of linear tail biting trellises tail biting trellises are important graphical representations of codes which can be simpler to decode than conventional trellises while conventional trellises are well understood the general theory of tail biting trellises is still under development in this paper we first develop a new algebraic framework for a systematic analysis of linear trellises which enables us to address open foundational questions in particular we present a useful and powerful characterization of linear trellis isomorphy we also obtain a new proof of the linear factorization theorem of koetter vardy and point out unnoticed problems for the group case next we apply our framework to 1 describe all the factorizations of linear trellises 2 determine and classify all the minimal linear trellises for a given linear code and 3 prove that nonmergeable one to one linear trellises are determined by the edge label sequences of certain closed paths we also provide new insight into mergeability and state how results on reduced linear trellises can be extended to nonreduced ones our classification results are relevant to iterative linear programming decoding as we show that minimal linear trellises can yield different pseudocodewords even if they have the same graph structure

bounds on the state complexity of codes from the hermitian function field and its subfields an upper bound on the minimal state complexity of codes from the hermitian function field and some of its subfields is derived coordinate orderings under which the state complexity of the codes is not above the bound are specified for the self dual hermitian code it is proved that the bound coincides with the minimal state complexity of the code finally it is shown that hermitian codes over fields of characteristic 2 admit a recursive twisted squaring construction

low floor decoders for ldpc codes one of the most significant impediments to the use of ldpc codes in many communication and storage systems is the error rate floor phenomenon associated with their iterative decoders the error floor has been attributed to certain subgraphs of an ldpc code s tanner graph induced by so called trapping sets we show in this paper that once we identify the trapping sets of an ldpc code of interest a sum product algorithm spa decoder can be custom designed to yield floors that are orders of magnitude lower than floors of the the conventional spa decoder we present three classes of such decoders 1 a bi mode decoder 2 a bit pinning decoder which utilizes one or more outer algebraic codes and 3 three generalized ldpc decoders we demonstrate the effectiveness of these decoders for two codes the rate 1 2 2640 1320 margulis code which is notorious for its floors and a rate 0 3 640 192 quasi cyclic code which has been devised for this study although the paper focuses on these two codes the decoder design techniques presented are fully generalizable to any ldpc code

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

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

decoding of convolutional codes using a syndrome trellis soft decision maximum likelihood decoding of convolutional codes using the viterbi algorithm with a syndrome trellis is proposed the parity check matrix of a convolutional code is used to construct the trellis this trellis is minimal the number of operations for the decoding of one block of a q ary rate k n convolutional code is spl sim nq sup min k n k q sup spl nu where spl nu is the memory size of the code when the code rate satisfies k n 1 2 the proposed algorithm is simpler than the classical viterbi algorithm that has complexity spl sim nq sup k q sup spl nu

optimal sectionalization of a trellis while the complexity of trellis decoding for a given block code is essentially a function of the number of states and branches in its trellis the decoding complexity may be often reduced by means of an appropriate sectionalization of the trellis notwithstanding the many examples of such sectionalizations for particular codes that appeared in the literature no systematic method for finding the best sectionalization of a given trellis is presently known we present a polynomial time algorithm which produces the optimal sectionalization of a given trellis t in time o n sup 2 where n is the length of the code generated by t the algorithm is developed in a general setting of certain operations and functions defined on the set of trellises it therefore applies to both linear and nonlinear codes and easily accommodates a broad range of optimality criteria the particular optimality criterion based on minimizing the total number of additions and comparisons required for maximum likelihood trellis decoding is investigated in detail several different methods for decoding a given trellis are discussed and compared in a number of examples finally analysis of the dynamical properties of certain optimal sectionalizations is presented

on n n 1 punctured convolutional codes and their trellis modules it is known that an n n 1 non catastrophic antipodal punctured convolutional encoder of memory m is minimal that is the corresponding code cannot be produced by an encoder of smaller memory size in this letter it is shown that the trellis module of a code produced by an n n 1 non catastrophic punctured convolutional encoder is optimum if and only if the encoder is antipodal

on n n 1 convolutional codes with low trellis complexity we show that the state complexity profile of a convolutional code c is the same as that of the reciprocal of the dual code of c in case that minimal encoders for both codes are used then we propose an optimum permutation for any given n n 1 binary convolutional code that will yield an equivalent code with the lowest state complexity with this permutation we are able to find many n n 1 binary convolutional codes which are better than punctured convolutional codes of the same code rate and memory size by either lower decoding complexity or better weight spectra

on the bcjr trellis for linear block codes in this semi tutorial paper we will investigate the computational complexity of an abstract version of the viterbi algorithm on a trellis and show that if the trellis has e edges the complexity of the viterbi algorithm is spl theta e this result suggests that the best trellis representation for a given linear block code is the one with the fewest edges we will then show that among all trellises that represent a given code the original trellis introduced by bahl cocke jelinek and raviv in 1974 and later rediscovered by wolf 1978 massey 1978 and forney 1988 uniquely minimizes the edge count as well as several other figures of merit following forney and kschischang and sorokine 1995 we will also discuss trellis oriented or minimal span generator matrices which facilitate the calculation of the size of the bcjr trellis as well as the actual construction of it

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

on hybrid stack decoding algorithms for block codes this correspondence presents sequential algorithms for soft decision decoding of linear block codes they use a stack algorithm based on the trellis of the code we are interested in the trellis as a means to avoid path decoding repetitions as well the possibility of bidirectional decoding offers a chance to increase the likelihood of explored paths we have developed three successive algorithms that offer a good decrement in the overall complexity and mainly in the most complex decoding case while giving near maximum likelihood performance this is important since it determines the maximum buffer size necessary in the decoder

codes on graphs constraint complexity of cycle free realizations of linear codes cycle free graphical realizations of linear codes generalize trellis realizations given a linear code c and a cycle free graph topology there exists a well defined minimal realization for c on that graph in which each constraint is a linear code with a well defined length and dimension the constraint complexity of the realization is defined as maximum dimension of any constraint code there exists a graph that minimizes constraint complexity in which all internal nodes have degree 3 and all interface nodes have degree 2 and which moreover can be put in the form of a tree structured trellis realization the constraint complexity of a general cycle free graph realization can be less than that of any conventional trellis realization but not by very much such realizations can yield reductions in decoding complexity even when they do not reduce constraint complexity

minimal trellis modules and equivalent convolutional codes in this correspondence it is shown that some convolutional codes with distinct memory sizes of minimal encoders are equivalent in the sense that the minimal trellises of these codes are the shifted versions of one another for an n k binary convolutional code the weight spectrum obtained from the minimal trellis may be slightly different from that obtained from the conventional code trellis with n bit branches code search is conducted to find some good n n 1 binary convolutional codes bounds on the trellis complexity measured by the number of states and the number of branches in the minimal trellis module of any convolutional code and its equivalent codes are also derived

trellis decoding complexity of linear block codes in this partially tutorial paper we examine minimal trellis representations of linear block codes and analyze several measures of trellis complexity maximum state and edge dimensions total span length and total vertices edges and mergers we obtain bounds on these complexities as extensions of well known dimension length profile dlp bounds codes meeting these bounds minimize all the complexity measures simultaneously conversely a code attaining the bound for total span length vertices or edges must likewise attain it for all the others we define a notion of uniform optimality that embraces different domains of optimization such as different permutations of a code or different codes with the same parameters and we give examples of uniformly optimal codes and permutations we also give some conditions that identify certain cases when no code or permutation can meet the bounds in addition to dlp based bounds we derive new inequalities relating one complexity measure to another which can be used in conjunction with known bounds on one measure to imply bounds on the others as an application we infer new bounds on maximum state and edge complexity and on total vertices and edges from bounds on span lengths

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

on the optimum bit orders with respect to the state complexity of trellis diagrams for binary linear codes it was shown earlier that for a punctured reed muller rm code or a primitive bch code which contains a punctured rm code of the same minimum distance as a large subcode the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions the problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized hamming weight hierarchy of a code and it is shown that for rm codes the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to v k wei 1991 the state complexity of the trellis diagram for the extended and permuted 64 24 bch code is discussed

trellis complexity and minimal trellis diagrams of lattices this paper presents results on trellis complexity and low complexity trellis diagrams of lattices we establish constructive upper bounds on the trellis complexity of lattices these bounds both improve and generalize the similar results of tarokh and vardy see ibid vol 43 p 1294 1300 1997 we also construct trellis diagrams with minimum number of paths for some important lattices such trellises are called minimal the constructed trellises which are novel in many cases can be employed to efficiently decode the lattices via the viterbi algorithm in particular a general structure for minimal trellis diagrams of d sub n lattices is obtained this structure corresponds to a new code formula for d sub n moreover we develop some important duality results which are used in both deriving the upper bounds and finding the minimal trellises all the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases

unifying views of tail biting trellis constructions for linear block codes in this paper we present new ways of describing and constructing linear tail biting trellises for block codes we extend the well known bahl cocke jelinek raviv bcjr construction for conventional trellises to tail biting trellises the bcjr like labeling scheme yields a simple specification for the tail biting trellis for the dual code with the dual trellis having the same state complexity profile as that of the primal code finally we show that the algebraic specification of forney for state spaces of conventional trellises has a natural extension to tail biting trellises

path partitions and forward only trellis algorithms this is a semitutorial paper on trellis based algorithms we argue that most decoding detection algorithms described on trellises can be formulated as path partitioning algorithms with proper definitions of mappings from subsets of paths to metrics of subsets thereby the only two operations needed are path concatenation and path collection which play the roles of multiplication and addition respectively furthermore we show that the trellis structure permits the path partitioning algorithms to be formulated as forward only algorithms with structures resembling the viterbi 1967 algorithm thus eliminating the need for backward computations regardless of what task needs to be performed on the trellis while all of the actual decoding detection algorithms presented here are rederivations of variations of previously known methods we believe that the exposition of the algorithms in a unified manner as forward only path partitioning algorithms is the most intuitive manner in which to generalize the viterbi algorithm we also believe that this approach may in fact influence the practical implementation of the algorithms as well as influence the construction of other forward only algorithms e g byte wise forward only detection algorithms

the extraction and complexity limits of graphical models for linear codes two broad classes of graphical modeling problems for codes can be identified in the literature constructive and extractive problems the former class of problems concern the construction of a graphical model in order to define a new code the latter class of problems concern the extraction of a graphical model for a fixed given code the design of a new low density parity check code for some given criteria e g target block length and code rate is an example of a constructive problem the determination of a graphical model for a classical linear block code that implies a decoding algorithm with desired performance and complexity characteristics is an example of an extractive problem this work focuses on extractive graphical model problems and aims to lay out some of the foundations of the theory of such problems for linear codes the primary focus of this work is a study of the space of all graphical models for a fixed given code the tradeoff between cyclic topology and complexity in this space is characterized via the introduction of a new bound the forest inducing cut set bound fi csb the proposed bound provides a more precise characterization of this tradeoff than that which can be obtained using existing tools e g the csb and can be viewed as a generalization of the square root bound for tail biting trellises to graphical models with arbitrary cyclic topologies searching the space of graphical models for a given code is then enabled by introducing a set of basic graphical model transformation operations that are shown to span this space finally heuristics for extracting novel graphical models for linear block codes using these transformations are investigated

common fixed point theorems of compatible and weakly compatible maps in menger spaces abstract in this paper we prove some new common fixed point theorems for compatible and weakly compatible self maps on menger space under contractive conditions our results improve and generalize the results of singh and jain b singh s jain a fixed point theorem in menger space through weak compatibility j math anal appl 301 2005 439 448 razani and shirdaryazdi a razani m shirdaryazdi a common fixed point theorem of compatible maps in menger space chaos solitons fractals 32 2007 26 34 and others

a low weight trellis based iterative soft decision decoding algorithm for binary linear block codes this paper presents a new low weight trellis based soft decision iterative decoding algorithm for binary linear block codes the algorithm is devised based on a set of optimality conditions and the generation of a sequence of candidate codewords for an optimality test the initial candidate codeword is generated by a simple decoding method the subsequent candidate codewords if needed are generated by a chain of low weight trellis searches one at a time each search is conducted through a low weight trellis diagram centered around the latest candidate codeword and results in an improvement over the previous candidate codewords that have been already tested this improvement is then used as the next candidate codeword for a test of optimality the decoding iteration stops whenever a candidate codeword is found to satisfy a sufficient condition on optimality or the latest low weight trellis search results in a repetition of a previously generated candidate codeword a divide and conquer technique is also presented for codes that are not spanned by their minimum weight codewords the proposed decoding algorithm has been applied to some well known codes of lengths 48 64 and 128 simulation results show that the proposed algorithm achieves either practically optimal error performance for the example codes of length 48 and 64 or near optimal error performance for the 128 29 32 rm code with a significant reduction in computational decoding complexity

suboptimal decoding of linear codes partition technique general symmetric channels are introduced and near maximum likelihood decoding in these channels is studied first we define a class of suboptimal decoding algorithms based on an incomplete search through the code trellis it is proved that the decoding error probability of suboptimal decoding is bounded above for any q ary code of length n and code rate r by twice the error probability of its maximum likelihood decoding and tends to the latter as n grows second we design a suboptimal trellis like algorithm which reduces the known decoding complexity of the order of q sup n min r 1 r operations to that of q sup nr i r operations for all cyclic codes and virtually all long linear codes we also consider the corresponding bounds for concatenated codes an important corollary is that this suboptimal decoding can provide complexity below the lower bounds on trellis complexity at a negligible expense in terms of decoding error probability

designing lexicographic codes with a given trellis complexity we generalize constructions of lexicographic codes to produce locally optimal codes with a desired trellis decoding complexity these constructions are efficient for high rate codes and provide a means for automated code design as a byproduct we improve known bounds on the parameters of lexicodes

asymptotically good codes have infinite trellis complexity the trellis complexity s c of a block code c is defined as the logarithm of the maximum number of states in the minimal trellis realization of the code the parameter s c governs the complexity of maximum likelihood decoding and is considered a fundamental descriptive characteristic of the code in a number of recent works we derive a new lower bound on s c which implies that asymptotically good codes have infinite trellis complexity more precisely for i spl ges 1 let c sub i be a code over an alphabet of size q of length n sub i rate r sub i and minimum distance d sub i the infinite sequence of codes c sub c sub 2spl middotspl middotspl middot such that n sub ispl rarrspl infin when i spl rarrspl infin is said to be asymptotically good if both r sub i and d sub in sub i are bounded away from zero as i spl rarrspl infin we prove that the complexity s c sub i increases linearly with n sub i in any asymptotically good sequence of codes

proof of a conjecture of mceliece regarding the expansion index of the minimal trellis we prove a conjecture of mceliece establishing that for each fixed order of positions of a linear code c the minimal trellis minimizes the quantity e v 1 where e and v stand for the number of edges and vertices in the trellis respectively as a consequence it follows that the minimal trellis uniquely minimizes the total number of operations required for viterbi decoding of a given code moreover we show that these results extend to the general class of rectangular codes namely the class of codes that admit a biproper trellis presentation which includes all group codes and many other useful nonlinear codes

a class of linear codes with good parameters from algebraic curves a class of linear codes with good parameters is constructed in this correspondence it turns out that linear codes of this class are subcodes of the subfield subcodes of goppa s geometry codes in particular we find 61 improvements on brouwer s table based on our codes

linear tail biting trellises characteristic generators and the bcjr construction this paper investigates the constructions of tail biting trellises for linear block codes as introduced by koetter and vardy 2003 and nori and shankar 2006 for a given code the sets of characteristic generators are defined slightly more generally than by koetter and vardy in particular they are not uniquely determined by the code the effect of the choice of characteristic generators on the resulting product trellises called kv trellises is discussed in detail it is shown that each kv trellis is a span based bcjr trellis and that the latter are always nonmergeable finally a duality conjecture posed by koetter and vardy is addressed by making use of a dualization technique of bcjr trellises the conjecture is proven for minimal trellises

data detection and kalman estimation for multiple space time trellis codes this work presents a joint channel estimation and data detection algorithm over high doppler fading channels for space time trellis codes sttcs having more than one code vector per trellis branch such codes are referred to as multiple sttcs msttcs this algorithm belongs to the per survivor processing psp family where a survivor sequence associated with a trellis state can be considered as the data sequence aiding per path channel estimation in a joint detection and estimation procedure we propose a smoothed data detection technique that utilizes past present and future received symbols to increase the probability of the survivor being a correct data sequence a symbol by symbol maximum a posteriori probability map criterion with a fixed delay d is employed for data detection combined with a kalman predictor kp for channel estimation this novel algorithm called smoothed data detection and kalman estimation sdd ke provides a significant performance gain when d 0 over d 0 with a linear increase in complexity when d increases comparison with the delayed mixture kalman filtering d mkf technique shows that the sdd ke algorithm provides important performance and complexity advantages