on some applications of cellular automata abstract cellular automata model corresponding to cattaneo s diffusion is constructed its phase space is given delay is shown to decrease the chaotic damage spread region then cellular automata are used to study a stochastic minority game payoff memory approach introduced by smale is used a corresponding dynamical system is also derived cellular automata model for an epidemic in a population whose individuals are cycling between two places is constructed both cases of uniform and random susceptibilities are included

truncation length for viterbi decoding a bound is derived and analyzed for the bit error rate ber of a viterbi decoder with survivor truncation estimates of the snr signal to noise ratio loss on the awgn additive white gaussian noise channel due to truncation are obtained for convolutional codes larger truncation lengths are required than the smallest value that does not effectively decrease the code s free distance especially at low e sub b n sub 0

epidemic models and percolation the authors argue that local epidemic models with immunisation are in the same universality class as percolation cluster growth models and show that the static exponents are equal to all orders in the epsilon expansion they calculate the dynamic exponent nu t 1 1 28 epsilon o epsilon 2 and relate this to other exponents involving the chemical distance

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

periodic travelling waves in a family of deterministic cellular automata abstract reaction diffusion equations whose kinetics contain a stable limit cycle are an established class of models for a range of biological and chemical systems in this paper i construct a family of deterministic cellular automata with nine states which are qualitatively similar to oscillatory reaction diffusion equations in that their rules reflect both local oscillations and spatial diffusion the automata can be crudely interpreted as models of predator prey interactions and i show that the behaviour following local perturbation of the prey only state in one space dimension is very similar in the automata and in standard reaction diffusion models for predator prey systems in particular in many cases invasion of prey by predators leaves behind periodic travelling waves in the wake of invasion i study in detail these periodic plane waves in the automaton by explicitly investigating periodic solutions of the difference equation governing travelling waves i show that the automaton has many different periodic wave solutions and i compare their properties with those of periodic wave solutions of reaction diffusion systems the basic conclusion is that included amongst the periodic waves in the automaton are a family of solutions which mimic quite closely the properties of reaction diffusion periodic waves

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

a new class of automata networks a new class of automata networks is defined their evolution rules are determined by a probability measure p on the set of all integers z and an indicator function ia on the interval 0 1 it is shown that any cellular automaton rule can be represented by a nonunique rule formulated in terms of a pair p ia this new class of automata networks contains discrete systems which are not cellular automata for a given p a metric can be defined on the space of all rules which induces a metric on the space of all cellular automata rules

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

bounds on the decoding error probability of binary linear codes via their spectra bounds on the error probability of maximum likelihood decoding of a binary linear code are considered the bounds derived use the weight spectrum of the code and they are tighter than the conventional union bound in the case of large noise in the channel the bounds derived are applied to a code with an average spectrum and the result is compared to the random coding exponent the author shows that the bound considered for the binary symmetrical channel case coincides asymptotically with the random coding bound for the case of awgn channel the author shows that berlekamp s 1980 tangential bound can be improved but even this improved bound does not coincide with the random coding bound although it can be very close to it

automata network sir models for the spread of infectious diseases in populations of moving individuals automata network sir models for the spread of infectious diseases are studied the local rule consists of two subrules the first one applied sequentially describes the motion of the individuals the second is synchronous and models infection and removal or recovery the spatial correlations created by the application of the second subrule are partially destroyed according to the degree of mixing of the population which follows from the application of the first subrule one and two population models are considered in the second case individuals belonging to one population may be infected only by individuals belonging to the other population as is the case for example for the heterosexual propagation of a venereal disease it is shown that the occurrence of the epidemic in one population may be triggered by the occurrence of the epidemic in the other population the emphasis is on the influence of the degree of mixing of the individuals which follows from their diffusive motion in particular the asymptotic behaviours for very small and very large mixing are determined when the degree of mixing tends to infinity the correlations are completely destroyed and the time evolution of the epidemic is then correctly predicted by the mean field approximation

finite dimensional approximations to wiener measure and path integral formulas on manifolds abstract certain natural geometric approximation schemes are developed for wiener measure on a compact riemannian manifold these approximations closely mimic the informal path integral formulas used in the physics literature for representing the heat semi group on riemannian manifolds the path space is approximated by finite dimensional manifolds h p m consisting of piecewise geodesic paths adapted to partitions p of 0 1 the finite dimensional manifolds h p m carry both an h 1 and a l 2 type riemannian structures g 1 p and g 0 p respectively it is proved that 1 z i p e 1 2 e d vol g i p i d as mesh p 0 where e is the energy of the piecewise geodesic path h p m and for i 0 and 1 z i p is a normalization constant vol g i p is the riemannian volume form relative to g i p and is wiener measure on paths on m here 1 1 and 0 exp 1 6 1 0 scal s ds where scal is the scalar curvature of m these results are also shown to imply the well known integration by parts formula for the wiener measure

optimizing transmission lengths for limited feedback with nonbinary ldpc examples this paper presents a general approach for optimizing the number of symbols in increments packets of incremental redundancy in a feedback communication system with a limited number of increments this approach is based on a tight normal approximation on the rate for successful decoding applying this approach to a variety of feedback systems using nonbinary nb low density parity check ldpc codes shows that greater than 90 of capacity can be achieved with average blocklengths fewer than 500 transmitted bits one result is that the performance with ten increments closely approaches the performance with an infinite number of increments the paper focuses on binary input additive white gaussian noise bi awgn channels but also demonstrates that the normal approximation works well on examples of fading channels as well as high snr awgn channels that require larger qam constellations this paper explores both variable length feedback codes with termination vlft and the more practical variable length feedback vlf codes without termination that require no assumption of noiseless transmitter confirmation for vlf we consider both a two phase scheme and crc based scheme

route to chaos for a global variable of a two dimensional game of life type automata network we consider a two dimensional cellular automaton whose rule consists of two subrules the first applied synchronously is a local rule inspired from the game of life with a larger neighbourhood the second applied sequentially describes the motion of a fraction m of individuals such rules appear to be useful for modelling complex systems in ecology such as natural populations of animals in which the motion of the individuals is believed to play an important role if the motion is long range the density of individuals exhibits a sequence of period doubling bifurcations and behaves chaotically when m is large enough if the motion is short range i e restricted to first neighbours patterns become inhomogeneous spatial correlations decay with a finite correlation length xi of the order of square root m we observe the formation of domains of mean width xi with a chaotic behaviour of the local density of individuals but the collective behaviour is stationary the global density tends to a fixed value when the lattice size is much larger than xi xi

channel coding rate in the finite blocklength regime this paper investigates the maximal channel coding rate achievable at a given blocklength and error probability for general classes of channels new achievability and converse bounds are given which are tighter than existing bounds for wide ranges of parameters of interest and lead to tight approximations of the maximal achievable rate for blocklengths n as short as 100 it is also shown analytically that the maximal rate achievable with error probability isclosely approximated by c v n q 1 where c is the capacity v is a characteristic of the channel referred to as channel dispersion and q is the complementary gaussian cumulative distribution function

a study on universal codes with finite block lengths based on random codes and typical set decoding an alternative proof of root and varaiya s compound channel coding theorem for linear gaussian channels is presented the performance limit of codes with finite block length under a compound channel is studied through error bounds and simulation although the theorem promises uniform convergence of the probability of error as the block length approaches infinity with short block lengths the performance can differ considerably for individual channels simulation results show that universal performance can be a practical goal as the block lengths become large

parisi s mean field solution for spin glasses as an analytic continuation in the replica number parisi s replica symmetry breaking solution for spin glasses is extended to finite replica number n the free energy fp n obtained this way as well as its first two derivatives with respect to n are shown to join the corresponding values in the sherrington kirkpatrick sk solution at a characteristic value ns t where stability breaks down in the latter the continuation composed of the sk branch fsk n for n or ns t and the parisi branch fp n for 0 or n or ns t fulfils the requirements of convexity monotonicity and stability for all n

stationary states and spatial patterning in an sis epidemiology model with implicit mobility abstract by means of the asynchronous cellular automata algorithm we study stationary states and spatial patterning in an s i s model in which the individuals are attached to the vertices of a graph and their mobility is mimicked by varying the neighbourhood size q here we consider the following cases q is fixed at certain value and q is taken at random at each step and for each individual the obtained numerical data are then mapped onto the solution of its version corresponding to the limit q this allows for deducing an explicit form of the dependence of the fraction of infected individuals on the curing rate a detailed analysis of the appearance of spatial patterns of infected individuals in the stationary state is performed

a probabilistic automata network epidemic model with births and deaths exhibiting cyclic behaviour a probabilistic automata network model for the spread of an infectious disease in a population of moving individuals is studied the local rule consists of two subrules the first one applied synchronously models infection birth and death processes it is a probabilistic cellular automaton rule the second applied sequentially describes the motion of the individuals the model contains six parameters the probabilities p for a susceptible to become infected by contact with an infective the respective birth rates bs and bi of the susceptibles from either a susceptible or an infective parent the respective death rates ds and di of susceptibles and infectives and a parameter m characterizing the motion of the individuals the model has three fixed points the first is trivial it describes a stationary state with no living individuals the second corresponds to a disease free state with no infectives the third and last one characterizes an endemic state with non zero densities of susceptibles and infectives moreover the model may exhibit oscillatory behaviour of the susceptible and infective densities as functions of time through a hopf type bifurcation the influence of the different parameters on the stability of all these states is studied with a particular emphasis on the influence of motion which has been found to be a stabilizing factor of the cyclic behaviour

estimations of the transfer functions of noncatastrophic convolutional encoders a computational method which allows us to upper bound the solution of a wide class of systems of linear recurrent equations is proposed this method is used to estimate the transfer functions of noncatastrophic convolutional encoders

finite length analysis of low density parity check codes on the binary erasure channel in this paper we are concerned with the finite length analysis of low density parity check ldpc codes when used over the binary erasure channel bec the main result is an expression for the exact average bit and block erasure probability for a given regular ensemble of ldpc codes when decoded iteratively we also give expressions for upper bounds on the average bit and block erasure probability for regular ldpc ensembles and the standard random ensemble under maximum likelihood ml decoding finally we present what we consider to be the most important open problems in this area

limited search trellis decoding of convolutional codes the least storage and node computation required by a breadth first tree or trellis decoder that corrects t errors over the binary symmetric channels is calculated breadth first decoders work with code paths of the same length without backtracking the viterbi algorithm is an exhaustive trellis decoder of this type other schemes look at a subset of the tree or trellis paths for random tree codes theorems about the asymptotic number of paths required and their depth are proved for concrete convolutional codes the worst case storage for t error sequences is measured in both cases the optimal decoder storage has the same simple dependence on t the m algorithm and algorithms proposed by g j foschini ibid vol it 23 p 605 9 sept 1977 and by s j simmons phd diss queens univ kingston ont canada are optimal or nearly so they are all far more efficient than the viterbi algorithm

mean field type equations for spread of epidemics the small world model abstract in the paper we study a cellular automata ca model of epidemic dynamics the effects of local spatial correlations on a temporal aggregated spread of single epidemics are studied as a function of increasing proportion of global contacts small world model we conjecture that even in the presence of high local correlations the aggregated mean field type models can be quite successful if the contact rate is treated as a free parameter the dependence of the estimated contact rate on the mixing parameter can be understood in terms of a simple probabilistic model the contact rate reflects not only a microscopic and epidemiological situation but also a complicated social pattern including short and long range contacts as well as a possibly hierarchical structure of human society

performance of viterbi decoding with and without arq on rician fading channels in this paper we investigate the performance of the viterbi decoding algorithm with without automatic repeat request arq over a rician flat fading channel with unlimited interleaving we show that the decay rate of the average bit error probability with respect to the bit energy to noise ratio is at least equal to d f at high bit energy to noise ratio for both cases with arq and without arq where d f is the free distance of the convolutional code the yamamoto itoh flag helps to reduce the average bit error probability by a factor of 4 d f with a negligible retransmission rate we also prove an interesting result that the average bit error probability decays exponentially fast with respect to the rician factor for any fixed bit energy per noise ratio in addition the average bit error exponent with respect to the rician factor is shown to be d f

sphere packing bounds revisited for moderate block lengths the main reference of this paper is the sphere packing bound of 1967 sp67 derived by shannon gallager and berlekamp it offers a lower bound on the decoding error probability over a very large variety of channels if it has failed so far to provide any usable material in practical implementation of telecommunication systems it is due to an original focus on asymptotic results making it inapplicable for moderate code lengths and to the difficulty of the involved methods which makes the derivation of sp67 quite hermetic and uninspiring for further research the purpose of this paper is two fold 1 to stir up some renewed interest in the topic on which shannon concluded his career in information theory thanks to a qualitative rather than technical review of the derivation of sp67 introduced by a review of the simpler sphere packing bound derived by shannon in 1959 2 to prove the practical interest of sp67 by extending its field of application to continuous output channels and particularly the additive white gaussian noise awgn channel used with any particular modulation scheme and by improving its lower bound for the moderate code length case so that it becomes the best lower bound for most iteratively decodable codes turbo codes low density parity check ldpc codes repeat accumulate ra codes etc of usual lengths

trellis codes for periodic erasures this paper describes techniques for the design and analysis of trellis codes that provide reliable communication over every channel in a specified set of possible channels where each channel is characterized by additive white gaussian noise with a distinct periodic variation in signal to noise ratio an important practical application for such trellis codes is the periodic erasure channel produced by partial band interference dispersed by a block interleaver we present trellis codes that provide reliable communication over all periodic erasure patterns of a given period for which the number of unerased coded bits per period is at least equal to the number of information bits per period

a note on tailbiting codes and their feedback encoders tailbiting codes encoded by feedback convolutional encoders are studied a condition for when tailbiting will work is given and it is described how the encoder starting state can be obtained for feedback encoders in both controller and observer canonical forms finally results from a search for systematic feedback encoders that encode tailbiting codes with good decoding bit error probabilities are presented

critical behaviour of a probabilistic automata network sis model for the spread of an infectious disease in a population of moving individuals a probabilistic automata network sis susceptible infective susceptible model for the spread of an infectious disease in a population of moving individuals is studied the local rule consists of two subrules the first one applied synchronously models infection and recovery it is a probabilistic cellular automaton rule the second applied sequentially describes the motion of the individuals the model contains three parameters the probabilities pi to get infected and pr to recover and the average number of tentative moves per individual m depending upon the values of these parameters in the infinite time limit the system is either in the disease free state or in the endemic state it goes from one state to the other through a transcritical bifurcation similar to a second order phase transition characterized by a non negative order parameter whose role is played in this model by the stationary density of infected individuals the pi pr phase diagram and the critical behaviour of the stationary density of infectives in the neighbourhood of the phase transition are studied as a function of m according to whether the individuals perform short or long range moves it is found that the parameters characterizing the transition have a qualitatively different behaviour as m varies when m is very large the correlations created by the application of the subrule modelling infection and recovery are destroyed and as expected the behaviour of the system is then correctly predicted by a mean field type approximation which assumes a homogeneous mixing of the individuals when m is not large this assumption is no longer correct

feedback in the non asymptotic regime without feedback the backoff from capacity due to non asymptotic blocklength can be quite substantial for blocklengths and error probabilities of interest in many practical applications in this paper novel achievability bounds are used to demonstrate that in the non asymptotic regime the maximal achievable rate improves dramatically thanks to variable length coding and feedback for example for the binary symmetric channel with capacity 1 2 the blocklength required to achieve 90 of the capacity is smaller than 200 compared to at least 3100 for the best fixed blocklength code even with noiseless feedback virtually all the advantages of noiseless feedback are shown to be achievable even if the feedback link is used only to send a single signal informing the encoder to terminate the transmission stop feedback it is demonstrated that the non asymptotic behavior of the fundamental limit depends crucially on the particular model chosen for the end of packet control signal fixed blocklength codes and related questions concerning communicating with a guaranteed delay are discussed in which situation feedback is demonstrated to be almost useless even non asymptotically

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

reliability output decoding of tail biting convolutional codes we present extensions to raghavan and baum s reliability output viterbi algorithm rova to accommodate tail biting convolutional codes these tail biting reliability output algorithms compute the exact word error probability of the decoded codeword after first calculating the posterior probability of the decoded tail biting codeword s starting state one approach employs a state estimation algorithm that selects the maximum a posteriori state based on the posterior distribution of the starting states another approach is an approximation to the exact tail biting rova that estimates the word error probability a comparison of the computational complexity of each approach is discussed in detail the presented reliability output algorithms apply to both feedforward and feedback tail biting convolutional encoders these tail biting reliability output algorithms are suitable for use in reliability based retransmission schemes with short blocklengths in which terminated convolutional codes would introduce rate loss

universal trellis codes for diagonally layered space time systems foschini s diagonally layered space time transmission system known as d blast is an architecture designed for a rayleigh fading environment using multiple element antenna arrays at both the transmit and receive sites to achieve very high spectral efficiencies in this paper we propose a simple coding technique for d blast that uses a single trellis code with finite traceback viterbi decoding we examine the performance of universal trellis codes that are designed to have a distance structure that is matched to the periodic signal to noise ratio snr variation of the channel created by d blast under the assumption that the channel is static during one burst but may change from burst to burst we show that a universal 64 state trellis code on a 2 spl times 2 d blast system with long enough block lengths displays universal behavior working on almost every 2 spl times 2 channel with at least the mutual information required by a standard 64 state awgn trellis code the only 2 spl times 2 channel where more mutual information is required is a certain rotation of the zero eigenvalue channel we also present 4 spl times 4 and 8 spl times 8 examples

the chern gauss bonnet theorem via supersymmetric euclidean field theories we prove the chern gauss bonnet theorem using sigma models whose source supermanifolds have super dimension 0 2 along the way we develop machinery for understanding manifold invariants encoded by families of 0 dimensional euclidean field theories and their quantization

an optimal circular viterbi decoder for the bounded distance criterion we propose a viterbi type decoder for tailbiting trellis codes that works by traversing the tailbiting circle somewhat more than once the traversal is the least possible for any bounded distance viterbi decoder procedures are given that compute this minimum unlike previous decoders of the type the new scheme does not suffer limit cycles or from pseudocodewords the bit error rate is compared to that of bahl cocke jelinek raviv and maximum likelihood decoding

an improved sphere packing bound for finite length codes over symmetric memoryless channels this paper derives an improved sphere packing isp bound for finite length error correcting codes whose transmission takes place over symmetric memoryless channels and the codes are decoded with an arbitrary list decoder we first review classical results i e the 1959 sphere packing sp59 bound of shannon for the gaussian channel and the 1967 sphere packing sp67 bound of shannon et al for discrete memoryless channels an improvement on the sp67 bound as suggested by valembois and fossorier is also discussed these concepts are used for the derivation of a new lower bound on the error probability of list decoding referred to as the isp bound which is uniformly tighter than the sp67 bound and its improved version the isp bound is applicable to symmetric memoryless channels and some of its applications are presented its tightness under maximum likelihood ml decoding is studied by comparing the isp bound to previously reported upper and lower bounds on the ml decoding error probability and also to computer simulations of iteratively decoded turbo like codes this paper also presents a technique which performs the entire calculation of the sp59 bound in the logarithmic domain thus facilitating the exact calculation of this bound for moderate to large block lengths without the need for the asymptotic approximations provided by shannon

supersymmetric path integrals the supersymmetric path integral is constructed for quantum mechanical models on flat space as a supersymmetric extension of the wiener integral it is then pushed forward to a compact riemannian manifold by means of a malliavin type construction the relation to index theory is discussed

evidence for a new phase in the domany kinzel cellular automaton we consider a generalized version including anisotropy of the stochastic one dimensional cellular automaton studied by domany and kinzel it recovers wolfram like deterministic cellular automata as particular cases the phase diagram presents three and not two as previously suggested phases which were detected through the numerical study of both the order parameter and the sensitivity to initial conditions the various universality classes are exhibited as well cellular automata ca are totally discrete dynamical systems discrete space discrete time and discrete number of states which find general applications in physics chemistry biology computer science etc with each site noted i is associated a variable cr which can be in k different states cr 0 1 k 1 the dynamics is defined at each time step by rules depending on the values at a previous time of o j associated with a given number of r arbitrary sites called inputs usually one considers regular lattices and the inputs refer to the sites on the local neighborhood only the local rules of a ca may be probabilistic or deterministic and the sites are simultaneously updated thus we recognize the glauber dynamics for example as a special case of a ca however in the more general context of ca the dynamics is not restricted to the usual boltzmann weight and detailed balance therefore ca do not necessarily evolve towards equilibrium thus they can model chemical reactions crystal growth models turbulence biological problems or other nonlinear processes far from thermal equilibrium

decision depths of convolutional codes the decision depth function of a convolutional code l sub d w is the first trellis depth at which all unmerged incorrect subset paths exceed weight w the authors tabulate this function for most codes in common use l sub d w specifies the path memory length of a decoder it follows a clear asymptotic growth rule

information spectrum approach to second order coding rate in channel coding in this paper second order coding rate of channel coding is discussed for general sequence of channels the optimum second order transmission rate with a constant error constraint epsiv is obtained by using the information spectrum method we apply this result to the discrete memoryless case the discrete memoryless case with a cost constraint the additive markovian case and the gaussian channel case with an energy constraint we also clarify that the gallager bound does not give the optimum evaluation in the second order coding rate

a lower bound on the probability of error in multihypothesis testing consider two random variables x and y where x is finitely or countably infinitely valued and where y is arbitrary let spl epsiv denote the minimum probability of error incurred in estimating x from y it is shown that spl epsiv spl ges sub 0 spl les spl alpha spl les 1 sup sup 1 spl alpha p spl pi x y spl les spl alpha where spl pi x y denotes the posterior probability of x given y this bound finds information theoretic applications in the proof of converse channel coding theorems it generalizes and strengthens previous lower bounds due to shannon and to verdu and han 1994