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 a family of abelian codes and their state complexities we study reed muller codes and berman codes as abelian codes we show that the duals of berman codes and reed muller codes can be considered as belonging to the same family of abelian codes we also determine the minimum distance and state complexity of the duals of berman codes each of the classical parameters generalizes that of reed muller codes in the obvious way but the state complexity does not we conclude by comparing the asymptotic behavior of the state complexity of the duals of berman codes with that of the obvious generalization of the state complexity of reed muller codes

epidemic analysis of the second order transition in the ziff gulari barshad surface reaction model we study the dynamic behavior of the ziff gulari barshad irreversible surface reaction model around its kinetic second order phase transition using both epidemic and poisoning time analyses we find that the critical point is given by p 1 0 3873682 ifmmode pm else textpm fi 0 0000015 which is lower than the previously determined value we also obtain precise values of the dynamical critical exponents z ensuremath delta and ensuremath eta which provide further numerical evidence that this transition is in the same universality class as directed percolation

directed percolation in a two dimensional stochastic fire diffuse fire model in this paper we establish from extensive numerical experiments that the two dimensional stochastic fire diffuse fire model belongs to the directed percolation universality class this model is an idealized model of intracellular calcium release that retains the both the discrete nature of calcium stores and the stochastic nature of release it is formed from an array of noisy threshold elements that are coupled only by a diffusing signal the model supports spontaneous release events that can merge to form spreading circular and spiral waves of activity the critical level of noise required for the system to exhibit a non equilibrium phase transition between propagating and non propagating waves is obtained by an examination of the textit local slope delta t of the survival probability pi t propto exp delta t for a wave to propagate for a time t

swinging control of nonlinear oscillations the speed gradient method of control design used previously for problems of regulation and tracking is extended to oscillating systems with energy based objective functions the concept of swinging control is introduced meaning achievement of arbitrary large level of the objective function by arbitrary small control level the existence of swinging control for hamiltonian systems is established simulation results for pendulum swinging problem are demonstrated

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 sequential decoder for linear block codes with a variable bias term metric a sequential decoder for linear block codes that performs maximum likelihood soft decision decoding is described the decoder uses a metric computed from a lower bound on the cost of the unexplored portion of the code tree it is shown that for certain block codes the average computational complexity of this metric is superior to that of the fano metric a new function the cumulative column distance function is introduced for linear block codes this function is an important factor that determines the average computational effort of a sequential decoder for a linear block code with an arbitrary maximum likelihood metric simulation results show that a sequential decoder for linear block codes with a fast growing cumulative column distance function achieves a low computational complexity a result analogous to that for convolutional codes

maximum likelihood soft decoding of binary block codes and decoders for the golay codes maximum likelihood soft decision decoding of linear block codes is addressed a binary multiple check generalization of the wagner rule is presented and two methods for its implementation one of which resembles the suboptimal forney chase algorithms are described besides efficient soft decoding of small codes the generalized rule enables utilization of subspaces of a wide variety thereby yielding maximum likelihood decoders with substantially reduced computational complexity for some larger binary codes more sophisticated choice and exploitation of the structure of both a subspace and the coset representatives are demonstrated for the 24 12 golay code yielding a computational gain factor of about 2 with respect to previous methods a ternary single check version of the wagner rule is applied for efficient soft decoding of the 12 6 ternary golay code

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

on repeated root cyclic codes a parity check matrix for a q ary repeated root cyclic code is derived using the hasse derivative then the minimum distance of a q ary repeated root cyclic code is expressed in terms of the minimum distance of a certain simple root cyclic code with the help of this result several binary repeated root cyclic codes of lengths up to n 62 are shown to contain the largest known number of codewords for their given length and minimum distance the relative minimum distance d sub min n of q ary repeated root cyclic codes of rate r or r is proven to tend to zero as the largest multiplicity of a root of the generator g x increases to infinity it is further shown that repeated root cycle codes cannot be asymptotically better than simple root cyclic codes

repeated root cyclic codes in the theory of cyclic codes it is common practice to require that n q 1 where n is the word length and f sub q is the alphabet it is shown that the even weight subcodes of the shortened binary hamming codes form a sequence of repeated root cyclic codes that are optimal in nearly all other cases one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime this statement is based on an analysis for lengths up to 100 a theorem shows why this was to be expected but it also leads to low complexity decoding methods this is an advantage especially for the codes that are not much worse than corresponding codes of odd length it is demonstrated that a binary cyclic code of length 2n n odd can be obtained from two cyclic codes of length n by the well known mod u mod u v mod construction this leads to an infinite sequence of optimal cyclic codes with distance 4 furthermore it is shown that low complexity decoding methods can be used for these codes the structure theorem generalizes to other characteristics and to other lengths some comparisons of the methods using earlier examples are given

on the trellis representation of the delsarte goethals codes in this correspondence the trellis representation of the kerdock and delsarte goethals codes is addressed it is shown that the states of a trellis representation of dg m spl delta under any bit order are either strict sense nonmerging or strict sense nonexpanding except maybe at indices within the code s distance set for spl delta spl ges 3 and for m spl ges 6 the state complexity s sub max dg m spl delta is found for all values of m and spl delta a formula for the number of states and branches of the biproper trellis diagram of dg m spl delta is given for some of the indices and upper and lower bounds are given for the remaining indices the formula and the bounds refer to the delsarte goethals codes when arranged in the standard bit order

the treewidth of mds and reed muller codes the constraint complexity of a graphical realization of a linear code is the maximum dimension of the local constraint codes in the realization the treewidth of a linear code is the least constraint complexity of any of its cycle free graphical realizations this notion provides a useful parameterization of the maximum likelihood decoding complexity for linear codes in this paper we show the surprising fact that for maximum distance separable codes and reed muller codes treewidth equals trelliswidth which for a code is defined to be the least constraint complexity or branch complexity of any of its trellis realizations from this we obtain exact expressions for the treewidth of these codes which constitute the only known explicit expressions for the treewidth of algebraic codes

optimal soft decision block decoders based on fast hadamard transform an approach for efficient utilization of fast hadamard transform in decoding binary linear block codes is presented computational gain is obtained by employing various types of concurring codewords and memory reduction is also achieved by appropriately selecting rows for the generator matrix the availability of these codewords in general and particularly in some of the most frequently encountered codes is discussed

on the generalized hamming weights of product codes the rth generalized hamming weight of a linear code is the minimum support size of any r dimensional subcode it has been found useful in the studies of cryptography and trellis coding the authors derive several results on expressing the generalized hamming weights of a product code in terms of those of its component codes they also formulate a general conjecture

a goppa like bound on the trellis state complexity of algebraic geometric codes for a linear code c of length n and dimension k wolf 1978 noticed that the trellis state complexity s c of c is upper bounded by w c min k n k we point out some new lower bounds for s c in particular if c is an algebraic geometric code then s c spl ges w c g a where g is the genus of the underlying curve and a is the abundance of the code

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

bounds on the dimension of codes and subcodes with prescribed contraction index abstract let c be a linear code over gf q spanned by the rows of a matrix g of rank k a nonnegative integer is said to be the contraction index of c if a maximal set of pairwise linearly independent columns of g has k elements we derive several upper and lower bounds on the dimension of a proper subcode of c with a prescribed contraction index v q of length n minimum hamming distance d and contraction index for certain values of n and d the latter bound is shown to be tight for all q and this substantially generalizes the results obtained by delsarte and by duc for 1

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

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

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

critical dynamics in systems controlled by fractional kinetic equations abstract the field theory renormalization group is used for analyzing the fractional langevin equation with the order of the temporal derivative 0 1 fractional laplacian of the order and gaussian noise correlator the case of non linearity m with odd m 3 is considered it is proved that the model is multiplicatively renormalizable propagators were found in the momentum and coordinate representation expressed in terms of fox s h functions existence of the dissipative scaling regime in the framework of the e expansion for 2 1 l l 1 2 is proved requirement of the continuous dependence of the critical exponents on imposes the condition m 3 the main quantitative result is the calculation of the dynamical critical exponent z for 1 2 up to e 2 we have obtained for it the expression z 1 2 4 0 1555 e 2 o e 3

controlled lagrangians and the stabilization of mechanical systems i the first matching theorem we develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled lagrangians the procedure involves making structured modifications to the lagrangian for the uncontrolled system thereby constructing the controlled lagrangian the euler lagrange equations derived from the controlled lagrangian describe the closed loop system where new terms in these equations are identified with control forces since the controlled system is lagrangian by construction energy methods can be used to find control gains that yield closed loop stability we use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group the procedure is demonstrated for several underactuated balance problems including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

trellis oriented decomposition and trellis complexity of composite length cyclic codes the trellis complexity of composite length cyclic codes clcc s is addressed we first investigate the trellis properties of concatenated and product codes in general known factoring of clcc s into concatenated subcodes is thereby employed to derive upper bounds on the minimal trellis size and state space profile new decomposition of clcc s into product subcodes is established and utilized to derive further upper hounds on the trellis parameters the coordinate permutations that correspond to these bounds are exhibited additionally new results on the generalized hamming weights of clcc s are obtained the reduction in trellis complexity of many clcc s leads to soft decision decoders with relatively low complexity

bit level soft decision decoding of reed solomon codes a reed solomon decoder that makes use of bit level soft decision information is presented a reed solomon generator matrix that possesses a certain inherent structure in gf 2 is derived this structure allows the code to be represented as a union of cosets each coset being an interleaver of several binary bch codes such partition into cosets provides a clue for efficient bit level soft decision decoding two decoding algorithms are derived in the development of the first algorithm a memoryless channel is assumed making the value of this algorithm more conceptual than practical the second algorithm which is obtained as a modification of the first does account for channel memory and thus accommodates a bursty channel both decoding algorithms are in many cases orders of magnitude more efficient than conventional techniques

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

on the problem of finding zero concurring codewords zero concurring codewords disclose a certain structure of the code that may be used for efficient soft decision decoding and for designing dc free codes methods for constructing sets of zero concurring codewords are presented for several families of codes for the general case an algorithm solution of the problem is offered a table of results obtained using the proposed techniques is supplied for all the primitive narrow sense binary bch codes of length up to 127

control of nonlinear underactuated systems in this paper we introduce a new method to design control laws for nonlinear underactuated systems our method produces an infinite dimensional family of control laws whereas most control techniques only produce a finite dimensional family these control laws each come with a natural lyapunov function the inverted pendulum cart is used as an example in addition we construct an abstract system that is open loop unstable and cannot be stabilized using any linear control law and demonstrate that our method produces a stabilizing control law 2000 john wiley sons inc

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

the twisted squaring construction trellis complexity and generalized weights of bch and qr codes the structure of the twisted squaring construction a generalization of the squaring construction is studied with respect to trellis diagrams and complexity we show that binary affine invariant codes which include the extended primitive bch codes and the extended binary quadratic residue codes are equivalent to twisted squaring construction codes in particular a recursive symmetric reversible design of the bch codes is derived using these constructions the parameters of the minimal trellis diagram of the bch codes are determined including the componentwise state space profile and trellis complexity new designs and permutations that yield low trellis complexity for the quadratic residue codes are presented generalized hamming weights are derived from these constructions as an example the 48 24 12 quadratic residue code is analyzed a strictly componentwise optimal permutation is derived and the corresponding state space profile and complete generalized hamming weight hierarchy are obtained