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

levy walk with multiple internal states levy walk with multiple internal states can effectively model the motion of particles that don t immediately move back to the directions or areas which they come from when the levy walk behaves superdiffusion it is discovered that the non immediately repeating property characterized by the constructed transition matrix has no influence on the particle s mean square displacement msd or pearson coefficient this is a kind of stable property of levy walk however if the levy walk shows the dynamical behaviors of normal diffusion then the effect of non immediately repeating emerges for the levy walk with some particular transition matrices it may display nonsymmetric dynamics in these cases the behaviors of their variances are detailedly discussed especially some comparisons with the ones of the continuous time random walks are made a striking difference is the changes of the exponents of the variances the first passage time distribution and its average of levy walks are simulated the results of which turn out that the first passage time can distinguish levy walks with different transition matrices while the msd can not

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

large deviations estimates for systems with small noise effects and applications to stochastic systems theory for typical stochastic systems e g tracking systems estimates of the behavior are hard to get if the noise effects are small then asymptotic methods are appealing to get for example estimates of times required to lose track etc in this paper systems with small noise effects and wide bandwidth noise inputs are analyzed via large deviations methods inputs to many systems in control communication or in physics are not white noise but of certain wide bandwidth types since large deviations results can be sensitive to the actual noise model used working with a model that is close to the physical form is important several such models are dealt with where the bandwidth is large but the intensity small for the models chosen the action functionals turn out to be the same as for the small gaussian white noise models thus it is actually feasible to do computations with them estimates of the probability that the path lies in various sets are obtained the formula for the mean esca

generalized hamming weights of nonlinear codes and the relation to the z sub 4 linear representation we give a new definition of generalized hamming weights of nonlinear codes and a new interpretation connected with it these generalized weights are determined by the entropy length profile of the code we show that this definition characterizes the performance of nonlinear codes on the wire tap channel of type ii the new definition is invariant under translates of the code it satisfies the property of strict monotonicity and the generalized singleton bound we check the relations between the generalized weight hierarchies of z sub 4 linear codes and their binary image under the gray map we also show that the binary image of a z sub 4 linear code is a symmetric not necessarily rectangular code moreover if this binary image is a linear code then it admits a twisted squaring construction

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

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

nonlinear equations and weighted norm inequalities we study connections between the problem of the existence of positive solutions for certain nonlinear equations and weighted norm inequalities in particular we obtain explicit criteria for the solvability of the dirichlet problem u v u w u 0 on u 0 on on a regular domain in rn in the superlinear case q 1 the coefficients v w are arbitrary positive measurable functions or measures on we also consider more general nonlinear differential and integral equations and study the spaces of coefficients and solutions naturally associated with these problems as well as the corresponding capacities our characterizations of the existence of positive solutions take into account the interplay between v w and the corresponding green s kernel they are not only sufficient but also necessary and are established without any a priori regularity assumptions on v and w we also obtain sharp two sided estimates of solutions up to the boundary some of our results are new even if v 1 and is a ball or half space the corresponding weighted norm inequalities are proved for integral operators with kernels satisfying a refined version of the so called 3g inequality by an elementary integration by parts argument this also gives a new unified proof for some classical inequalities including the carleson measure theorem for poisson integrals and trace inequalities for riesz potentials and green potentials

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

boundary singularities of solutions to elliptic viscous hamilton jacobi equations abstract we study the boundary value problem with measures for e1 u g u 0 in a bounded domain in r n satisfying e2 u on and prove that if g l 1 1 t 2 n 1 n d t is nondecreasing e1 e2 can be solved with any positive bounded measure when g r r q with q 1 we prove that any positive function satisfying e1 admits a boundary trace which is an outer regular borel measure not necessarily bounded when g r r q with 1 q q c n 1 n we prove the existence of a positive solution with a general outer regular borel measure as boundary trace and characterize the boundary isolated singularities of positive solutions when g r r q with q c q 2 we prove that a necessary condition for solvability is that must be absolutely continuous with respect to the bessel capacity c 2 q q q we also characterize boundary removable sets for moderate and sigma moderate solutions

the preparata and goethals codes trellis complexity and twisted squaring constructions the trellis complexity of the preparata and goethals codes is examined it is shown that at least for a given set of permutations these codes are rectangular upper bounds on the state complexity profiles of the preparata and goethals codes are given the upper bounds on the state complexity of the preparata and goethals codes are determined by the dimension length profiles dlp of the extended primitive double and triple error correcting bch codes respectively a twisted squaring construction for the preparata and goethals codes is given based on the double and triple error correcting extended primitive bch codes respectively

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

diffusion across characteristic boundaries with critical points we consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with i an asymptotically stable equilibrium point or limit cycle and ii an equilibrium point surrounded by closed trajectories the mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points uniformly valid asymptotic solutions are constructed for each of the problems for the asymptotically stable equilibrium point the method of matched asymptotic expansions with the integral condition of matkowsky and schuss is employed a method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories the influence on the solutions of the critical points on the boundary is exhibited and explained an application to the physical pendulum is given finally our r

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

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

boundary singularities of solutions of semilinear elliptic equations with critical hardy potentials we study the boundary behaviour of positive functions u satisfying e gd u frac xk d 2 x u g u 0 in a bounded domain gw of bbr n where 0 xk leq frac 1 4 g is a continuous nonndecreasing function and d is the distance function to prt gw we first construct the martin kernel associated to the the linear operator cl xk gd frac xk d 2 x and give a general condition for solving equation e with any radon measure gm for boundary data when g u u q 1 u we show the existence of a critical exponent q c q c n xk 1 whith the following properties when 0 q q c any measure is eligible for solving e with gm for boundary data if q geq q c a necessary and sufficient condition is expressed in terms of the absolute continuity of gm with respect to some besov capacity the same capacity characterizes the removable compact boundary sets at end any positive solution f gd u frac xk d 2 x u u q 1 u 0 with q 1 admits a boundary trace which is a positive outer regular borel measure when 1 q q c we prove that to any positive outer regular borel measure we can associate a positive solutions of f with this boundary trace

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 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

sharp two sided heat kernel estimates for critical schrodinger operators on bounded domains on a smooth bounded domain omega subset bf rm r n we consider the schrodinger operators v with v being either the critical borderline potential v x n 2 2 4 x 2 or v x 1 4 dist x 2 under dirichlet boundary conditions in this work we obtain sharp two sided estimates on the corresponding heat kernels to this end we transform the schrodinger operators into suitable degenerate operators for which we prove a new parabolic harnack inequality up to the boundary to derive the harnack inequality we have established a series of new inequalities such as improved hardy logarithmic hardy sobolev hardy moser and weighted poincare as a byproduct of our technique we are able to answer positively to a conjecture of e b davies

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

existence and nonexistence of solutions to elliptic equations involving the hardy potential abstract the purpose of this paper is to study the nonexistence of nonnegative super solutions to the problem 0 1 u x 2 u q u p in r n k where 0 1 r p 0 k is a compact set in r n with n 1 and q is a potential in r n k satisfying that lim inf x q x x 0 for some 2 when 1 is the laplacian operator and when 0 1 it is the fractional laplacian which is a typical nonlocal operator in this paper we find the critical exponent p 1 depending on and such that problem 0 1 has no nontrivial nonnegative super solutions for 0 p p furthermore we also consider the existence and nonexistence of isolated singular solutions to the equation u x 2 u q u p in r n 0 lim x u x 0 where 0 p 0 and q x 1 x with 0 2

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 the existence of weak solutions of semilinear elliptic equations and systems with hardy potentials abstract let r n n 3 be a bounded c 2 domain and x dist x put l 2 with 0 in this paper we provide various necessary and sufficient conditions for the existence of weak solutions to l u u p in u on where 0 p 0 and are measures on and respectively we then establish existence results for the system l u v p in l v u p in u v on where 1 p 0 p 0 and are measures on and are measures on we also deal with elliptic systems where the nonlinearities are more general

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 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

an elliptic semilinear equation with source term and boundary measure data the supercritical case we give new criteria for the existence of weak solutions to equation with source term begin align delta u u q text in omega u sigma text on partial omega end align where q 1 omega is a either a bounded smooth domain or mathbb r n and sigma in mathfrak m partial omega is a nonnegative radon measure on partial omega in particular one of the criteria is expressed in terms of some bessel capacities on partial omega we also give a sufficient condition for the existence of weak solutions to equation with source mixed term begin align delta u u q 1 1 u nabla u q 2 text in omega u sigma text on partial omega end align where q 1 q 2 geq 0 q 1 q 2 1 q 2 2 sigma in mathfrak m partial omega is a radon measure on partial omega

conditionally cycle free graphical models for coset codes conditionally cycle free graphical models i e cyclic graphical models which become cycle free after conditioning on a subset of the hidden variables are constructed for coset codes following the description of a general construction procedure examples of a number of families of codes including first order reed muller rm and the delsarte goethals codes are provided for which the proposed procedure yields optimal soft in soft out siso decoding algorithms that are less complex than the best known trellis based algorithms in the case of the first order rm codes which have a recursive coset construction the optimal siso decoding algorithm that results when the proposed construction is applied repeatedly is denoted recursive coset representation rcr decoding connections are made between rcr decoding and existing algorithms that exploit fast hadamard transforms finally the utility of the proposed decoding algorithms are supported by a practically motivated application the construction of serially concatenated codes that have high rates and low error floors extended hamming codes are proposed as outer codes in such constructions with efficient decoding employing the siso algorithms developed herein

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

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

large deviations results for the exit problem with characteristic boundary abstract we consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region d we show the standard large deviations results for the exit distribution and mean exit time as obtained by wentzell and freidlin under the assumption of nontangential drift b n

limiting exit location distributions in the stochastic exit problem consider a two dimensional continuous time dynamical system with an attracting fixed point s if the deterministic dynamics are perturbed by white noise random perturbations of strength epsilon the system state will eventually leave the domain of attraction omega of s we analyze the case when as epsilon to0 the exit location on the boundary partial omega is increasingly concentrated near a saddle point h of the deterministic dynamics we show using formal methods that the asymptotic form of the exit location distribution on partial omega is generically non gaussian and asymmetric and classify the possible limiting distributions a key role is played by a parameter mu equal to the ratio lambda s h lambda u h of the stable and unstable eigenvalues of the linearized deterministic flow at h if mu 1 then the exit location distribution is generically asymptotic as epsilon to0 to a weibull distribution with shape parameter 2 mu on the cal o epsilon mu 2

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

a singular perturbation method for the computation of the mean first passage time in a nonlinear filter we give a new application of recently developed singular perturbation methods in the area of mathematical theory of nonlinear filtering we consider the phenomenon of cycle slipping in a second order phase locked loop pll which serves as demodulator for a random fm message we introduce new scaling parameters into the ito system of stochastic differential equations describing the pll thus identifying the phenomenon of cycle slipping with kolmogorov s exit problem we use singular perturbation methods to obtain an explicit expression for the mean time between cycle slips furthermore we describe the mechanism of cycle slips and identify new parameters which determine the probability of their occurrence

the exit distribution on the stochastic separatrix in kramers exit problem kramers exit problem is concerned with noise activated escape from a potential well in the case when the noise strength varepsilon temperature measured in units of potential barrier height is small this becomes a singular perturbation problem unexpectedly the noise induced escape from the domain of attraction of the stable equilibrium at the bottom of the well does not occur at points on the separatrix where the energy is minimal rather the distribution of exit points on the separatrix in the phase plane is shown to be spread away from the saddle point where the energy is minimal it is shown that actually most of the probability of the exit points on the separatrix is located at a distance o sqrt varepsilon from the saddle point and the probability vanishes altogether at the saddle point in this problem large deviations theory fails to predict the distribution of the exit point for finite noise we construct an asymptotic approximation to the solution of the fokker planck eq