darboux transformations and supersymmetry for the generalized schrodinger equations in 1 1 dimensions we study intertwining relations supersymmetry and darboux transformations for generalized schrodinger equations in 1 1 dimensions we obtain intertwiners in an explicit form by means of which we construct arbitrary order darboux transformations for our class of equations we develop a corresponding supersymmetric formulation and prove equivalence of the darboux transformations with the supersymmetry formalism finally we show that our darboux transformations can also be constructed by means of point transformations avoiding the use of intertwiners

constacyclic codes cocycles and a u v u v construction a connection between cohomology cocycles and constacyclic codes is explored it suggests an isomorphism between cyclic codes of length mn and a direct sum of m constacyclic codes of length n the isomorphism is used i to study the discrete fourier transforms and the decomposition of group ring codes ii to give a u v u v construction over gf q when q is odd the u v u v construction gives some of the best ternary cyclic codes and classes of nearly mds cyclic codes of length 2q 2 the symmetry of the construction is used to give a new proof that there are no cyclic self dual codes over gf q when q is odd

improved bounded strength decoupling schemes for local hamiltonians we address the task of switching off the hamiltonian of a system by removing all internal and system environment couplings we propose dynamical decoupling schemes that use only bounded strength controls for quantum many body systems with local system hamiltonians and local environmental couplings to do so we introduce the combinatorial concept of balanced cycle orthogonal arrays boas and show how to construct them from classical error correcting codes the derived decoupling schemes may be useful as a primitive for more complex schemes e g for hamiltonian simulation for the case of n qubits and a two local hamiltonian the length of the resulting decoupling scheme scales as o n log n improving over the previously best known schemes that scaled quadratically with n more generally using boas constructed from families of bose chaudhuri hocquenghem bch codes we show that bounded strength decoupling for any ell local hamiltonian where ell geq 2 can be achieved using decoupling schemes of length at most o n ell 1 log n

on the algebraic structure of quasi cyclic codes i finite fields a new algebraic approach to quasi cyclic codes is introduced the key idea is to regard a quasi cyclic code over a field as a linear code over an auxiliary ring by the use of the chinese remainder theorem crt or of the discrete fourier transform dft that ring can be decomposed into a direct product of fields that ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the u spl upsi u spl upsi and vandermonde constructions all binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction quinting and septing constructions are introduced other results made possible by the ring decomposition are a characterization of self dual quasi cyclic codes and a trace representation that generalizes that of cyclic codes

how many weights can a quasi cyclic code have we investigate the largest number of nonzero weights of quasi cyclic codes in particular we focus on the function gamma q n ell k q that is defined to be the largest number of nonzero weights a quasi cyclic code of index gcd ell n length n and dimension k over mathbb f q can have and connect it to similar functions related to linear and cyclic codes we provide several upper and lower bounds on this function using different techniques and studying its asymptotic behavior moreover we determine the smallest index for which a q ary reed muller code is quasi cyclic a result of independent interest

a bounded distance decoding algorithm for the leech lattice with generalizations an algorithm is given that decodes the leech lattice with not much more than twice the complexity of soft decision decoding of the golay code the algorithm has the same effective minimum distance as maximum likelihood decoding and increases the effective error coefficient by less than a factor or two the algorithm can be recognized as a member of the class of multistage algorithms that are applicable to hierarchical constructions it is readily generalized to lattices that can be expressed in terms of binary code formulas and in particular to construction b lattices

multilevel codes based on partitioning following v v ginzburg 1984 a hierarchy of codes is proposed to match the geometric partitioning of a signal set the authors show that coset codes including ungerboeck lattice and binary codes and indeed any codes which rely on partitioning of the signal set are all subclasses of the proposed coding scheme the combination of such codes in a multilevel scheme often leads to reduced complexity in comparison with previously published schemes a variety of decoder structures is discussed

new optimal ternary linear codes the class of quasi twisted qt codes is a generalization of the class of quasi cyclic codes similar to the way constacyclic codes are a generalization of cyclic codes in this paper rate 1 p qt codes over gf 3 are presented which have been constructed using integer linear programming and heuristic combinatorial optimization many of these attain the maximum possible minimum distance for any linear code with the given parameters and several improve the maximum known minimum distances two of these new codes namely 90 6 57 and 120 6 78 are optimal and so prove that d sub 3 90 6 57 and d sub 3 120 6 78

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

how many weights can a cyclic code have upper and lower bounds on the largest number of weights in a cyclic code of given length dimension and alphabet are given an application to irreducible cyclic codes is considered sharper upper bounds are given for the special cyclic codes called here strongly cyclic whose nonzero codewords have period equal to the length of the code asymptotics are derived on the function gamma k q that is defined as the largest number of nonzero weights a cyclic code of dimension k over mathbb f q can have and an algorithm to compute it is sketched the nonzero weights in some infinite families of reed muller codes either binary or q ary as well as in the q ary hamming code are determined two difficult results of independent interest

soft decoding techniques for codes and lattices including the golay code and the leech lattice two kinds of algorithms are considered 1 if is a binary code of length n a soft decision decoding algorithm for changes an arbitrary point of r n into a nearest codeword nearest in euclidean distance 2 similarly a decoding algorithm for a lattice lambda in r n changes an arbitrary point of r n into a closest lattice point some general methods are given for constructing such algorithms ami are used to obtain new and faster decoding algorithms for the gosset lattice e 8 the golay code the leech lattice

efficiency and proper efficiency in nonlinear vector maximum problems abstract a nonlinear vector maximum problem nlvmp is considered an equivalent vector maximum problem vmp is formulated by linearizing the objectives of nlvmp assuming generalized convexity and a constraint qualification necessary and sufficient conditions in both kuhn tucker and isermann form are obtained for an efficient solution of nlvmp also if objectives are pseudolinear and satisfy a certain boundedness condition it is proved that every efficient solution of nlvmp is properly efficient finally a numerical example is discussed to illustrate the results

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

z p sup k 1 linear codes we characterize codes over z sub p which are the gray images of 1 p sup k cyclic codes or cyclic codes over z p sup k l k spl ges 1 a necessary and sufficient condition for the gray image of a z p sup 2 linear 1 p cyclic code to be linear is given in many cases this yields an explicit description of the gray image of a linear 1 p cyclic code over z p sup 2 of length relatively prime to p linear cyclic codes over z p sup 2 whose gray images are linear cyclic codes over z sub p have been characterized some generalizations of these results to the case of z p sup k 1 where k spl ges 2 are also obtained

a class of optimum block codes in signal space by using convolutional coding coupled with expanded sets of channel signals ungerboeck showed that it is possible to obtain coding gains of several decibels at a given data rate without increasing the transmission bandwidth in this paper we construct a class of optimum signal space codes using short binary block as opposed to convolutional codes our results are compared to those obtained by ungerboeck a simplified decoding procedure that is asymptotically optimum is also presented

linear constructions for dna codes in this paper we translate in terms of coding theory constraints that are used in designing dna codes for use in dna computing or as bar codes in chemical libraries we propose new constructions for dna codes satisfying either a reverse complement constraint a gc content constraint or both that are derived from additive and linear codes over four letter alphabets we focus in particular on codes over gf 4 and we construct new dna codes that are in many cases better sometimes far better than previously known codes we provide updated tables up to length 20 that include these codes as well as new codes constructed using a combination of lexicographic techniques and stochastic search

multilevel trellis coded modulations for the rayleigh fading channel the authors present coded 8 phase shift keyed 8 psk modulations for the rayleigh fading channel the schemes are based on multilevel trellis coded modulation constructions and utilize maximum free hamming distance binary convolutional codes as building blocks a suboptimal multistage decoder that utilizes interstage interleaving and iterative decoding is proposed and evaluated examples are constructed to show that the proposed schemes outperform the best modified codes of the ungerboeck type due to significantly higher implicit time diversity yielding seven branches of built in time diversity whereas the ungerboeck code yields four branches of time diversity for a 64 state system the transmission delay is higher however the new schemes can provide three levels of unequal error protection when 8 psk or 8 differential phase shift keying 8 dpsk modulations are used they provide 10 14 db channel signal to noise ratio gain over uncoded 4 dpsk at a bit error rate of 10 sup 3 for a modest decoding complexity

finding cyclic redundancy check polynomials for multilevel systems this letter describes a technique for finding cyclic redundancy check polynomials for systems for transmission over symmetric channels which encode information in multiple voltage levels so that the resulting redundancy check gives good error protection and is efficient to implement the codes which we construct have a hamming distance of 3 or 4 we discuss a way to reduce burst error in parallel transmissions and some tricks for efficient implementation of the shift register for these polynomials we illustrate our techniques by discussing a particular example where the number of levels is 9 but they are applicable in general

channel coding with multilevel phase signals a coding technique is described which improves error performance of synchronous data links without sacrificing data rate or requiring more bandwidth this is achieved by channel coding with expanded sets of multilevel phase signals in a manner which increases free euclidean distance soft maximum likelihood ml decoding using the viterbi algorithm is assumed following a discussion of channel capacity simple hand designed trellis codes are presented for 8 phase shift keying psk and 16 quadrature amplitude shift keying qask modulation these simple codes achieve coding gains in the order of 3 4 db it is then shown that the codes can be interpreted as binary convolutional codes with a mapping of coded bits into channel signals which we call mapping by set partitioning based on a new distance measure between binary code sequences which efficiently lower bounds the euclidean distance between the corresponding channel signal sequences a search procedure for more powerful codes is developed codes with coding gains up to 6 db are obtained for a variety of multilevel phase modulation schemes simulation results are presented and an example of carrier phase tracking is discussed

optimality theorems for convex semidefinite vector optimization problems abstract in this paper we consider a convex semidefinite vector optimization problem sdvp involving a convex objective vector function a matrix linear inequality constraint and a geometric constraint and define properly weakly efficient solutions for sdvp as we do for ordinary vector optimization problems we present necessary and sufficient optimality conditions for efficient solutions of sdvp which hold without any constraint qualification also we present necessary and sufficient optimality conditions for properly efficient solutions and weakly efficient solutions for sdvp which hold without any constraint qualification furthermore we give a sufficient condition for an efficient solution of sdvp to be properly efficient

coset codes i introduction and geometrical classification practically all known good constructive coding techniques for bandlimited channels including lattice codes and various trellis coded modulation schemes can be characterized as coset codes a coset code is defined by a lattice partition lambda lambda and by a binary encoder c that selects a sequence of cosets of the lattice lambda the fundamental coding gain of a coset code as well as other important parameters such as the error coefficient the decoding complexity and the constellation expansion factor are purely geometric parameters determined by c lambda lambda the known types of coset codes as well as a number of new classes that systematize and generalize known codes are classified and compared in terms of these parameters

a vandermonde code construction we use a vandermonde matrix to construct a code of length mn from m codes of length n a distance estimate for the construction is given when the starting codes are suitable constacyclic codes the constructed code is cyclic and from this we obtain new cyclic codes the results generalize those of hughes see ibid vol 46 p 674 80 2000 on the u spl upsi u spl upsi construction

new quasi twisted quaternary linear codes let n k d sub q codes be linear codes of length n dimension k and minimum hamming distance d over gf q in this article 31 new quaternary q 4 codes are constructed which improve the respective lower bounds on minimum distance in brouwer s tables

pseudolinearity and efficiency first order and second order characterizations of pseudolinear functions are derived for a nonlinear programming problem involving pseudolinear functions only it is proved that every efficient solution is properly efficient under some mild conditions

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

new multilevel codes over gf q set partitioning is applied to multidimensional signal spaces over gf q i e gf sup n1 q n1 or d are presented these codes use reed solomon codes as component codes longer multilevel block codes are also constructed using q ary block codes with block length longer than q 1 as component codes some quaternary multilevel block codes are presented with the same length and number of information symbols as but larger distance than the best previously known quaternary one level block codes it is proved that if all the component block codes are linear the multilevel block code is also linear low rate q ary convolutional codes word error correcting convolutional codes and binary to q ary convolutional codes can also be used to construct multilevel trellis codes over gf q or binary to q ary trellis codes

green s functions sum rules and matrix elements for susy partners it is now well known that the solutions in the schrodinger equations for two potentials which are supersymmetric partners are linked by intertwining relations involving first order differential operators in this paper we explore the consequence of this linkage for the green s functions corresponding to susy partners and the relation between the sums over the inverses of the eigenvalues for the two potentials we also establish some relations between the matrix elements of certain operators evaluated between the eigenstates of two partner potentials we show that there can be circumstances where some matrix elements vanish as a consequence of the presence of supersymmetry

some ternary and quaternary codes and associated sphere packings tables of good ternary and quaternary codes are presented and they are used in the construction of dense sphere packings results include 1 tables of the best ternary and quaternary constacyclic codes including cyclic codes up to block length 50 2 a class of optimal n 2 codes over gf q 3 the u v w mod 2u v mod u construction a new ternary code construction technique that can be used to construct the ternary reed muller codes and others and 4 tables of linear ternary and quaternary codes obtained by modifying and combining various codes here and in the literature packings are generated in even dimensions up to 100 using these codes and a sphere packing construction in dimensions 36 and 60 new record densities appear to have been achieved

new ternary and quaternary linear block codes we present several new ternary and quaternary codes found with a heuristic search procedure originally proposed by the authors to find good binary codes and which is based on combinatorial optimization

a trace formula for dirac green s functions related by darboux transformations we construct green s functions and a trace formula for pairs of stationary dirac equations under sturm liouville boundary conditions where the equations are related to each other by a darboux transformation our findings generalize former results pozdeeva e 2008 j phys a math theor at press

trellis coded modulation with multidimensional constellations trellis coded modulation schemes using four eight or 16 dimensional constellations have a number of potential advantages over the usual two dimensional schemes a smaller constituent two dimensional constellation easier tolerance to phase ambiguities and a better trade off between complexity and coding gain a number of such schemes are presented and evaluated starting with a variety of multidimensional lattices we show how to select multidimensional constellations how to partition them into subsets how to construct trellis codes using those subsets and how to map hits to constellation points simplifications of the viterbi decoding algorithm are presented we conclude that there are multidimensional trellis coded modulation schemes that perform better for the same complexity than do two dimensional schemes

bounds on mixed binary ternary codes upper and lower bounds are presented for the maximal possible size of mixed binary ternary error correcting codes a table up to length 13 is included the upper bounds are obtained by applying the linear programming bound to the product of two association schemes the lower bounds arise from a number of different constructions

statistical analysis of the product high order ambiguity function the high order ambiguity function haf was introduced for the estimation of polynomial phase signals pps embedded in noise since the haf is a nonlinear operator it suffers from noise masking effects and from the appearance of undesired cross terms and possibly spurious harmonics in the presence of multicomponent mc signals the product haf phaf was then proposed as a way to improve the performance of the haf in the presence of noise and to solve the ambiguity problem in this correspondence we derive a statistical analysis of the phaf in the presence of additive white gaussian noise awgn valid for high signal to noise ratio snr and a finite number of data samples the analysis is carried out in detail for single component pps but the multicomponent case is also discussed error propagation phenomena implicit in the recursive structure of the phaf based estimator are explicitly taken into account the analysis is validated by simulation results for both single and multicomponent ppss

on the performance evaluation of trellis codes generating function techniques for analyzing the error event and the bit error probabilities for trellis codes are considered the conventional state diagram approach for linear codes where the number of states is equal to the number of trellis states cannot be applied directly to arbitrary trellis codes and instead a state diagram where the number of states is equal to the square of the number of trellis states must be used it is shown that for an interesting class of trellis codes a modified generating function can be defined for which the number of states em is equal to the number of trellis states the class of codes considered includes trellis codes of rate r n 1 n based upon set partitioning whenever the first partition breaks the signal constellation into two subsets which have the same configuration matrix i e the same ordered set of mutual distances the complexity of calculating this modified generating function is the same as for the ordinary generating function of a convolutional code with the same number of trellis states bounds on the performance of some interesting codes are given based upon this method

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

coded phase frequency modulation channel coding combined with expanded signal sets has been shown 1 to improve error performance over uncoded modulation without expanding the bandwidth of the transmitted signals in this paper new coded modulation formats defined over an expanded set of signals varying both in phase and frequency are presented the new schemes combine fsk and psk modulation and make use of trellis coding and viterbi decoding to improve error performance over uncoded modulation the free euclidean distance is calculated for several classes of codes and upper bounds and simulation results are also presented for some simple codes the spectral characteristics of the proposed coded modulation formats are evaluated and compared to conventional two dimensional modulation formats differential encoding and various extensions of the basic scheme are also discussed

green s functions and trace formulas for generalized sturm liouville problems related by darboux transformations we study green s functions of the generalized sturm liouville problems that are related to each other by darboux equivalently supersymmetrical transformations we establish an explicit relation between the corresponding green s functions and derive a simple formula for their trace the class of equations considered here includes the conventional schrodinger equation and generalizations such as for position dependent mass and with linearly energy dependent potential as well as the stationary fokker planck equation

new trellis codes based on lattices and cosets a new technique is proposed for constructing trellis codes which provides an alternative to ungerboeck s method of set partitioning the new codes use a signal constellation consisting of points from an n dimensional lattice lambda with an equal number of points from each coset of a sublattice lambda one part of the input stream drives a generalized convolutional code whose outputs are cosets of lambda while the other part selects points from these cosets several of the new codes are better than those previously known

construction of fixed length insertion deletion correcting runlength limited codes an algorithm is presented for the construction of fixed length insertion deletion correcting runlength limited rll codes in this construction binary d k constrained codewords are generated by codewords of a q ary lee metric based code it is shown that this new construction always yields better codes than known constructions the use of a q ary lee 1987 metric code q odd is based on the assumption that an error insertion deletion or peak shift has maximal size q 1 2 it is shown that a decoding algorithm for the lee metric code can be extended so that it can also be applied to insertion deletion correcting rll codes furthermore such an extended algorithm can also correct some error patterns containing errors of size more than q 1 2 as a consequence if s denotes the maximal size of an error then in some cases the alphabet size of the generating code can be s 1 instead of 2 spl middot s 1

explicit darboux transformations of arbitrary order for generalized time dependent schrodinger equations we construct darboux transformations of arbitrary order for a generalized linear time dependent schrodinger equation special cases of which correspond to time dependent hamiltonians coupled to a magnetic field with position dependent mass and with weighted energy our darboux transformation reduces correctly to these known cases and also to new generalized schrodinger equations furthermore fundamental properties of the conventional darboux transformation are maintained such as factorization of the nth order transformation into first order transformations and existence of a reality condition for the transformed potentials

multilevel codes and multistage decoding h imai and s hirakawa have proposed 1977 a multilevel coding method based on binary block codes that admits a staged decoding procedure the author extends the coding method to coset codes and shows how to calculate minimum squared distance and path multiplicity in terms of the norms and multiplicities of the different cosets the multilevel structure allows the redundancy in the coset selection procedure to be allocated efficiently among the different levels it also allows the use of suboptimal multistage decoding procedures that have performance complexity advantages over maximum likelihood decoding

new optimal quaternary linear codes of dimension 5 in this correspondence new optimal quaternary linear codes of dimension 5 are presented which extend previously known results these codes belong to the class of quasi twisted qt codes and have been obtained using a greedy local search algorithm other codes are also given which provide a lower bound on the maximum possible minimum distance the generator polynomials for these codes are tabulated and the minimum distances of known qt codes are given

some best rate 1 p and rate p 1 p systematic quasi cyclic codes over gf 3 and gf 4 the class of quasi cyclic qc codes has been proven to contain many good codes to date the known qc codes are primarily rate 1 p and p 1 p binary codes constructed from circulant matrices these results are extended to qc codes over gf 3 and gf 4 codes are constructed using integer linear programming and heuristic combinatorial optimization many of these attain the maximum possible minimum distance for any linear code with the same parameters and several improve the maximum known distances the link between power residue pr codes and qc codes is exploited as a means of constructing new qc codes and to initialize the search algorithm previously unknown minimum distances for nonbinary quadratic residue codes are given the minimum distances for the pr codes and the maximum known distances for the qc codes are tabulated

an approach to ungerboeck coding for rectangular signal sets corresp a method of finding good ungerboeck codes for large rectangular quadrature amplitude modulation qam signal sets is described using the concept of euclidean weights due to ungerboeck we prove that a 2 n point basic constellation may be employed to determine exactly the free distance for an ungerboeck coded rectangular 2 m point set when m n 1 bits are uncoded and the remaining bits pass through a rate n 1 n convolutionai encoder it is shown that rate 2 3 encoders may be used to achieve most of the theoretically possible coding gain in the proposed scheme where the effect of the error coefficient on the coding gain has been considered

susy transformation of the green function and a trace formula an integral relation is established between the green functions corresponding to two hamiltonians which are supersymmetric susy partners and in general may possess both discrete and continuous spectra it is shown that when the continuous spectrum is present the trace of the difference of the green functions for susy partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each green function our findings are illustrated by using the free particle example considered both on the whole real line and on a half line

reduced lists of error patterns for maximum likelihood soft decoding a method whereby a substantially reduced family of error patterns called survivors may be created for maximum likelihood soft decoding is introduced the family of survivor depends on the received word a decoder based on this approach first forms the survivors then scores them rather than obtaining the survivors by online elimination of error patterns the use of predetermined lists that represent all families of survivors is proposed