self orthogonality of q ary images of q m ary codes and quantum code construction a code over gf can be imaged or expanded into a code over gf using a basis for the extension field over the base field the properties of such an image depend on the original code and the basis chosen for imaging problems relating the properties of a code and its image with respect to a basis have been of great interest in the field of coding theory in this work a generalized version of the problem of self orthogonality of the q ary image of a qm ary code has been considered given an inner product more generally a bi additive form necessary and sufficient conditions have been derived for a code over a field extension and an expansion basis so that an image of that code is self orthogonal the conditions require that the original code be self orthogonal with respect to several related bi additive forms whenever certain power sums of the dual basis elements do not vanish numerous interesting corollaries have been derived by specializing the general conditions an interesting result for the canonical or regular inner product in fields of characteristic two is that only self orthogonal codes result in self orthogonal images another result is that image of a code is self orthogonal for all bases if and only if trace of the code is self orthogonal except for the case of binary images of 4 ary codes the conditions are particularly simple to state and apply for cyclic codes to illustrate a possible application new quantum error correcting codes have been constructed with larger minimum distance than previously known

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

efficient iterative techniques for soft decision decoding of reed solomon codes two new iterative soft decision decoding methods for reed solomon rs codes are proposed these methods are based on bit level belief propagation bp decoding in order to make bp decoding effective for rs codes we use an extended binary parity check matrix with a lower density and reduced number of 4 cycles compared to the original binary parity check matrix of the code in the first proposed method we take advantage of the cyclic structure of rs codes based on this property we can apply the belief propagation algorithm on any cyclically shifted version of the received symbols with the same binary parity check matrix for each shifted version of received symbols the distribution of reliability values will change and deterministic errors can be avoided this method results in considerable performance improvement of rs codes compared to hard decision decoding the performance is also superior to some popular soft decision decoding methods the second method is based on information correction in bp decoding it means that we determine least reliable bits and by changing their channel information the convergence of the decoder is improved compared to the first method this method needs less bp iterations less complexity but its performance is not as good

applying ant system for solving unequal area facility layout problems ant colony optimization aco is a young metaheuristic algorithm which has shown promising results in solving many optimization problems to date a formal aco based metaheuristic has not been applied for solving unequal area facility layout problems ua flps this paper proposes an ant system as one of the aco variants to solve them as a discrete optimization algorithm the proposed algorithm uses slicing tree representation to easily represent the problems without too restricting the solution space it uses several types of local search to improve its search performance it is then tested using several case problems with different size and setting overall the proposed algorithm shows encouraging results in solving ua flps

a unified method for evaluating the error correction radius of reliability based soft decision algorithms for linear block codes this paper presents a unified method for evaluating the error correction radii of many reliability based soft decision decoding algorithms for binary linear block codes based on this unified method these decoding algorithms as well as their potential improvements can be compared directly the error correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft decision decoder to the transmitted signal sequence it is shown that this problem can be formulated as a constrained optimization problem although no general closed form expression is possible a simple algorithm that always converges to the optimum solution of this optimization problem is presented based on these results the error correction radii of some well known reliability based soft decision decoding algorithms are revisited and compared

filterbank decompositions for non systematic reed solomon codes with applications to soft decoding this paper focuses on reed solomon rs codes which are the most widespread classical error correcting codes recently we have shown that an finite impulse response fir critically subsampled filterbank representation can be derived for some rs codes however this work only addresses rs codes with a non coprime codeword and dataword length seriously limiting its practical usability in this paper an alternative purely algebraic method is presented to construct such a filterbank apart from providing additional insight into the algebraic structure of non systematic rs codes this method is suited to eliminate the non coprimeness constraint mentioned before using this filterbank decomposition a rs code is broken into smaller subcodes that can subsequently be used to build a soft in soft out siso rs decoder it is shown how any rs code written as an fir filterbank can be siso decoded using the filterbank based decoder owing to the importance of systematic rs codes it is shown that any systematic rs code can be decoded using the fir filterbank decomposition this leads to better decoding performance in addition with a slightly lower complexity a further extension towards systematic rs codes is also presented in this paper resulting in an infinite impulse response critically subsampled filterbank representation

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

embedding formulations and complexity for unions of polyhedra it is well known that selecting a good mixed integer programming mip formulation is crucial for effectively obtaining a solution with state of the art solvers although best practices and guidelines for constructing good formulations abound there is rarely a single systematic construction approach that leads to the best possible formulation here we introduce embedding formulations and complexity as a new mip formulation paradigm for systematically constructing formulations for disjunctive constraints that are optimal with respect to size more specifically they yield the smallest possible ideal formulation i e one whose lp relaxation has integral extreme points among all formulations that only use 0 1 auxiliary variables we use the paradigm to characterize optimal formulations for special ordered sets of type 2 and certain piecewise linear functions of two variables we also show that the resultant formulations can provide a significant computational advantage over all known formulations for pi

normalization of complex valued planar vector fields which degenerate along a real curve abstract taking as a start point the recent article of meziani 7 we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve we mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle some related questions e g on semi global solvability and on the normalization of a class of generalized mizohata operators are also discussed

combining wang landau sampling algorithm and heuristics for solving the unequal area dynamic facility layout problem abstract the dynamic facility layout problem dflp is the problem of placing facilities in a certain plant floor for multiple stages so that facilities do not overlap and the sum of the material handling and rearrangement costs are minimized we describe a model where the facilities have unequal areas and the layout for each stage is produced on the continuous plant floor the most difficulty of solving this problem consists in the lack of a powerful optimization method wang landau wl sampling algorithm is an improved monte carlo method and has been successfully applied to solve many optimization problems in this paper we combine the wl sampling algorithm and some heuristic strategies to solve the unequal area dflp in the wl sampling algorithm a vacant point strategy is applied to update layout at one stage to prevent overlapping of facilities and reduce the empty space among facilities a pushing strategy and a pressuring strategy are applied we have tested the proposed algorithm on four groups of cases and the computational results show that the proposed algorithm is effective in solving the unequal area dflp

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

progressive algebraic soft decision decoding of reed solomon codes the algebraic soft decision decoding asd algorithm is a polynomial time soft decoding algorithm for reed solomon rs codes it outperforms both the algebraic hard decision decoding ahd and the conventional unique decoding algorithms but with a high computational cost this paper proposes a progressive asd pasd algorithm that enables the conventional asd algorithm to perform decoding with an adjustable designed factorization output list size ols the ols is enlarged progressively leading to an incremental computation for the interpolation and an enhanced error correction capability multiple factorizations are performed in order to find out the intended message polynomial which will be validated by a cyclic redundant check crc code the incremental interpolation constraints are introduced to characterize the progressive decoding the validity analysis of the algorithm shows the pasd algorithm is a natural and computationally saving generalization of the asd algorithm delivering the same interpolation solution the average decoding complexity of the algorithm is further theoretically characterized revealing its dependence on the channel condition the simulation results further validate the analysis by showing that the average decoding complexity can be converged to the minimal level in a good channel condition finally performance evaluation shows the pasd algorithm preserves the error correction capability of the asd algorithm

improving discrete model representations via symmetry considerations in this paper we focus on a useful modeling concept that is frequently ignored while formu lating discrete optimization problems very often there exists a natural symmetry inherent in the problem itself that if propagated to the model can hopelessly mire a branch and bound solver by burdening it to explore and eliminate such alternative symmetric solutions we discuss three applications where such a symmetry arises a telecommunications network design problem a noise pollution problem and a machine procurement and operation problem for each case we identify the indistinguishable objects in the model that create the problem symmetry and show how imposing certain decision hierarchies within the model significantly enhances its solvability while using a popular modern day commercial branch and cut software cplex 6 5

decoding the golay codes we introduce exceptionally simple decoding algorithms for the two extended golay codes the algorithms are based on recent methods of conway and curtis of finding the unique blocks containing five points in either the 5 8 24 steiner system or the 5 6 12 steiner system these decoding methods are simple enough to enable decoding extended golay codes by hand both of the methods involve relations between the extended golay codes and other self dual codes proofs are given explaining these relationships and why the decoding methods work the decoding algorithms are explained and illustrated with many examples 3 chap 12 has facts about the mathieu group and some details about decoding the golay codes

a convex optimisation framework for the unequal areas facility layout problem the unequal areas facility layout problem is concerned with finding the optimal arrangement of a given number of non overlapping indivisible departments with unequal area requirements within a facility we present a convex optimisation based framework for efficiently finding competitive solutions for this problem the framework is based on the combination of two mathematical programming models the first model is a convex relaxation of the layout problem that establishes the relative position of the departments within the facility and the second model uses semidefinite optimisation to determine the final layout aspect ratio constraints frequently used in facility layout methods to restrict the occurrence of overly long and narrow departments in the computed layouts are taken into account by both models we present computational results showing that the proposed framework consistently produces competitive and often improved layouts for well known large instances when compared with other approaches in the literature

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

efficient bounded distance decoding of the hexacode and associated decoders for the leech lattice and the golay code two soft decision decoding algorithms for the 6 3 4 quaternary code hexacode are presented both algorithms realize half the minimum euclidean distance of the code the proposed algorithms are most practical in using them bounded distance decoding of the golay code and the leech lattice are performed with at most 187 and 519 real number operations respectively compare this to 651 respectively 3595 operations required by the best known maximum likelihood decoders vardy and be ery 1991 1993 and 431 respectively 1007 operations required by the bounded distance decoders amrani et al 1994 we present some simulation results for the proposed leech lattice decoders revealing near optimal performance a comparison to known trellis codes is also provided

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

local solvability inlpof first order linear operators we prove that first order partial differential operators of principal type with smooth coefficients are locally solvable inlp 1 p if they satisfy condition p

the q ary image of some q sup m ary cyclic codes permutation group and soft decision decoding using a particular construction of generator matrices of the q ary image of q sup m ary cyclic codes it is proved that some of these codes are invariant under the action of particular permutation groups the equivalence of such codes with some two dimensional 2 d abelian codes and cyclic codes is deduced from this property these permutations are also used in the area of the soft decision decoding of some expanded reed solomon rs codes to improve the performance of generalized minimum distance decoding

stats a slicing tree and tabu search based heuristic for the unequal area facility layout problem in this paper a slicing tree based tabu search heuristic for the rectangular continual plane facility layout problem flp is presented in addition to the incorporation of facilities with unequal areas we also integrate the possibility to specify various requirements regarding rectangular shape and dimensions of each individual facility by using bounding curves therefore it is possible to solve problems containing facilities of fixed and facilities of flexible shapes at the same time we present a procedure that calculates the layout corresponding to a given slicing tree on the basis of bounding curves these layouts are slicing structures which are able to contain empty spaces to guarantee that stringent shape restrictions of facilities are kept due to these features this approach is better suited for practical use than so far existing ones the effectiveness of our approach in terms of objective function value is shown by comparing our results to those found in the literature even a large problem instance comprised of 62 facilities has been solved

a new treatment of priority first search maximum likelihood soft decision decoding of linear block codes in this correspondence we present a new method to convert the maximum likelihood soft decision decoding problem for linear block codes into a graph search problem where the generalized dijkstra s algorithm can still be applied to the decoding procedure the cost assigned to every branch in the graph is based on a generalization of the wagner rule which is an equivalent form of the maximum likelihood decoding rule the new decoding algorithm uses the properties of error patterns to reduce the search space

performance of reed solomon coded frequency hop spread spectrum communications in partial band interference this paper is concerned with the performance of a communications system which utilizes frequency hop spread spectrum diversity transmission reed solomon coding and parallel error correction and erasure correction decoding both binary signaling and m ary orthogonal signaling are considered the goals are twofold first it is desirable to provide good performance in partial band gaussian noise interference by use of coding and diversity with an efficient error correction algorithm second it is necessary to totally neutralize narrow band interference regardless of the power level or statistical distribution of the interference in order to have an effective spread spectrum system through an analysis of the effects of partial band interference on a frequency hop spread spectrum system with diversity it is shown that the use of reedsolomon coding with a parallel errors and erasures decoding algorithm accomplishes these goals the paper also investigates the accuracy of the chernoff bound as an approximation to the true performance of a frequency hop spreadspectrum communication system with diversity side information m ary orthogonal signaling and reed solomon coding the performance results presented in the paper are based on analysis and computer evaluation approximate results based on the chernoff bound are also given it is shown that the chernoff bound for m ary orthogonal signaling gives a very poor approximation for many cases of interest this is largely due to the looseness of the union bound

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

decoding linear block codes using a priority first search performance analysis and suboptimal version an efficient maximum likelihood soft decision decoding algorithm for linear block codes using a generalized dijkstra s algorithm was proposed by han hartmann and chen 1993 we prove that this algorithm is efficient for most practical communication systems where the probability of error is less than 10 sup 3 by finding an upper bound of the computational effort of the algorithm a suboptimal decoding algorithm is also proposed the performance of this suboptimal decoding algorithm is within 0 25 db of the performance of an optimal decoding algorithm for the 104 52 binary extended quadratic residue code and within 0 5 db of the optimal performance for the 128 64 binary bch code respectively

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

iterative soft decision decoding of hermitian codes this paper proposes an iterative soft decision decoding algorithm for one of the most popular algebraic geometric ag codes hermitian codes the algorithm is designed by integrating the two most powerful soft decision decoding algorithms the adaptive belief propagation abp algorithm and the koetter vardy kv list decoding algorithm the abp algorithm performs iterative decoding based on an adapted parity check matrix of a hermitian code to enhance the reliability of the soft received information with the enhanced reliability the kv algorithm performs soft decision list decoding to obtain the original message since the matrix adaptation relies on bit reliabilities regrouping of the unreliable bits is introduced to assist the abp decoding a complexity reducing abp kv decoding approach is proposed based on assessing the soft information provided by the abp algorithm and determining whether the following kv decoding steps should be carried out geometric interpretation of the abp algorithm is presented demonstrating the necessity of performing matrix adaptation our performance analysis shows the proposed iterative decoding algorithm outperforms both the existing decoding approaches for hermitian codes and the abp kv decoding of reed solomon rs codes

algebraic soft decision decoding of reed solomon codes using bit level soft information the performance of algebraic soft decision decoding of reed solomon codes using bit level soft information is investigated optimal multiplicity assignment strategies for algebraic soft decision decoding sdd with infinite cost are first studied over erasure channels and the binary symmetric channel the corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived the multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic sdd over a mixed error and bit level erasure channel the bit level decoding region of the proposed multiplicity assignment strategy is shown to be significantly larger than that of conventional berlekamp massey decoding as an application a bit level generalized minimum distance decoding algorithm is proposed the proposed decoding compares favorably with many other reed solomon sdd algorithms over various channels moreover owing to the simplicity of the proposed bit level generalized minimum distance decoding its performance can be tightly bounded using order statistics

on mathematical optimization for the visualization of frequencies and adjacencies as rectangular maps in this paper we address the problem of visualizing a frequency distribution and an adjacency relation attached to a set of individuals we represent this information using a rectangular map i e a subdivision of a rectangle into rectangular portions so that each portion is associated with one individual their areas reflect the frequencies and the adjacencies between portions represent the adjacencies between the individuals due to the impossibility of satisfying both area and adjacency requirements our aim is to fit as well as possible the areas while representing as many adjacent individuals as adjacent rectangular portions as possible and adding as few false adjacencies i e adjacencies between rectangular portions corresponding to non adjacent individuals as possible we formulate this visualization problem as a mixed integer linear programming milp model we propose a matheuristic that has this milp model at its heart our experimental results demonstrate that our matheuristic provides rectangular maps with a good fit in both the frequency distribution and the adjacency relation

enhanced model formulations for optimal facility layout this paper presents an improved mixed integer programming mip model and effective solution strategies for the facility layout problem and is motivated by the work of meller et al 1999 this class of problems seeks to determine a least cost layout of departments having various size and area requirements within a rectangular building and it is challenging even for small instances the difficulty arises from the disjunctive constraints that prevent departmental overlaps and the nonlinear area constraints for each department which existing models have failed to approximate with adequate accuracy we develop several modeling and algorithmic enhancements that are demonstrated to produce more accurate solutions while also decreasing the solution effort required we begin by deriving a novel polyhedral outer approximation scheme that can provide as accurate a representation of the area requirements as desired we also design alternative methods for reducing problem symmetry evaluate the performance of several classes of valid inequalities explore the construction of partial convex hull representations for the disjunctive constraints and investigate judicious branching variable selection priority schemes the results indicate asubstantial increase in the accuracy of the layout produced while at the same time providing adramatic reduction in computational effort in particular three previously unsolved test problems from the literature for which meller et al s algorithm terminated prematurely after 24 cpu hours of computation on a sun ultra 2 workstation with 390 mb ram with respective optimality gaps of 10 14 26 45 and 40 have been solved to exact optimality with reasonable effort using our proposed approach

encoding and decoding binary product lattices a binary product lattice is generated from two binary component lattices of lower dimensions by employing the kronecker product this work focuses on codes carved from binary product lattices defined as such an intriguing problem is that of effectively mapping independent data sequences onto a selected subset of lattice points a novel approach is disclosed yielding an explicit connection between source bits and lattice points decoding methods typically used for binary product codes do not apply for product lattices several alternative decoding approaches are discussed in particular a provably bounded distance decoder is presented it relies on the fact that a product lattice code point may be regarded as a two dimensional array whose rows and columns are points in the component lattices the obtained results are compared with classical lattices known in the art

code automorphisms and permutation decoding of certain reed solomon binary images we consider primitive reed solomon rs codes over the field f2m of length n 2m 1 building on lacan s results for the case of binary extension fields we show that the binary images of certain two parity symbol rs n n 2 3 code have a code automorphism subgroup related to the general linear group gl m 2 for these codes we obtain a code automorphism subgroup of order m gl m 2 an explicit algorithm is given to compute a code automorphism if it exists that sends a particular choice of m binary positions into binary positions that correspond to a single symbol of the rs code if one such automorphism exists for a particular choice of m binary symbol positions we show that there are at least m of them computationally efficient permutation decoders are designed for the two parity symbol rs n n 2 3 codes simulation results are shown for the additive white gaussian noise awgn channel for the finite fields f23 and f24 we go on to derive subgroups of code automorphisms belonging to binary images of certain rs codes that have three parity symbols a table of code automorphism subgroup orders computed using the groups algorithms and programming gap software is tabulated for the fields f23 f24 and f25

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

bounds on the bit error probability of a linear cyclic code over gf 2 sup l and its extended code an upper bound on the bit error probability bep of a linear cyclic code over gf 2 sup l with hard decision hd maximum likelihood ml decoding on memoryless symmetric channels is derived performance results are presented for reed solomon codes on gf 32 gf 64 and gf

efficient models for the facility layout problem abstract in this paper two new models of the facility layout problem are presented linear continuous with absolute values in the objective function and constraints and linear mixed integer the linear mixed integer models have lesser number of integer variables than any other existing formulation for the facility layout problem while most other linear mixed integer models available in the literature have been obtained through a linearization of the quadratic assignment problem the ones presented in this paper are not the continuous models have an even more compact form an advantage of the formulations presented in this paper is that the location of sites need not be known a priori more importantly two of the formulations model the layout problem with facilities of unequal area solving the models presented with an unconstrained optimization algorithm yields good quality suboptimal solutions in a relatively low computation time the continuous models appear to be more useful for solving the facility layout problem than other models published in the literature

on expanded cyclic and reed solomon codes this paper offers new insights into expanded cyclic and reed solomon codes first we present a new description of general expanded codes defined over gf qm the proposed explicit description of the expanded generator matrix and the expanded parity check matrix reserves the symbol wise algebraic structure we reveal that the dual of an expanded code is the symbol wise dual code expanded by the dual basis second we characterize expanded cyclic codes utilizing the proposed expanded generator matrix a similar analysis may be carried out over the proposed expanded parity check matrix we focus on exploring the properties of component words of a codeword as well as their reciprocal relations we reveal that the existence of subspace subcodes is dictated by conjugate roots of its dual generator polynomial but not by its nonconjugate roots we further show that each generator parity check matrix can be divided into block bidiagonal submatrices associated with conjugate roots of the dual generator generator polynomial third we shed light on binary expanded reed solomon codes which are of particular mathematical and practical interest we first show an explicit way of constructing superior subspace subcodes utilizing the product basis which exhibit a higher dimension than the counterpart subfield subcodes we next demonstrate that the effective dimension of the expanded parity check generator matrix which governs the complexity of maximum likelihood soft decision decoding can be noticeably reduced by employing block bidiagonal structures of submatrices associated with conjugate roots of the generator dual generator polynomial we finally conclude that the fixed rate binary expanded reed solomon codes are asymptotically bad regardless of basis realization in the sense that the ratio of minimum distance over code length diminishes with code length going to infinity we assert badness in three scenarios namely 1 conventional fixed starting spectral null for all rates 2 centered spectral null for rates above 1 3 and 3 dynamic spectral null for rates above 11 21 our conclusions overturn the prevailing conjecture that binary expanded reed solomon codes particularly high rate ones are asymptotically good codes while deviating from the ensemble of generalized reed solomon codes which asymptotically achieve the gilbert varshamov bound

iterative soft input soft output decoding of reed solomon codes by adapting the parity check matrix an iterative algorithm is presented for soft input soft output siso decoding of reed solomon rs codes the proposed iterative algorithm uses the sum product algorithm spa in conjunction with a binary parity check matrix of the rs code the novelty is in reducing a submatrix of the binary parity check matrix that corresponds to less reliable bits to a sparse nature before the spa is applied at each iteration the proposed algorithm can be geometrically interpreted as a two stage gradient descent with an adaptive potential function this adaptive procedure is crucial to the convergence behavior of the gradient descent algorithm and therefore significantly improves the performance simulation results show that the proposed decoding algorithm and its variations provide significant gain over hard decision decoding hdd and compare favorably with other popular soft decision decoding methods

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

soft decision decoding of reed solomon codes this paper presents a maximum likelihood decoding mld and a suboptimum decoding algorithm for reed solomon rs codes the proposed algorithms are based on the algebraic structure of the binary images of rs codes theoretical bounds on the performance are derived and shown to be consistent with simulation results the proposed suboptimum algorithm achieves near mld performance with significantly lower decoding complexity it is also shown that the proposed suboptimum algorithm has better performance compared with generalized minimum distance decoding while the proposed mld algorithm has significantly lower decoding complexity than the well known vardy be ery 1991 mld algorithm

exploiting special structures in constructing a hierarchy of relaxations for 0 1 mixed integer problems a new hierarchy of relaxations is presented that provides a unifying framework for constructing a spectrum of continuous relaxations spanning from the linear programming relaxation to the convex hull representation for linear mixed integer 0 1 problems this hierarchy is an extension of the reformulation linearization technique rlt of sherali and adams 1990 1994a and is particularly designed to exploit special structures specifically inherent special structures are exploited by identifying particular classes of multiplicative factors that are applied to the original problem to reformulate it as an equivalent polynomial programming problem and subsequently this resulting problem is linearized to produce a tighter relaxation in a higher dimensional space this general framework permits us to generate an explicit hierarchical sequence of tighter relaxations leading up to the convex hull representation a similar hierarchy can be constructed for polynomial mixed integer 0 1 problems additional ideas for further strengthening rlt based constraints by using conditional logical implications as well as relationships with sequential lifting are also explored several examples are presented to demonstrate how underlying special structures including generalized and variable upper bounding covering partitioning and packing constraints as well as sparsity can be exploited within this framework for some types of structures low level relaxations are exhibited to recover the convex hull of integer feasible solutions

turbo decoding of product codes using adaptive belief propagation the adaptive belief propagation abp algorithm was recently proposed by jiang and narayanan for the soft decoding of reed solomon rs codes in this paper simplified versions of this algorithm are investigated for the turbo decoding of product codes the complexity of the turbo oriented adaptive belief propagation tab algorithm is significantly reduced by moving the matrix adaptation step outside of the belief propagation iteration loop a reduced complexity version of the tab algorithm that offers a trade off between performance and complexity is also proposed simulation results for the turbo decoding of product codes show that belief propagation based on adaptive parity check matrices is a practical alternative to the currently very popular chase pyndiah algorithm

the impact of distribution system characteristics on computational tractability mathematical programming models that seek optimal design and operational plans for distribution systems can be computationally intractable this paper examines the extent to which distribution configuration and demand characteristics affect the ease of obtaining an optimal solution problem characteristics which are reflected in a model by the parameter values can render the model for one distribution scenario to be computationally intractable and that for another to yield an optimal solution easily we introduce echelon flow based valid inequalities and use them to explicate the extent to which problem characteristics impact computational tractability

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding

extensions to stats for practical applications of the facility layout problem abstract we consider a very general case of the facility layout problem which allows incorporating various aspects appearing in real life applications these aspects include loose requirements on facilities footprints each of which only needs to be of rectangular shape and can optionally be restricted concerning the surface area or the aspect ratio compared to former approaches other generalizations of practical relevance are multiple not necessarily rectangular workshops exclusion zones in workshops predefined positions of facilities the consideration of aisles and the adherence of further restrictions such as the enforced placement of certain facilities next to an exterior wall or a minimum distance between certain pairs of facilities although different objectives could be applied we especially focus on the most relevant one in practice the minimization of transportation costs we show that this problem can heuristically be solved using an extension of the slicing tree and tabu search stats based approach the application of this algorithm on practical data shows its effectiveness the paper concludes with a step by step guide for the application of stats in practice

a β accurate linearization method of euclidean distance for the facility layout problem with heterogeneous distance metrics abstract most existing research on facility layout problems flps considers a single distance metric mainly rectilinear distance in the calculation of the material handling cost between departments however there are many industrial cases in which heterogeneous distance metrics may need to be used simultaneously to cater for different styles of material handling such as the euclidean distance metric for conveyor belts and the tchebychev distance metric for overhead cranes in this paper we study the unequal area facility layout problem with heterogeneous distance metrics ua flp hdm considering a hybrid use of three metrics i e rectilinear euclidean and tchebychev as distance measures of different styles of material handling in the production system we propose a accurate linearization method that uses a set of tangent planes to convert the non linear euclidean distance constraint into a set of linear constraints that guarantee the approximation error within a given percentage e g as small as 0 01 in our experiments and develop linear constraints for the tchebychev distance metric as well based on these contributions we present a mixed integer linear programming milp model for the ua flp hdm computational experiments are carried out to test the performance of the milp model with five benchmark problems in the literature and compare the layout designs using different distance metrics numerical results indicate that different distance metrics may lead to significantly different solutions and a hybrid use of heterogeneous distance metrics fits better for real industrial applications