26 research outputs found

    Some constructions of superimposed codes in Euclidean spaces

    Get PDF
    AbstractWe describe three new methods for obtaining superimposed codes in Euclidean spaces. With help of them we construct codes with parameters improving upon known constructions. We also prove that the spherical simplex code is not optimal as superimposed code at least for dimensions greater than 9

    Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes

    Full text link
    An optimal constant-composition or constant-weight code of weight ww has linear size if and only if its distance dd is at least 2w−12w-1. When d≥2wd\geq 2w, the determination of the exact size of such a constant-composition or constant-weight code is trivial, but the case of d=2w−1d=2w-1 has been solved previously only for binary and ternary constant-composition and constant-weight codes, and for some sporadic instances. This paper provides a construction for quasicyclic optimal constant-composition and constant-weight codes of weight ww and distance 2w−12w-1 based on a new generalization of difference triangle sets. As a result, the sizes of optimal constant-composition codes and optimal constant-weight codes of weight ww and distance 2w−12w-1 are determined for all such codes of sufficiently large lengths. This solves an open problem of Etzion. The sizes of optimal constant-composition codes of weight ww and distance 2w−12w-1 are also determined for all w≤6w\leq 6, except in two cases.Comment: 12 page

    A distributed multi-threaded data partitioner with space-filling curve orders

    Get PDF
    The problem discussed in this thesis is distributed data partitioning and data re-ordering on many-core architectures. We present extensive literature survey, with examples from various application domains - scientific computing, databases and large-scale graph processing. We propose a low-overhead partitioning framework based on geometry, that can be used to partition multi-dimensional data where the number of dimensions is >=2. The partitioner linearly orders items with good spatial locality. Partial output is stored on each process in the communication group. Space-filling curves are used to permute data - Morton order is the default curve. For dimensions <=3, we have options to generate Hilbert-like curves. Two metrics used to determine partitioning overheads are memory consumption and execution time, although these two factors are dependent on each other. The focus of this thesis is to reduce partitioning overheads as much as possible. We have described several optimizations to this end - incremental adjustments to partitions, careful dynamic memory management and using multi-threading and multi-processing to advantage. The quality of partitions is an important criteria for evaluating a partitioner. We have used graph partitioners as base-implementations against which our partitions are compared. The degree and edge-cuts of our partitions are comparable to graph partitions for regular grids. For irregular meshes, there is still room for improvement. No comparisons have been made for evaluating partitions of datasets without edges. We have deployed these partitions on two large applications - atmosphere simulation in 2D and adaptive mesh refinement in 3D. An adaptive mesh refinement benchmark was built to be part of the framework, which later became a testcase for evaluating partitions and load-balancing schemes. The performance of this benchmark is discussed in detail in the last chapter

    Incidence geometry from an algebraic graph theory point of view

    Get PDF
    The goal of this thesis is to apply techniques from algebraic graph theory to finite incidence geometry. The incidence geometries under consideration include projective spaces, polar spaces and near polygons. These geometries give rise to one or more graphs. By use of eigenvalue techniques, we obtain results on these graphs and on their substructures that are regular or extremal in some sense. The first chapter introduces the basic notions of geometries, such as projective and polar spaces. In the second chapter, we introduce the necessary concepts from algebraic graph theory, such as association schemes and distance-regular graphs, and the main techniques, including the fundamental contributions by Delsarte. Chapter 3 deals with the Grassmann association schemes, or more geometrically: with the projective geometries. Several examples of interesting subsets are given, and we can easily derive completely combinatorial properties of them. Chapter 4 discusses the association schemes from classical finite polar spaces. One of the main applications is obtaining bounds for the size of substructures known as partial m- systems. In one specific case, where the partial m-systems are partial spreads in the polar space H(2d − 1, q^2) with d odd, the bound is new and even tight. A variant of the famous Erdős-Ko-Rado problem is considered in Chapter 5, where we study sets of pairwise non-trivially intersecting maximal totally isotropic subspaces in polar spaces. A combination of geometric and algebraic techniques is used to obtain a classification of such sets of maximum size, except for one specific polar space, namely H(2d − 1, q^2) for odd rank d ≥ 5. Near polygons, including generalized polygons and dual polar spaces, are studied in the last chapter. Several results on substructures in these geometries are given. An inequality of Higman on the parameters of generalized quadrangles is generalized. Finally, it is proved that in a specific dual polar space, a highly regular substructure would yield a distance- regular graph, generalizing a result on hemisystems. The appendix consists of an alternative proof for one of the main results in the thesis, a list of open problems and a summary in Dutch

    ISCR Annual Report: Fical Year 2004

    Full text link

    Applications of polynomials to spherical codes and designs

    Get PDF
    In dit proefschrift wordt onderzoek gedaan naar een aantal problemen die verwantschap hebben met sferische codes en designs. In het eerste hoofdstuk wordt een inleiding gegeven tot sferische codes en designs. Er zijn twee belangrijke problemen te onderscheiden. Enerzijds willen we de precieze waarde (of een boven- en ondergrens) van de grootst mogelijke kardinaliteit (i.e. A(n; s)) van een sferische code vaststellen, indien de dimensie n en de maximale cosinus s zijn gegeven. Aan de andere kant willen we de grootte van een sferisch design minimaliseren voor vaste dimensie n en sterkte ¿ . De kleinst mogelijke kardinaliteit van een ¿ -design in n dimensies wordt aangegeven met B(n; ¿ ). Het probleem is boven- en ondergrenzen voor B(n; ¿ ) te vinden (of de precieze waarde). Het tweede hoofdstuk behandelt de lineaire programmeer technieken die gebruikt worden voor het vinden van een bovengrens voor A(n; s) en een ondergrens voor B(n; ¿ ). De beste bovengrens voor A(n; s) werd ontdekt door Levenshtein. Een uitleg van de logica van deze bound, samen met de eigenschappen van de betrokkene parameters wordt gegeven. In het derde hoofdstuk worden noodzakelijke en voldoende voorwaarden gegeven voor het bestaan van verbeteringen van de Levenshtein bounds voor A(n; s). Verder wordt er onderzoek gedaan naar deze voorwaarden en wordt er aangetoond dat betere grenzen vrij vaak bestaan. In het vierde hoofdstuk worden beperkingen afgeleid op de distributie van de optredende inprodukten van een spferisch design met een relatief kleine kardinaliteit (i.e. dicht bij de klassieke grenzen). Deze condities blijken voldoende te zijn voor non-existentie in veel gevallen. Onze methode werkt efficient zowel in kleine dimensies als asymptotisch voor grote n. Voor ¿ = 3 en ¿ = 5 worden nieuwe asymptotische grenzen op de kleinst mogelijke oneven grootte van ¿ -designs afgeleid. Het vijfde en laatste hoofdstuk introduceert en bestudeert bepaalde invarianten van sferische codes die momenten genoemd worden. Zulk onderzoek zou informatie kunnen geven over de structuur van sferische codes en designs
    corecore