15,976 research outputs found

    Spherical two-distance sets

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    A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.Comment: 9 pages, (v2) several small changes and corrections suggested by referees, accepted in Journal of Combinatorial Theory, Series

    Bounds on three- and higher-distance sets

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    A finite set X in a metric space M is called an s-distance set if the set of distances between any two distinct points of X has size s. The main problem for s-distance sets is to determine the maximum cardinality of s-distance sets for fixed s and M. In this paper, we improve the known upper bound for s-distance sets in n-sphere for s=3,4. In particular, we determine the maximum cardinalities of three-distance sets for n=7 and 21. We also give the maximum cardinalities of s-distance sets in the Hamming space and the Johnson space for several s and dimensions.Comment: 12 page

    Stabbing line segments with disks: complexity and approximation algorithms

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    Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii r>0r>0 where the set of segments forms a straight line drawing G=(V,E)G=(V,E) of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for r[dmin,ηdmax]r\in [d_{\min},\eta d_{\max}] and some constant η\eta where dmaxd_{\max} and dmind_{\min} are Euclidean lengths of the longest and shortest graph edges respectively. Fast O(ElogE)O(|E|\log|E|)-time O(1)O(1)-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality rηdmaxr\geq \eta d_{\max} holds uniformly for some constant η>0,\eta>0, i.e. when lengths of edges of GG are uniformly bounded from above by some linear function of r.r.Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International Conference on Analysis of Images, Social Networks and Texts (AIST-2017

    Density of Spherically-Embedded Stiefel and Grassmann Codes

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    The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum-minimum-distance codes. Computing a code's density follows from computing: i) the normalized volume of a metric ball and ii) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally-equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.Comment: Two-column version (24 pages, 6 figures, 4 tables). To appear in IEEE Transactions on Information Theor

    A generalization of Larman-Rogers-Seidel's theorem

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    A finite set X in the d-dimensional Euclidean space is called an s-distance set if the set of Euclidean distances between any two distinct points of X has size s. Larman--Rogers--Seidel proved that if the cardinality of a two-distance set is greater than 2d+3, then there exists an integer k such that a^2/b^2=(k-1)/k, where a and b are the distances. In this paper, we give an extension of this theorem for any s. Namely, if the size of an s-distance set is greater than some value depending on d and s, then certain functions of s distances become integers. Moreover, we prove that if the size of X is greater than the value, then the number of s-distance sets is finite.Comment: 12 pages, no figur

    On a generalization of distance sets

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    A subset XX in the dd-dimensional Euclidean space is called a kk-distance set if there are exactly kk distinct distances between two distinct points in XX and a subset XX is called a locally kk-distance set if for any point xx in XX, there are at most kk distinct distances between xx and other points in XX. Delsarte, Goethals, and Seidel gave the Fisher type upper bound for the cardinalities of kk-distance sets on a sphere in 1977. In the same way, we are able to give the same bound for locally kk-distance sets on a sphere. In the first part of this paper, we prove that if XX is a locally kk-distance set attaining the Fisher type upper bound, then determining a weight function ww, (X,w)(X,w) is a tight weighted spherical 2k2k-design. This result implies that locally kk-distance sets attaining the Fisher type upper bound are kk-distance sets. In the second part, we give a new absolute bound for the cardinalities of kk-distance sets on a sphere. This upper bound is useful for kk-distance sets for which the linear programming bound is not applicable. In the third part, we discuss about locally two-distance sets in Euclidean spaces. We give an upper bound for the cardinalities of locally two-distance sets in Euclidean spaces. Moreover, we prove that the existence of a spherical two-distance set in (d1)(d-1)-space which attains the Fisher type upper bound is equivalent to the existence of a locally two-distance set but not a two-distance set in dd-space with more than d(d+1)/2d(d+1)/2 points. We also classify optimal (largest possible) locally two-distance sets for dimensions less than eight. In addition, we determine the maximum cardinalities of locally two-distance sets on a sphere for dimensions less than forty.Comment: 17 pages, 1 figur
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