Let k3(n) denote the minimal cardinality of a ternary code of length n and covering radius one. In this paper we show k3(7) ≥ 156 and k3(8) ≥ 402 improving on the best previously known bounds k3(7) ≥ 153 and k3(8) ≥ 398. The proofs are founded on a recent technique of the author for dealing with systems of linear inequalities satisfied by the number of elements of a covering code, that lie in k...

In a recent article 19], Tarnai and GG aspp ar used computer simulations to nd thin coverings of a square with up to ten equal circles. We will give improved coverings with six and eight circles and a new, thin covering with eleven circles, found by the use of simulated annealing. Furthermore, we present a combinatorial method for constructing lower bounds for the optimal covering radius.

Upper bounds on Kq (n; R), the minimum number of codewords in a q-ary code of length n and covering radius R, are improved. Such bounds are obtained by constructing corresponding covering codes. In particular, codes of length q + 1 are discussed. Good such codes can be obtained from maximum distance separable (MDS) codes. Furthermore, they can often be combined eeectively with other covering co...

A new algorithm for Support Vector regression is described. For a priori chosen 1/, it automatically adjusts a flexible tube of minimal radius to the data such that at most a fraction 1/ of the data points lie outside. Moreover, it is shown how to use parametric tube shapes with non-constant radius. The algorithm is analysed theoretically and experimentally.

New constructions of linear nonbinary codes with covering radius R = 2 are proposed. They are in part modifications of earlier constructions by the author and in part are new. Using a starting code with R = 2 as a “seed” these constructions yield an infinite family of codes with the same covering radius. New infinite families of codes with R = 2 are obtained for all alphabets of size q 4 and al...

We investigate minimax Latin hypercube designs in two dimensions for several distance measures. For the `-distance we are able to construct minimax Latin hypercube designs of n points, and to determine the minimal covering radius, for all n. For the `1-distance we have a lower bound for the covering radius, and a construction of minimax Latin hypercube designs for (infinitely) many values of n....

We consider the indirect covering subtree problem (Kim et al., 1996). The input is an edge weighted tree graph along with customers located at the nodes. Each customer is associated with a radius and a penalty. The goal is to locate a tree-shaped facility such that the sum of setup and penalty cost is minimized. The setup cost equals the sum of edge lengths taken by the facility and the penalty...

Spherical t-designs on the unit sphere S ⊂ R, introduced by Delsarte, Goethals, and Seidel (1977), are equal weight numerical integration rules that are exact for all polynomials of degree at most t on S. This talk considers the calculation and properties of of spherical t-designs, in particular for S where most applications reside. Bondarenko, Radchenko, and Viazovska (2013) proved that there ...

This paper discusses the problem of covering and hitting a set of line segments L in R by a pair of axis-parallel squares such that the side length of the larger of the two squares is minimized. We also discuss the restricted version of covering, where each line segment in L is to be covered completely by at least one square. The proposed algorithm for the covering problem reports the optimum r...