Lattices with exponentially large kissing numbers

We give an extension of the table of Odlyzko-Sloane [1] with the best known lower and upper bounds for the kissing numbers τn, equal to the maximum number of nonoverlapping unit spheres that can simultaneously touch S n−1 . Some lower and upper bounds are presented in a table for the range 25 ≤ n ≤ 32.

Notices-American Mathematical Society, 2004

The ìkissing number problemî asks for the maximal number of white spheres that can touch a black sphere of the same size in n-dimensional space. The answers in dimensions one, two and three are classical, but the answers in dimensions eight and twenty-four were a big surprise in 1979, based on an extremely elegant method initiated by Philippe Delsarte in

European Journal of Combinatorics, 1996

A TTILA S ALI In this paper we reduce the intriguing conjecture dim( L ) ϭ o ( ͉ L ͉ ) for lattices to an extremal set-theoretical conjecture showing a possible way of solving it . Furthermore , the latter conjecture is interesting for its own sake , too .

Discrete & Computational Geometry, 2021

We show that the lattice Hadwiger (=kissing) number of superballs is exponential in the dimension. The same is true for some more general convex bodies.

Discrete & Computational Geometry, 1992

Let the lattice A have covering radius R, so that closed balls of radius R around the lattice points just cover the space. The covering multiplicity CM(A) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for an n-dimensional lattice is n if n < 8, and conjecture that it exceeds n in all other cases. We determine the covering multiplicity of the Leech lattice and of the lattices I,, A,, Dn, E~ and their duals for small values of n. Although it appears that CM(I,)=2 "-1 if n<33, as n~0o we have CM(I,,) ~ 2.089...". The results have application to numerical integration.

Mathematika, 1996

This paper leans on results of Baranovskii 1], 2]. The covering radius R(L) of a lattice L is the radius of smallest balls with centers in points of L which cover all the space spanned by L. R(L) is tightly related to minimal vectors of classes of the quotient 1 2 L=L.

Journal of Number Theory, 1991

The n-dimensional lattices that contain fewest distances are characterized for all n # 2.

Discrete Applied Mathematics, 2007

Determining the maximum number of D-dimensional spheres of radius r that can be adjacent to a central sphere of radius r is known as the Kissing Number Problem (KNP). The problem has been solved for 2, 3 and very recently for 4 dimensions. We present two nonlinear (nonconvex) mathematical programming models for the solution of the KNP. We solve the problem by using two stochastic global optimization methods: a Multi Level Single Linkage algorithm and a Variable Neighbourhood Search. We obtain numerical results for 2, 3 and 4 dimensions.

Mathematika, 1982

Barnes and Sloane recently described a “general construction” for lattice packings of equal spheres in Euclidean space. In the present paper we simplify and further generalize their construction, and make it suitable for iteration. As a result we obtain lattice packings in ℝm with density Δ satisfying , as m → ∞ where is the smallest value of k for which the k-th iterated logarithm of m is less than 1. These appear to be the densest lattices that have been explicitly constructed in high-dimensional space. New records are also established in a number of lower dimensions, beginning in dimension 96.

Discrete &amp; Computational Geometry, 1998

The translative kissing number H (K) of a d-dimensional convex body K is the maximum number of mutually nonoverlapping translates of K which touch K. In this paper we show that there exists an absolute constant c > 0 such that H (K) ≥ 2 cd for every positive integer d and every d-dimensional convex body K. We also prove a generalization of this result for pairs of centrally symmetric convex bodies.