New bounds for multiple packings of Euclidean sphere (original) (raw)

Sphere Packings and Error-Correcting Codes

Canadian Journal of Mathematics, 1971

Error-correcting codes are used in several constructions for packings of equal spheres in n-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m ≧ 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density Δ of the packings in En for n = 2m satisfiesIn this paper we make systematic use of error-correcting codes to obtain sphere packings in En, including several of the densest packings known and several new packings.

Sphere Packing and Error-Correcting Codes

Grundlehren der mathematischen Wissenschaften, 1999

Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces E n. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2 m for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density A of the packings in E n for n = 2 m satisfies log A ~-\n log log n as n-* oo.

A Lower Bound on the Density of Sphere Packings via Graph Theory

International Mathematics Research Notices

Using graph-theoretic methods we give a new proof that for all sufficiently large n, there exist sphere packings in R n of density at least cn2 −n exceeding the classical Minkowski bound by a factor linear in n. This matches up to a constant the best known lower bounds on the density of sphere packings due to Rogers [9], Davenport-Rogers [4], and Ball [2]. The suggested method makes it possible to describe the points of such a packing with complexity exp(n log n), which is significantly lower than in the other approaches.

Sphere Packing: Asymptotic Behavior and Existence of Solution

2000

Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(A k ) decreases to 0 if A k is a sequence of matrices in SL(n, R) with lim k→∞ A k = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained.

Bounds for the sum of distances of spherical sets of small size

ArXiv, 2021

We derive upper and lower bounds on the sum of distances of a spherical code of size N in n dimensions when N „ n, 0 ă α ď 2. The bounds are derived by specializing recent general, universal bounds on energy of spherical sets. We discuss asymptotic behavior of our bounds along with several examples of codes whose sum of distances closely follows the upper bound.

NEW UPPER BOUNDS FOR SOME SPHERICAL CODES

The maximal cardinality of a code W on the unit sphere in n dimen- sions with (x,y) ≤ s whenever x,y ∈ W, x 6= y, is denoted by A(n,s). We use two methods for obtaining new upper bounds on A(n,s) for some values of n and s. We find new linear programming bounds by suitable polynomials of degrees which are higher than the degrees of the previously known good polynomials due to Lev- enshtein (11, 12). Also we investigate the possibilities for attaining the Levenshtein bounds (11, 12). In such cases we find the distance distributions of the correspond- ing feasible maximal spherical codes. Usually this leads to a contradiction showing that such codes do not exist.

JAMS 1 Sphere Packing: asymptotic behavior and existence of solution

2003

Abstract. Lattices in n-dimensional Euclidean spaces may be parameterized by the non-compact symmetric space SL(n, R)/SO(n, R). We consider sphere packings determined by lattices and study the density function in the symmetric space, showing that the density function ρ(Ak) decreases to 0 if Ak is a sequence of matrices in SL(n, R) with limk→ ∞ ‖Ak ‖ = ∞. As a consequence, we give a simple prove that the optimal solution for the sphere packing problem is attained. The sphere packing problem is one of the famous open problems in mathematics. In short, it asks about the densest way a set of equal spheres can be packed in space n-dimensional Euclidean space R n, without overlapping one the other. In this context, the density means the proportion between the covered and the uncovered amount of space. It has many variations: one could replace spheres of equal radii by spheres of radii 0 < a ≤ r ≤ b bounded from above and below, replace spheres by a collection of identical (preferably c...

On Bounds on the Minimum Distance of Spherical Codes

We derive new upper bounds on the maximal squared minimum distance of some spherical codes with fixed cardinality. To do this we rearrange a technique for obtaining improvements on the Levenshtein upper bounds on the size of spherical codes. In three dimensions our method gives some progress in a problem from the classical geometry.