scholarly journals A Gilbert–Varshamov-type bound for lattice packings

2011 ◽  
Vol 118 (3) ◽  
pp. 938-948
Author(s):  
Chaoping Xing ◽  
Sze Ling Yeo
Keyword(s):  
Lattice Packings

Some Sphere Packings in Higher Space

1964 ◽  
Vol 16 ◽  
pp. 657-682 ◽  
Author(s):  
John Leech
Keyword(s):  
Two Dimensions ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice Packings

This paper is concerned with the packing of equal spheres in Euclidean spaces [n] of n > 8 dimensions. To be precise, a packing is a distribution of spheres any two of which have at most a point of contact in common. If the centres of the spheres form a lattice, the packing is said to be a lattice packing. The densest lattice packings are known for spaces of up to eight dimensions (1, 2), but not for any space of more than eight dimensions. Further, although non-lattice packings are known in [3] and [5] which have the same density as the densest lattice packings, none is known which has greater density than the densest lattice packings in any space of up to eight dimensions, neither, for any space of more than two dimensions, has it been shown that they do not exist.

scholarly journals A NEW INEQUALITY FOR THE HERMITE CONSTANTS

2008 ◽  
Vol 04 (03) ◽  
pp. 363-386
Author(s):  
ROLAND BACHER
Keyword(s):  
Dimensional Lattice ◽  
Maximal Density ◽  
Hermite’S Constant ◽  
Increasing Functions ◽  
Lattice Packings

We describe continuous increasing functions Cn(x) such that γn ≥ Cn(γn-1) where γm is Hermite's constant in dimension m. This inequality yields a new proof of the Minkowski–Hlawka bound Δn ≥ ζ(n)21-n for the maximal density Δn of n-dimensional lattice packings.

Finite Lattice Packings and the Wulff‐Shape

Mathematika ◽  
2005 ◽  
Vol 52 (1-2) ◽  
pp. 17-29
Author(s):  
Ulrich Betke ◽  
Károly Böröczky
Keyword(s):  
Finite Lattice ◽  
Wulff Shape ◽  
Lattice Packings

Six and Seven Dimensional Non-Lattice Sphere Packings

1969 ◽  
Vol 12 (2) ◽  
pp. 151-155 ◽  
Author(s):  
John Leech
Keyword(s):  
Sphere Packing ◽  
Lattice Packing ◽  
Sphere Packings ◽  
Euclidean Spaces ◽  
Lattice Packings

The densest lattice packings of equal spheres in Euclidean spaces En of n dimensions are known for n ⩽ 8. However, it is not known for any n ⩾ 3 whether there can be any non-lattice sphere packing with density exceeding that of the densest lattice packing. W. Barlow described [1] a non-lattice packing in E3 with the same density as the densest lattice packing, and I described [6] three non-lattice packings in E5 which also have this property.

scholarly journals On sequences of lattice packings

1976 ◽  
Vol 15 (1) ◽  
pp. 133-139 ◽  
Author(s):  
Joseph Hammer ◽  
Denis Dwyer
Keyword(s):  
Lattice Packings

In this note we establish theorems on compactness of lattice packings.

CLASSICAL METHODS IN THE THEORY OF LATTICE PACKINGS

1979 ◽  
Vol 34 (4) ◽  
pp. 1-68 ◽  
Author(s):  
S S Ryshkov ◽  
E P Baranovskii
Keyword(s):  
Lattice Packings
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