Integer codes and lattice packings by cubes of sidelength k

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.

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Integer codes correcting burst asymmetric within a byte and double asymmetric errors

Cryptography and Communications ◽

10.1007/s12095-019-00388-0 ◽

2019 ◽

Vol 12 (2) ◽

pp. 221-230

Author(s):

Aleksandar Radonjic ◽

Vladimir Vujicic

Keyword(s):

Asymmetric Errors ◽

Integer Codes

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Lattice Packings of Mirror Symmetric or Centrally Symmetric Three-Dimensional Convex Bodies

Journal of Mathematical Sciences ◽

10.1007/s10958-016-2683-7 ◽

2016 ◽

Vol 212 (5) ◽

pp. 536-541

Author(s):

V. V. Makeev

Keyword(s):

Convex Bodies ◽

Three Dimensional ◽

Centrally Symmetric ◽

Lattice Packings

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A NEW INEQUALITY FOR THE HERMITE CONSTANTS

International Journal of Number Theory ◽

10.1142/s1793042108001390 ◽

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.

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