441 research outputs found
Weighted Coverings and Packings
In this paper we introduce a generalization of the concepts of coverings and packings in Hamming space called weighted coverings and packings. This allows us to formulate a number of well-known coding theoretical problems in a uniform manner. We study the existence of perfect weighted codes, discuss connections between weighted coverings and packings, and present many constructions for them
Counting packings of generic subsets in finite groups
A packing of subsets in a group is a
sequence such that are
disjoint subsets of . We give a formula for the number of packings if the
group is finite and if the subsets satisfy
a genericity condition. This formula can be seen as a generalization of the
falling factorials which encode the number of packings in the case where all
the sets are singletons
Notions of denseness
The notion of a completely saturated packing [Fejes Toth, Kuperberg and
Kuperberg, Highly saturated packings and reduced coverings, Monats. Math. 125
(1998) 127-145] is a sharper version of maximum density, and the analogous
notion of a completely reduced covering is a sharper version of minimum
density. We define two related notions: uniformly recurrent and weakly
recurrent dense packings, and diffusively dominant packings. Every compact
domain in Euclidean space has a uniformly recurrent dense packing. If the
domain self-nests, such a packing is limit-equivalent to a completely saturated
one. Diffusive dominance is yet sharper than complete saturation and leads to a
better understanding of n-saturation.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol4/paper9.abs.htm
Rotated sphere packing designs
We propose a new class of space-filling designs called rotated sphere packing
designs for computer experiments. The approach starts from the asymptotically
optimal positioning of identical balls that covers the unit cube. Properly
scaled, rotated, translated and extracted, such designs are excellent in
maximin distance criterion, low in discrepancy, good in projective uniformity
and thus useful in both prediction and numerical integration purposes. We
provide a fast algorithm to construct such designs for any numbers of
dimensions and points with R codes available online. Theoretical and numerical
results are also provided
LNCS
We address the problem of covering ℝ n with congruent balls, while minimizing the number of balls that contain an average point. Considering the 1-parameter family of lattices defined by stretching or compressing the integer grid in diagonal direction, we give a closed formula for the covering density that depends on the distortion parameter. We observe that our family contains the thinnest lattice coverings in dimensions 2 to 5. We also consider the problem of packing congruent balls in ℝ n , for which we give a closed formula for the packing density as well. Again we observe that our family contains optimal configurations, this time densest packings in dimensions 2 and 3
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