13,421 research outputs found
Covering Radius of Two-dimensional Lattices
The covering radius problem in any dimension is not known to be
solvable in nondeterministic polynomial time, but when in
dimension two, we give a deterministic polynomial time algorithm
by computing a reduced basis using Gauss\u27 algorithm in this paper
Computational Approaches to Lattice Packing and Covering Problems
We describe algorithms which address two classical problems in lattice
geometry: the lattice covering and the simultaneous lattice packing-covering
problem. Theoretically our algorithms solve the two problems in any fixed
dimension d in the sense that they approximate optimal covering lattices and
optimal packing-covering lattices within any desired accuracy. Both algorithms
involve semidefinite programming and are based on Voronoi's reduction theory
for positive definite quadratic forms, which describes all possible Delone
triangulations of Z^d.
In practice, our implementations reproduce known results in dimensions d <= 5
and in particular solve the two problems in these dimensions. For d = 6 our
computations produce new best known covering as well as packing-covering
lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our
approach leads to new best known covering lattices. Although we use numerical
methods, we made some effort to transform numerical evidences into rigorous
proofs. We provide rigorous error bounds and prove that some of the new
lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in
Discrete and Computational Geometry, see also
http://fma2.math.uni-magdeburg.de/~latgeo
Sphere Packing with Limited Overlap
The classical sphere packing problem asks for the best (infinite) arrangement
of non-overlapping unit balls which cover as much space as possible. We define
a generalized version of the problem, where we allow each ball a limited amount
of overlap with other balls. We study two natural choices of overlap measures
and obtain the optimal lattice packings in a parameterized family of lattices
which contains the FCC, BCC, and integer lattice.Comment: 12 pages, 3 figures, submitted to SOCG 201
Template-based searches for gravitational waves: efficient lattice covering of flat parameter spaces
The construction of optimal template banks for matched-filtering searches is
an example of the sphere covering problem. For parameter spaces with
constant-coefficient metrics a (near-) optimal template bank is achieved by the
A_n* lattice, which is the best lattice-covering in dimensions n <= 5, and is
close to the best covering known for dimensions n <= 16. Generally this
provides a substantially more efficient covering than the simpler hyper-cubic
lattice. We present an algorithm for generating lattice template banks for
constant-coefficient metrics and we illustrate its implementation by generating
A_n* template banks in n=2,3,4 dimensions.Comment: 10 pages, submitted to CQG for proceedings of GWDAW1
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