994 research outputs found
On upper bounds on the smallest size of a saturating set in a projective plane
In a projective plane (not necessarily Desarguesian) of order
a point subset is saturating (or dense) if any point of is collinear with two points in. Using probabilistic methods, the
following upper bound on the smallest size of a saturating set in
is proved: \begin{equation*} s(2,q)\leq 2\sqrt{(q+1)\ln
(q+1)}+2\thicksim 2\sqrt{q\ln q}. \end{equation*} We also show that for any
constant a random point set of size in with is a saturating
set with probability greater than Our probabilistic
approach is also applied to multiple saturating sets. A point set is -saturating if for every point of the number of secants of through is at least , counted with
multiplicity. The multiplicity of a secant is computed as
The following upper bound on the smallest
size of a -saturating set in is proved:
\begin{equation*} s_{\mu }(2,q)\leq 2(\mu +1)\sqrt{(q+1)\ln (q+1)}+2\thicksim
2(\mu +1)\sqrt{ q\ln q}\,\text{ for }\,2\leq \mu \leq \sqrt{q}. \end{equation*}
By using inductive constructions, upper bounds on the smallest size of a
saturating set (as well as on a -saturating set) in the projective
space are obtained.
All the results are also stated in terms of linear covering codes.Comment: 15 pages, 24 references, misprints are corrected, Sections 3-5 and
some references are adde
Intersection sets, three-character multisets and associated codes
In this article we construct new minimal intersection sets in
sporting three intersection numbers with hyperplanes; we
then use these sets to obtain linear error correcting codes with few weights,
whose weight enumerator we also determine. Furthermore, we provide a new family
of three-character multisets in with even and we
also compute their weight distribution.Comment: 17 Pages; revised and corrected result
Optimal Vertex Cover for the Small-World Hanoi Networks
The vertex-cover problem on the Hanoi networks HN3 and HN5 is analyzed with
an exact renormalization group and parallel-tempering Monte Carlo simulations.
The grand canonical partition function of the equivalent hard-core repulsive
lattice-gas problem is recast first as an Ising-like canonical partition
function, which allows for a closed set of renormalization group equations. The
flow of these equations is analyzed for the limit of infinite chemical
potential, at which the vertex-cover problem is attained. The relevant fixed
point and its neighborhood are analyzed, and non-trivial results are obtained
both, for the coverage as well as for the ground state entropy density, which
indicates the complex structure of the solution space. Using special
hierarchy-dependent operators in the renormalization group and Monte-Carlo
simulations, structural details of optimal configurations are revealed. These
studies indicate that the optimal coverages (or packings) are not related by a
simple symmetry. Using a clustering analysis of the solutions obtained in the
Monte Carlo simulations, a complex solution space structure is revealed for
each system size. Nevertheless, in the thermodynamic limit, the solution
landscape is dominated by one huge set of very similar solutions.Comment: RevTex, 24 pages; many corrections in text and figures; final
version; for related information, see
http://www.physics.emory.edu/faculty/boettcher
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
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