91 research outputs found
Semidefinite programming, harmonic analysis and coding theory
These lecture notes where presented as a course of the CIMPA summer school in
Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics.
This version is an update June 2010
Linear programming bounds for codes in Grassmannian spaces
We introduce a linear programming method to obtain bounds on the cardinality
of codes in Grassmannian spaces for the chordal distance. We obtain explicit
bounds, and an asymptotic bound that improves on the Hamming bound. Our
approach generalizes the approach originally developed by P. Delsarte and
Kabatianski-Levenshtein for compact two-point homogeneous spaces.Comment: 35 pages, 1 figur
New upper bounds for kissing numbers from semidefinite programming
Recently A. Schrijver derived new upper bounds for binary codes using
semidefinite programming. In this paper we adapt this approach to codes on the
unit sphere and we compute new upper bounds for the kissing number in several
dimensions. In particular our computations give the (known) values for the
cases n = 3, 4, 8, 24.Comment: 17 pages, (v4) references updated, accepted in Journal of the
American Mathematical Societ
Signal reconstruction from the magnitude of subspace components
We consider signal reconstruction from the norms of subspace components
generalizing standard phase retrieval problems. In the deterministic setting, a
closed reconstruction formula is derived when the subspaces satisfy certain
cubature conditions, that require at least a quadratic number of subspaces.
Moreover, we address reconstruction under the erasure of a subset of the norms;
using the concepts of -fusion frames and list decoding, we propose an
algorithm that outputs a finite list of candidate signals, one of which is the
correct one. In the random setting, we show that a set of subspaces chosen at
random and of cardinality scaling linearly in the ambient dimension allows for
exact reconstruction with high probability by solving the feasibility problem
of a semidefinite program
On the theta number of powers of cycle graphs
We give a closed formula for Lovasz theta number of the powers of cycle
graphs and of their complements, the circular complete graphs. As a
consequence, we establish that the circular chromatic number of a circular
perfect graph is computable in polynomial time. We also derive an asymptotic
estimate for this theta number.Comment: 17 page
On the density of sets avoiding parallelohedron distance 1
The maximal density of a measurable subset of R^n avoiding Euclidean
distance1 is unknown except in the trivial case of dimension 1. In this paper,
we consider thecase of a distance associated to a polytope that tiles space,
where it is likely that the setsavoiding distance 1 are of maximal density
2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n =
2, and for the Vorono\"i regions of the lattices An, n >= 2
Semidefinite programming, multivariate orthogonal polynomials, and codes in spherical caps
We apply the semidefinite programming approach developed in
arxiv:math.MG/0608426 to obtain new upper bounds for codes in spherical caps.
We compute new upper bounds for the one-sided kissing number in several
dimensions where we in particular get a new tight bound in dimension 8.
Furthermore we show how to use the SDP framework to get analytic bounds.Comment: 15 pages, (v2) referee comments and suggestions incorporate
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