1,932 research outputs found
Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
We introduce the problem of constructing weighted complex projective
2-designs from the union of a family of orthonormal bases. If the weight
remains constant across elements of the same basis, then such designs can be
interpreted as generalizations of complete sets of mutually unbiased bases,
being equivalent whenever the design is composed of d+1 bases in dimension d.
We show that, for the purpose of quantum state determination, these designs
specify an optimal collection of orthogonal measurements. Using highly
nonlinear functions on abelian groups, we construct explicit examples from d+2
orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,
and 12, for example, where no complete sets of mutually unbiased bases have
thus far been found.Comment: 28 pages, to appear in J. Math. Phy
Computing Optimal Designs of multiresponse Experiments reduces to Second-Order Cone Programming
Elfving's Theorem is a major result in the theory of optimal experimental
design, which gives a geometrical characterization of optimality. In this
paper, we extend this theorem to the case of multiresponse experiments, and we
show that when the number of experiments is finite, and optimal
design of multiresponse experiments can be computed by Second-Order Cone
Programming (SOCP). Moreover, our SOCP approach can deal with design problems
in which the variable is subject to several linear constraints.
We give two proofs of this generalization of Elfving's theorem. One is based
on Lagrangian dualization techniques and relies on the fact that the
semidefinite programming (SDP) formulation of the multiresponse optimal
design always has a solution which is a matrix of rank . Therefore, the
complexity of this problem fades.
We also investigate a \emph{model robust} generalization of optimality,
for which an Elfving-type theorem was established by Dette (1993). We show with
the same Lagrangian approach that these model robust designs can be computed
efficiently by minimizing a geometric mean under some norm constraints.
Moreover, we show that the optimality conditions of this geometric programming
problem yield an extension of Dette's theorem to the case of multiresponse
experiments.
When the number of unknown parameters is small, or when the number of linear
functions of the parameters to be estimated is small, we show by numerical
examples that our approach can be between 10 and 1000 times faster than the
classic, state-of-the-art algorithms
Linear Size Optimal q-ary Constant-Weight Codes and Constant-Composition Codes
An optimal constant-composition or constant-weight code of weight has
linear size if and only if its distance is at least . When , the determination of the exact size of such a constant-composition or
constant-weight code is trivial, but the case of has been solved
previously only for binary and ternary constant-composition and constant-weight
codes, and for some sporadic instances.
This paper provides a construction for quasicyclic optimal
constant-composition and constant-weight codes of weight and distance
based on a new generalization of difference triangle sets. As a result,
the sizes of optimal constant-composition codes and optimal constant-weight
codes of weight and distance are determined for all such codes of
sufficiently large lengths. This solves an open problem of Etzion.
The sizes of optimal constant-composition codes of weight and distance
are also determined for all , except in two cases.Comment: 12 page
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
Application of Expurgated PPM to Indoor Visible Light Communications - Part I: Single-User Systems
Visible light communications (VLC) in indoor environments suffer from the
limited bandwidth of LEDs as well as from the inter-symbol interference (ISI)
imposed by multipath. In this work, transmission schemes to improve the
performance of indoor optical wireless communication (OWC) systems are
introduced. Expurgated pulse-position modulation (EPPM) is proposed for this
application since it can provide a wide range of peak to average power ratios
(PAPR) needed for dimming of the indoor illumination. A correlation decoder
used at the receiver is shown to be optimal for indoor VLC systems, which are
shot noise and background-light limited. Interleaving applied on EPPM in order
to decrease the ISI effect in dispersive VLC channels can significantly
decrease the error probability. The proposed interleaving technique makes EPPM
a better modulation option compared to PPM for VLC systems or any other
dispersive OWC system. An overlapped EPPM pulse technique is proposed to
increase the transmission rate when bandwidth-limited white LEDs are used as
sources.Comment: Journal of Lightwave Technolog
Bounds for DNA codes with constant GC-content
We derive theoretical upper and lower bounds on the maximum size of DNA codes
of length n with constant GC-content w and minimum Hamming distance d, both
with and without the additional constraint that the minimum Hamming distance
between any codeword and the reverse-complement of any codeword be at least d.
We also explicitly construct codes that are larger than the best
previously-published codes for many choices of the parameters n, d and w.Comment: 13 pages, no figures; a few references added and typos correcte
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
- …