103,949 research outputs found
Construction of nested space-filling designs
New types of designs called nested space-filling designs have been proposed
for conducting multiple computer experiments with different levels of accuracy.
In this article, we develop several approaches to constructing such designs.
The development of these methods also leads to the introduction of several new
discrete mathematics concepts, including nested orthogonal arrays and nested
difference matrices.Comment: Published in at http://dx.doi.org/10.1214/09-AOS690 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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
Cyclotomic Constructions of Skew Hadamard Difference Sets
We revisit the old idea of constructing difference sets from cyclotomic
classes. Two constructions of skew Hadamard difference sets are given in the
additive groups of finite fields using unions of cyclotomic classes of order
, where is a prime and a positive integer. Our main tools
are index 2 Gauss sums, instead of cyclotomic numbers.Comment: 15 pages; corrected a few typos; to appear in J. Combin. Theory (A
Frequency permutation arrays
Motivated by recent interest in permutation arrays, we introduce and
investigate the more general concept of frequency permutation arrays (FPAs). An
FPA of length n=m lambda and distance d is a set T of multipermutations on a
multiset of m symbols, each repeated with frequency lambda, such that the
Hamming distance between any distinct x,y in T is at least d. Such arrays have
potential applications in powerline communication. In this paper, we establish
basic properties of FPAs, and provide direct constructions for FPAs using a
range of combinatorial objects, including polynomials over finite fields,
combinatorial designs, and codes. We also provide recursive constructions, and
give bounds for the maximum size of such arrays.Comment: To appear in Journal of Combinatorial Design
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