227,818 research outputs found
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
Problems on q-Analogs in Coding Theory
The interest in -analogs of codes and designs has been increased in the
last few years as a consequence of their new application in error-correction
for random network coding. There are many interesting theoretical, algebraic,
and combinatorial coding problems concerning these q-analogs which remained
unsolved. The first goal of this paper is to make a short summary of the large
amount of research which was done in the area mainly in the last few years and
to provide most of the relevant references. The second goal of this paper is to
present one hundred open questions and problems for future research, whose
solution will advance the knowledge in this area. The third goal of this paper
is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
Rotated sphere packing designs
We propose a new class of space-filling designs called rotated sphere packing
designs for computer experiments. The approach starts from the asymptotically
optimal positioning of identical balls that covers the unit cube. Properly
scaled, rotated, translated and extracted, such designs are excellent in
maximin distance criterion, low in discrepancy, good in projective uniformity
and thus useful in both prediction and numerical integration purposes. We
provide a fast algorithm to construct such designs for any numbers of
dimensions and points with R codes available online. Theoretical and numerical
results are also provided
-optimal designs for second-order response surface models
-optimal experimental designs for a second-order response surface model
with predictors are investigated. If the design space is the
-dimensional unit cube, Galil and Kiefer [J. Statist. Plann. Inference 1
(1977a) 121-132] determined optimal designs in a restricted class of designs
(defined by the multiplicity of the minimal eigenvalue) and stated their
universal optimality as a conjecture. In this paper, we prove this claim and
show that these designs are in fact -optimal in the class of all approximate
designs. Moreover, if the design space is the unit ball, -optimal designs
have not been found so far and we also provide a complete solution to this
optimal design problem. The main difficulty in the construction of -optimal
designs for the second-order response surface model consists in the fact that
for the multiplicity of the minimum eigenvalue of the "optimal information
matrix" is larger than one (in contrast to the case ) and as a consequence
the corresponding optimality criterion is not differentiable at the optimal
solution. These difficulties are solved by considering nonlinear Chebyshev
approximation problems, which arise from a corresponding equivalence theorem.
The extremal polynomials which solve these Chebyshev problems are constructed
explicitly leading to a complete solution of the corresponding -optimal
design problems.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1241 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Pooling designs with surprisingly high degree of error correction in a finite vector space
Pooling designs are standard experimental tools in many biotechnical
applications. It is well-known that all famous pooling designs are constructed
from mathematical structures by the "containment matrix" method. In particular,
Macula's designs (resp. Ngo and Du's designs) are constructed by the
containment relation of subsets (resp. subspaces) in a finite set (resp. vector
space). Recently, we generalized Macula's designs and obtained a family of
pooling designs with more high degree of error correction by subsets in a
finite set. In this paper, as a generalization of Ngo and Du's designs, we
study the corresponding problems in a finite vector space and obtain a family
of pooling designs with surprisingly high degree of error correction. Our
designs and Ngo and Du's designs have the same number of items and pools,
respectively, but the error-tolerant property is much better than that of Ngo
and Du's designs, which was given by D'yachkov et al. \cite{DF}, when the
dimension of the space is large enough
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