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1.
In an article of A. B. Owen (1998, J. Complexity14, 466–489) the question about the distribution properties of digital (t, m, s)-nets in small intervals was raised. We give upper and lower bounds for the maximum number of points of a (t, m, s)-net in these intervals and also provide a way of improving the distribution properties in some cases. 相似文献
2.
《Finite Fields and Their Applications》2002,8(4):444-454
Until now, the concept of digital (t,m,s)-nets is the most powerful concept for the construction of low-discrepancy point sets in the s-dimensional unit cube. In this paper we consider a special class of digital nets over 2, the so-called shift nets introduced by W. Ch. Schmid, and give bounds for the quality parameter t of such nets. 相似文献
3.
Summary We prove upper bounds on the star discrepancy of digital (t, m, 2)-nets and (t, 2)-sequences over Z2. The main tool is a decomposition lemma for digital (t, m, 2)-nets, which states that every digital (t, m, 2)-net is just the union of 2tdigitally shifted digital (0, m - t, 2)-nets. Using this result we generalize upper bounds on the star discrepancy of digital (0, m, 2) -nets and (0, 2) -sequences. 相似文献
4.
Until now, the concept of digital (t,m,s)-nets is the most powerful concept for the construction of low-discrepancy point sets in the s-dimensional unit cube. In this paper we consider a special class of digital nets over
2, the so-called shift nets introduced by W. Ch. Schmid, and give bounds for the quality parameter t of such nets. 相似文献
5.
Peter Kritzer Gerhard Larcher Friedrich Pillichshammer 《Annali di Matematica Pura ed Applicata》2007,186(2):229-250
We study the star discrepancy of Hammersley nets and van der Corput sequences which are important examples of low-dimensional
quasi-Monte Carlo point sets. By a so-called digital shift, the quality of distribution of these point sets can be improved.
In this paper, we advance and extend existing bounds on digitally shifted Hammersley and van der Corput point sets and establish
criteria for the choice of digital shifts leading to optimal results. Our investigations are partly based on a close analysis
of certain sums of distances to the nearest integer.
Mathematics Subject Classi cation (2000) 11K38; 11K09 相似文献
6.
The dyadic diaphony, introduced by Hellekalek and Leeb, is a quantitative measure for the irregularity of distribution of point sets in the unit-cube. In this paper we study the dyadic diaphony of digital nets over ℤ2. We prove an upper bound for the dyadic diaphony of nets and show that the convergence order is best possible. This follows from a relation between the dyadic diaphony and the
L2{\cal L}_2
discrepancy. In order to investigate the case where the number of points is small compared to the dimension we introduce the limiting dyadic diaphony, which is defined as the limiting case where the dimension tends to infinity. We obtain a tight upper and lower bound and we compare this result with the limiting dyadic diaphony of a random sample. 相似文献
7.
The theory of (t, m, s)-nets is useful in the study of sets of points in the unit cube with small discrepancy. It is known that the existence of a (0, 2,s)-net in baseb is equivalent to the existence ofs–2 mutually orthogonal latin squares of orderb. In this paper we generalize this equivalence by showing that fort0 the existence of a (t, t+2,s)-net in baseb is equivalent to the existence ofs mutually orthogonal hypercubes of dimensiont+2 and orderb. Using the theory of hypercubes we obtain upper bounds ons for the existence of such nets. Forb a prime power these bounds are best possible. We also state several open problems.This author would like to thank the Mathematics Department of the University of Tasmania for its hospitality during his sabbatical when this paper was written. The same author would also like to thank the NSA for partial support under grant agreement # MDA904-87-H-2023.This author's research was supported by a grant from the Commonwealth of Australia through the Australian Research Council. 相似文献
8.
This paper contains three parts where each part triggered and motivated the subsequent one. In the first part (Proper Secrets) we study the Shamir’s “k-out-of-n” threshold secret sharing scheme. In that scheme, the dealer generates a random polynomial of degree k−1 whose free coefficient is the secret and the private shares are point values of that polynomial. We show that the secret
may, equivalently, be chosen as any other point value of the polynomial (including the point at infinity), but, on the other
hand, setting the secret to be any other linear combination of the polynomial coefficients may result in an imperfect scheme.
In the second part ((t, k)-bases) we define, for every pair of integers t and k such that 1 ≤ t ≤ k−1, the concepts of (t, k)-spanning sets, (t, k)-independent sets and (t, k)-bases as generalizations of the usual concepts of spanning sets, independent sets and bases in a finite-dimensional vector
space. We study the relations between those notions and derive upper and lower bounds for the size of such sets. In the third
part (Linear Codes) we show the relations between those notions and linear codes. Our main notion of a (t, k)-base bridges between two well-known structures: (1, k)-bases are just projective geometries, while (k−1, k)-bases correspond to maximal MDS-codes. We show how the properties of (t, k)-independence and (t, k)-spanning relate to the notions of minimum distance and covering radius of linear codes and how our results regarding the
size of such sets relate to known bounds in coding theory. We conclude by comparing between the notions that we introduce
here and some well known objects from projective geometry.
相似文献
9.
We consider a hypoelliptic two-parameter diffusion. We first prove a sharp upper bound in small time (s, t)[0, 1]2 for the Lp-moments of the inverse of the Malliavin matrix of the diffusion process. Second, we establish the behaviour of22 log ps, t(x, y), as ↓0, where x is the initial condition of the diffusion, =
, and ps, t(x, y) is the density of the hypoelliptic two-parameter diffusion. 相似文献
10.
It is well known that there are close connections between low-discrepancy point sets and sequences on the one hand, and certain combinatorial and algebraic structures on the other hand. E. g., Niederreiter [1] showed the equivalence between (t, t + 2, s)-nets and orthogonal arrays of strength 2. Some years later this was generalized and made precise in the work of Lawrence [2] as well as Mullen and Schmid [3] by introducing ordered orthogonal arrays. This large class of combinatorial structures yields both new constructions and bounds for the existence of nets and sequences. The linear programming bound for ordered orthogonal arrays was first derived by Martin and Stinson [4]. As in the case of error-correcting codes and orthogonal arrays, it yields a very strong bound for ordered orthogonal arrays, and consequently for nets and sequences. Solving linear programming problems in exact arithmetics is very time-consuming. Using different approaches to reduce the computing time, we have calculated the linear programming bound for a wide parameter range. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献