On the existence and distribution quality of hyperplane sequences

Abstract It is well known that digital ( t , m , s ) -nets and ( T , s ) -sequences over a finite field have excellent properties when they are used as underlying nodes in quasi-Monte Carlo integration rules. One very general sub-class of digital nets are hyperplane nets which can be viewed as a generalization of cyclic nets and of polynomial lattice point sets. In this paper we introduce infinite versions of hyperplane nets and call these sequences hyperplane sequences. Our construction is based on the recent duality theory for digital sequences according to Dick and Niederreiter. We then analyze the equidistribution properties of hyperplane sequences in terms of the quality function T and the star discrepancy.

[1]  Josef Dick,et al.  Cyclic Digital Nets, Hyperplane Nets, and Multivariate Integration in Sobolev Spaces , 2006, SIAM J. Numer. Anal..

[2]  Harald Niederreiter,et al.  Duality for digital sequences , 2009, J. Complex..

[3]  H. Niederreiter,et al.  Duality for digital nets and its applications , 2001 .

[4]  Harald Niederreiter,et al.  Kronecker-type sequences and nonarchimedean diophantine approximations , 1993 .

[5]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[6]  H. Niederreiter.  Point sets and sequences with small discrepancy , 1987 .

[7]  Harald Niederreiter,et al.  Digital Nets and Coding Theory , 2004 .

[8]  Harald Niederreiter,et al.  Low-discrepancy point sets , 1986 .

[9]  H. Niederreiter,et al.  Low-Discrepancy Sequences and Global Function Fields with Many Rational Places , 1996 .

[10]  Shu Tezuka,et al.  Another Random Scrambling of Digital ( t , s )-Sequences , 2002 .

[11]  H. Niederreiter,et al.  A Kronecker Product Construction for Digital Nets , 2002 .

[12]  Y. Edel,et al.  Coding‐theoretic constructions for (t,m,s)‐nets and ordered orthogonal arrays , 2002 .

[13]  Harald Niederreiter,et al.  Constructions of (t, m, s)-nets and (t, s)-sequences , 2005, Finite Fields Their Appl..

[14]  Harald Niederreiter,et al.  Matrix-product constructions of digital nets , 2004, Finite Fields Their Appl..

[15]  G. Pirsic.  A Small Taxonomy of Integration Node Sets , 2005 .

[16]  Harald Niederreiter,et al.  Generalized $(t,s)$-sequences, Kronecker-type sequences, and Diophantine approximations of formal Laurent series , 1995 .

[17]  Harald Niederreiter,et al.  Nets, (t, s)-Sequences, and Codes , 2008 .

[18]  THE QUALITY PARAMETER OF CYCLIC NETS AND HYPERPLANE NETS , 2009 .

[19]  F. Pillichshammer,et al.  Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration , 2010 .