Coding-theoretic constructions for (t,m,s)-nets and ordered orthogonal arrays

✍ Scribed by Jürgen Bierbrauer; Yves Edel; Wolfgang Ch. Schmid

• Publisher: John Wiley and Sons
• Year: 2002
• Tongue: English
• Weight: 169 KB
• Volume: 10
• Category: Article
• ISSN: 1063-8539
• DOI: 10.1002/jcd.10015

✦ Synopsis

Abstract

(t,m,s)‐nets are point sets in Euclidean s‐space satisfying certain uniformity conditions, for use in numerical integration. They can be equivalently described in terms of ordered orthogonal arrays, a class of finite geometrical structures generalizing orthogonal arrays. This establishes a link between quasi‐Monte Carlo methods and coding theory. The ambient space is a metric space generalizing the Hamming space of coding theory. We denote it by NRT space (named after Niederreiter, Rosenbloom and Tsfasman). Our main results are generalizations of coding‐theoretic constructions from Hamming space to NRT space. These comprise a version of the Gilbert‐Varshamov bound, the (u,u+υ)‐construction and concatenation. We present a table of the best known parameters of q‐ary (t,m,s)‐nets for __q__ε{2,3,4,5} and dimension m≤50. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 403–418, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10015