Séminaire de Théorie Analytique des Nombres et Problèmes Diophantiens

Le 31 mai 2001

Wolfgang.Schmid (University of Salzburg)

Combinatorial and Coding-Theoretical Aspects of (t,m,s)-Nets.

 Résumé : After a short introduction to quasi-Monte Carlo methods for numerical integration we will talk on a class of low-discrepancy point sets, (t,m,s)-nets and especially on its linear sub-class called digital nets.

A highly interesting fact is the close relation to error-correcting codes and combinatorial objects. Corresponding combinatorial structures are so-called ordered orthogonal arrays (introduced in 1995) which are generalizations of orthogonal arrays.

The well-known duality between linear orthogonal arrays and linear codes carries over to the more general setting of linear ordered orthogonal arrays. This has been described by Rosenbloom and Tsfasman in 1997. They introduced the m-metric, a generalization of the usual Hamming metric, and gave upper and lower bounds for the parameters of codes in this metric.

We will use this setting to deduce an improved Gilbert-Varshamov bound for linear ordered orthogonal arrays (equivalently for digital nets) and stress the basic problem of net-embeddability (``when can an [s,s-m,k+1]-code be completed to a digital (m-k,m,s)-net''). This is a joint work with Y. Edel and J. Bierbrauer.

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