A higher order Blokh-Zyablov propagation rule for higher order nets

TL;DR

This paper extends the Blokh-Zyablov propagation rule to higher order nets, enabling the construction of higher quality point sets for quasi-Monte Carlo methods.

Contribution

The authors generalize the Blokh-Zyablov propagation rule to higher order nets, providing a new method to generate high-quality nets for numerical integration.

Findings

The extended propagation rule produces higher order nets with significantly improved quality.

Examples demonstrate the effectiveness of the new propagation rule in constructing better nets.

The method broadens the toolkit for designing efficient quasi-Monte Carlo point sets.

Abstract

Higher order nets were introduced by Dick as a generalisation of classical $(t,m,s)$-nets, which are point sets frequently used in quasi-Monte Carlo integration algorithms. Essential tools in finding such point sets of high quality are propagation rules, which make it possible to generate new higher order nets from existing higher order nets and even classical $(t,m,s)$-nets. Such propagation rules for higher order nets were first considered by the authors in [J. Dick, P. Kritzer. Duality theory and propagation rules for generalized digital nets. Math. Comp. 79, 993--1017, 2010] and further developed in [J. Baldeaux, J. Dick, F. Pillichshammer. Duality theory and propagation rules for higher order nets. Discrete Math. 311, 362--386, 2011]. In [E.L. Blokh, V.V. Zyablov. Coding of generalized concatenated codes. Problems of Information Transmission, 10, 218--222, 1974] Blokh and Zyablov established a very general propagation rule for linear codes. This propagation rule has been extended to $(t,m,s)$-nets by Sch\"urer and Schmid in [R. Sch\"{u}rer, W.Ch. Schmid. \textit{MinT---the database of optimal net, code, OA, and OOA parameters}. Available at: \texttt{http://mint.sbg.ac.at}]. In this paper we show that this propagation rule can also be extended to higher order nets. Examples indicate that this propagation rule yields new higher order nets with significantly higher quality.