Introduction to Quasi-Monte Carlo Integration and Applications by Gunther Leobacher & Friedrich Pillichshammer

Author:Gunther Leobacher & Friedrich Pillichshammer
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Now we come to the desired necessary condition for the existence of (0, m, s)-nets in base b.

Theorem 5.7 (Niederreiter).

Let , b ≥ 2. Assume that for with m ≥ 2 there exists a (0,m,s)-net in base b. Then s ≤ b + 1. □

Proof.

Assume that a (0, m, s)-net in base b where m ≥ 2 and s ≥ b + 2 exists. Then, according to Lemmas 5.4 and 5.5, there exists a (0, 2, b + 2)-net in base b, which contradicts Lemma 5.6. □

In other words, a (0, m, s)-net in base b with m ≥ 2 cannot exist as long as s ≥ b + 2. For example, there is no (0, m, s)-net in base 2 for m ≥ 2 and s ≥ 4.

As (0, m, s)-nets in base b do not exist for all parameters , b ≥ 2, we now weaken the condition from their definition.

Definition 5.8

Let , b ≥ 2, and let t ∈ { 0, …, m}. A (t,m,s)-net in base b is a b m -element point set in [0, 1) s which is fair with respect to every s-dimensional elementary interval in base b having volume . The parameter t is called the quality parameter of the net. Moreover, is called a strict (t, m, s)-net in base b, if t is the smallest number u ∈ { 0, …, m} such that is a (u, m, s)-net in base b.

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