Editing Sobol sequence (section)

== Good distributions in the ''s''-dimensional unit hypercube ==

Let ''I<sup>s</sup>'' = [0,1]<sup>''s''</sup> be the ''s''-dimensional unit [[hypercube]], and ''f'' a real integrable function over ''I<sup>s</sup>''. The original motivation of Sobol’ was to construct a sequence ''x<sub>n</sub>'' in ''I<sup>s</sup>'' so that
:<math> \lim_{n\to\infty} \frac{1}{n} \sum_{i=1}^n f(x_i) = \int_{I^s} f </math>
and the convergence be as fast as possible.

It is more or less clear that for the sum to converge towards the integral, the points ''x<sub>n</sub>'' should fill ''I<sup>s</sup>'' minimizing the holes. Another good property would be that the projections of ''x<sub>n</sub>'' on a lower-dimensional face of ''I<sup>s</sup>'' leave very few holes as well. Hence the homogeneous filling of ''I<sup>s</sup>'' does not qualify because in lower dimensions many points will be at the same place, therefore useless for the integral estimation.

These good distributions are called (''t'',''m'',''s'')-nets and (''t'',''s'')-sequences in base ''b''. To introduce them, define first an elementary ''s''-interval in base ''b'' a subset of ''I<sup>s</sup>'' of the form
:<math> \prod_{j=1}^s \left[ \frac{a_j}{b^{d_j}}, \frac{a_j+1}{b^{d_j}} \right], </math>
where ''a<sub>j</sub>'' and ''d<sub>j</sub>'' are non-negative integers, and <math> a_j < b^{d_j} </math> for all ''j'' in {1, ...,s}.

Given 2 integers <math>0\leq t\leq m</math>, a (''t'',''m'',''s'')-net in base ''b'' is a sequence ''x<sub>n</sub>'' of ''b<sup>m</sup>'' points of ''I<sup>s</sup>'' such that <math>\operatorname{Card} P \cap \{x_1, ..., x_{b^m}\} = b^t</math> for all elementary interval ''P'' in base ''b'' of hypervolume ''λ''(''P'') = ''b<sup>t−m</sup>''.

Given a non-negative integer ''t'', a (''t'',''s'')-sequence in base ''b'' is an infinite sequence of points ''x<sub>n</sub>'' such that for all integers <math>k \geq 0, m \geq t</math>, the sequence <math>\{x_{kb^m}, ..., x_{(k+1)b^m-1}\}</math> is a (''t'',''m'',''s'')-net in base ''b''.

In his article, Sobol’ described ''Π<sub>τ</sub>-meshes'' and ''LP<sub>τ</sub> sequences'', which are (''t'',''m'',''s'')-nets and (''t'',''s'')-sequences in base 2 respectively. The terms (''t'',''m'',''s'')-nets and (''t'',''s'')-sequences in base ''b'' (also called Niederreiter sequences) were coined in 1988 by [[Harald Niederreiter]].<ref name=Nied88>[[Harald Niederreiter|Niederreiter, H.]] (1988). "Low-Discrepancy and Low-Dispersion Sequences", ''Journal of Number Theory'' '''30''': 51–70.</ref> The term ''Sobol’ sequences'' was introduced in late English-speaking papers in comparison with [[Halton sequence|Halton]], Faure and other low-discrepancy sequences.