Editing Sobol sequence (section)

== Additional uniformity properties ==

Sobol’ introduced additional uniformity conditions known as property A and A’.<ref name=Sobol76>Sobol’, I. M. (1976) "Uniformly distributed sequences with an additional uniform property". ''Zh. Vych. Mat. Mat. Fiz.'' '''16''': 1332–1337 (in Russian); ''U.S.S.R. Comput. Maths. Math. Phys.'' '''16''': 236–242 (in English).</ref>

;Definition: A low-discrepancy sequence is said to satisfy '''Property A''' if for any binary segment (not an arbitrary subset) of the ''d''-dimensional sequence of length 2<sup>''d''</sup> there is exactly one draw in each 2<sup>''d''</sup> hypercubes that result from subdividing the unit hypercube along each of its length extensions into half.

;Definition: A low-discrepancy sequence is said to satisfy '''Property A’''' if for any binary segment (not an arbitrary subset) of the ''d''-dimensional sequence of length 4<sup>''d''</sup> there is exactly one draw in each 4<sup>''d''</sup> hypercubes that result from subdividing the unit hypercube along each of its length extensions into four equal parts.

There are mathematical conditions that guarantee properties A and A'.

;Theorem: The ''d''-dimensional Sobol’ sequence possesses Property A iff
::<math>
\det(\mathbf{V}_d) \equiv 1 (\mod 2),
</math>
:where '''V'''<sup>''d''</sup> is the ''d''&nbsp;×&nbsp;''d'' binary matrix defined by
::<math>
\mathbf{V}_d := \begin{pmatrix}
{v_{1,1,1}}&{v_{2,1,1}}&{\dots}&{v_{d,1,1}}\\ 
{v_{1,2,1}}&{v_{2,2,1}}&{\dots}&{v_{d,2,1}}\\ 
{\vdots}&{\vdots}&{\ddots}&{\vdots}\\ 
{v_{1,d,1}}&{v_{2,d,1}}&{\dots}&{v_{d,d,1}}
\end{pmatrix},
</math>
:with ''v''<sub>''k'',''j'',''m''</sub> denoting the ''m''-th digit after the binary point of the direction number ''v''<sub>''k'',''j''</sub> = (0.''v''<sub>''k'',''j'',1</sub>''v''<sub>''k'',''j'',2</sub>...)<sub>2</sub>.

;Theorem: The ''d''-dimensional Sobol’ sequence possesses Property A' iff
::<math>
\det(\mathbf{U}_d) \equiv 1 \mod 2,
</math>
:where '''U'''<sup>''d''</sup> is the 2''d''&nbsp;×&nbsp;2''d'' binary matrix defined by
::<math>
\mathbf{U}_d := \begin{pmatrix}
{v_{1,1,1}}&{v_{1,1,2}}&{v_{2,1,1}}&{v_{2,1,2}}&{\dots}&{v_{d,1,1}}&{v_{d,1,2}}\\ 
{v_{1,2,1}}&{v_{1,2,2}}&{v_{2,2,1}}&{v_{2,2,2}}&{\dots}&{v_{d,2,1}}&{v_{d,2,2}}\\ 
{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}&{\vdots}\\ 
{v_{1,2d,1}}&{v_{1,2d,2}}&{v_{2,2d,1}}&{v_{2,2d,2}}&{\dots}&{v_{d,2d,1}}&{v_{d,2d,2}}
\end{pmatrix},
</math>
:with ''v''<sub>''k'',''j'',''m''</sub> denoting the ''m''-th digit after the binary point of the direction number ''v''<sub>''k'',''j''</sub> = (0.''v''<sub>''k'',''j'',1</sub>''v''<sub>''k'',''j'',2</sub>...)<sub>2</sub>.

Tests for properties A and A’ are independent. Thus it is possible to construct the Sobol’ sequence that satisfies both properties A and A’ or only one of them.