The Initialization of Sobol Numbers
To construct a Sobol sequence a set of direction numbers vi,j needs to be selected. There is some freedom in the selection of initial direction numbers. Therefore, it is possible to receive different realisations of the Sobol sequence for selected dimensions. A bad selection of initial numbers can considerably reduce the efficiency of Sobol sequences when used for computation.
Arguably the easiest choice for the initialisation numbers is just to have the l-th leftmost bit set, and all other bits to be zero, i.e. mk,j = 1 for all k and j. This initialisation is usually called unit initialisation. However, such a sequence fails the test for Property A and A’ even for low dimensions and hence this initialisation is bad.
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