Construction of The Sobol Sequence
The algorithm for generating Sobol sequences is clearly explained in Bratley and Fox, Algorithm 659
To generate the j-th component of the points in a Sobol sequence, we need to choose a primitive polynomial of some degree sj over the field GF(2)
where the coefficients a1,j, a2,j, ..., asj−1,j are either 0 or 1. The error bounds for Sobol sequences given in indicate that we should use primitive polynomials of as low a degree as possible.
A sequence of positive integers {m1,j, m2,j, ...} are defined by the recurrence relation
where is the bit-by-bit exclusive-or operator. The initial values m1,j, m2,j, ..., msj,j can be chosen freely provided that each mk,j, 1 ≤ k ≤ sj, is odd and less than 2k.
The so-called direction numbers {v1,j, v2,j, . . .} are defined by
Then xi,j, the j-th component of the i-th point in a Sobol sequence, is given by
where ik is the k-th binary digit of i = (. . . i3i2i1)2. Here the notation (·)2 denotes the binary representation of numbers.
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