and kept in memory the values xn−1,j for all the required dimensions. Since the Gray code G(n) differs from that of the preceding one G(n − 1) by just a single, say the k-th, bit (which is a rightmost zero bit of n − 1), all that needs to be done is a single XOR operation for each dimension in order to propagate all of the xn−1 to xn, i.e. Additional uniformity properties Sobol’ introduced additional uniformity conditions known as property A and A’. 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 2d there is exactly one draw in each 2d 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 4d there is exactly one draw in each 4d 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 where Vd is the d × d binary matrix defined by with vk,j,m denoting the m-th digit after the binary point of the direction number vk,j = (0.vk,j,1vk,j,2...)2. Theorem The d-dimensional Sobol’ sequence possesses Property A' iff where Ud is the 2d × 2d binary matrix defined by with vk,j,m denoting the m-th digit after the binary point of the direction number vk,j = (0.vk,j,1vk,j,2...)2. 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. The initialisation 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