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  "body": "\n## Description\n\nA *Sobol sequence* is a sequence of points in $[0,1)^s$ built from binary digits.\nFor each coordinate $j = 1, \\ldots, s$ one fixes direction numbers $v_{j,1}, v_{j,2}, \\ldots \\in (0,1)$, constructed as follows.\n\n1. Choose a [[primitive polynomial over $\\F_2$|HQ7KMZYPSQ]] $x^{d} + c_1 x^{d-1} + \\cdots + c_{d-1} x + 1$ of degree $d = d_j$, with different polynomials for different $j$.\n2. Choose odd integers $m_1, \\ldots, m_d$ with $m_k \u003c 2^k$.\n3. For $k \u003e d$ set $m_k = 2 c_1 m_{k-1} \\oplus 2^2 c_2 m_{k-2} \\oplus \\cdots \\oplus 2^{d-1} c_{d-1} m_{k-d+1} \\oplus 2^d m_{k-d} \\oplus m_{k-d}$, where $\\oplus$ is the bitwise exclusive or of binary integers.\n4. Set $v_{j,k} = m_k / 2^k$.\n\nWrite $n \\ge 0$ in binary as $n = \\sum_{k \\ge 1} \\beta_k 2^{k-1}$ with $\\beta_k \\in \\set{0,1}$.\nThe $j$-th coordinate of the $n$-th point is\n$$x_{n,j} = \\beta_1 v_{j,1} \\oplus \\beta_2 v_{j,2} \\oplus \\cdots,$$\nwhere $\\oplus$ now acts digit by digit on binary expansions.\nThe first coordinate usually takes $v_{1,k} = 2^{-k}$, which gives the van der Corput sequence in base $2$.\n\n## Remarks\n\nTake $s = 2$, the van der Corput sequence in the first coordinate, and the polynomial $x + 1$ with $m_1 = 1$ in the second.\nThe recurrence gives $m_k = 2 m_{k-1} \\oplus m_{k-1}$, so $m_1, m_2, m_3 = 1, 3, 5$ and $v_{2,k} = 1/2, 3/4, 5/8$.\nThe points for $n = 0, 1, 2, 3$ are $(0,0)$, $(1/2, 1/2)$, $(1/4, 3/4)$, $(3/4, 1/4)$; for $n = 3$ the second coordinate is $0.10_2 \\oplus 0.11_2 = 0.01_2 = 1/4$.\nEach of the four squares of side $1/2$ contains exactly one of these points.\nSo does each vertical strip of width $1/4$ and each horizontal strip of height $1/4$.\nIn general Sobol sequences are $(t,s)$-sequences in base $2$ in the sense of Niederreiter, and their star discrepancy is $O\\big((\\log N)^s / N\\big)$.\nSo they are [[low-discrepancy sequences|XSPX1KW9AQ]], widely used in [[quasi-Monte Carlo methods|ABSRB2KXA6]].\nThe quality of the points in practice depends on the choice of the initial numbers $m_k$.\nBecause the construction uses only bit operations, the points are very fast to generate, especially in Gray code order.\n\n## Explanations\n\n### L1 human\n\nA Sobol sequence is a list of points that fills a square or a cube very evenly.\nEach coordinate is made by combining a few fixed binary fractions, chosen by the binary digits of the point's index.\nThe construction makes the first $2^m$ points, for each $m$, spread out evenly over grids of small boxes.\nFor example, the first four points in the plane put one point in each quarter of the unit square.\nThese points serve as sample points for computing averages of functions with high accuracy.\n",
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