## 1 Introduction: Problem Setting
### Sobol' Sequences  
\(\Lambda\Pi_{\tau}\)-sequences are _uniformly distributed sequences_ (UDSs) The term _UDS_ was introduced by [PERSON] in 1916 [10].  
For practical applications, it is essential to identify a uniformly distributed sequence (UDS) that satisfies three key requirements [11; 12]:  
1. **Best Asymptotic Behavior**: As \(n\) approaches infinity, the sequence should exhibit optimal asymptotic behavior.
2. **Well-Distributed Points**: For smaller values of \(n\), the points should be evenly distributed across the domain.
3. **Computational Efficiency**: The algorithm used to generate the sequence should be computationally inexpensive.  
By ensuring that these criteria are met, we can guarantee the effectiveness and practicality of the chosen _UDSs_ in various computational tasks.  
All \(\Lambda\Pi_{\tau}\)-sequences given in [12] satisfy the first requirement. Suitable distributions such as \(\Lambda\Pi_{\tau}\)-sequences are also called \((t,m,s)\)-nets and \((t,s)\)-sequences in base \(b\geq 2\). To introduce them, define first an elementary \(s\)-interval in base \(b\) as a subset of \(U^{s}\) of the form \(E=\prod_{j=1}^{s}\left[\frac{a_{j}}{b^{j}},\frac{a_{j}+1}{b^{j}}\right]\), where \(a_{j},d_{j}\geq 0\) are integers and \(a_{j}<b^{d_{j}}\) for all \(j\in\{1,\ldots,s\}\). Given two integers \(0\leq t\leq m\), a \((t,m,s)\)-net in base \(b\) is a sequence \(\mathrm{x}^{(i)}\) of \(b^{m}\) points of \(U^{s}\) such that \(Card\ E\cap\{\mathrm{x}^{(1)},\ldots,\mathrm{x}^{(m)}\}=b^{t}\) for any elementary interval \(E\) in base \(b\) of hypervolume \(\Lambda(E)=b^{t-m}\). Given a non-negative integer \(t\), a \((t,s)\)-sequence in base \(b\) is an infinite sequence of points \(\mathrm{x}^{(i)}\) such that, for all integers \(k\geq 0,m\geq t\), the sequence \(\{\mathrm{x}^{(kb^{m})},\ldots,\mathrm{x}^{((k+1)b^{m}-1)}\}\) is a \((t,m,s)\)-net in base \(b\).  
[PERSON]' [11] defines his \(\Pi_{\tau}\)-meshes and \(\Lambda\Pi_{\tau}\)-sequences, which are \((t,m,s)\)-nets and \((t,s)\)-sequences in base \(2\), respectively. The terms \((t,m,s)\)-nets and \((t,s)\)-sequences in base \(b\) (also called Niederreiter sequences) were introduced in 1988 by [PERSON] [13].  
To generate the \(j\)-th component of the points in a Sobol' sequence, we need to choose a primitive polynomial of some degree \(s_{j}\) over the Galois field of two elements GF(2) \(P_{j}=x^{s_{j}}+a_{1,j}x^{s_{j}-1}+a_{2,j}x^{s-2}+\ldots+a_{s_{j}-1,j}x+1\), where the coefficients \(a_{1,j},\ldots,a_{s_{j}-1,j}\) are either \(0\) or \(1\). GF(2) is the unique field with two elements \(\{0;1\}\) where the addition is defined equivalently to the logical XOR operation and the multiplication to the logical AND operation, respectively. A sequence of positive integers \(\{m_{1,j},m_{2,j},\ldots\}\) are defined by the recurrence relation  
\[m_{k,j}=2a_{1,j}m_{k-1,j}\oplus 2^{2}a_{2,j}m_{k-2,j}\oplus\cdots\oplus 2^{s_{j }}m_{k-s_{j}j}\oplus m_{k-s_{j}j},\]  
where \(\oplus\) is the bit-by-bit _exclusive-or_ operator. The values \(m_{1,j},\ldots,m_{s_{j},j}\) can be chosen freely provided that each \(m_{k,j},1\leq k\leq s_{j}\) is odd and less than \(2^{k}\). Therefore, it is possible to construct different Sobol' sequences for the fixed dimension \(s\). In practice, these numbers must be chosen very carefully to obtain very efficient Sobol' sequence generators [14]. The so-called direction numbers \(\{v_{1,j},v_{2,j},\ldots\}\) are defined by \(v_{k,j}=\frac{m_{k,j}}{2^{k}}\). Then, the \(j\)-th component of the \(i\)-th point in a Sobol' sequence is given by \(x_{i,j}=i_{1}v_{1,j}\oplus i_{2}v_{2,j}\oplus\ldots\), where \(i_{k}\) is the \(k\)-th binary digit of \(i=(\ldots i_{3}i_{2}i_{1})_{2}\).  
Subroutines to compute these points can be found in [15; 16]. The work [17] contains more details.