Let \(p_ 1,\dots, p_ s\in \mathbb{N}\) be each larger than \(s\) such that \((p_ 1\dots p_ s, s!) =1\) and \(p_ \nu \asymp p_ 1^{1/(s- \nu+1) (s-\nu +2)}\), \(\nu= 2,\dots,s\). The author constructs a mesh \(M\) consisting of \(N\) points \[ \begin{split} \Biggl( \left\{ {k_ 1 \over p_ 1} + {k_ 2 \over {p_ 1p_ 2}} +\cdots+ {k_ s \over {p_ 1\dots p_ s}} \right\}, \left\{ {k_ 1 \over p_ 1} + {{2k_ 2} \over {p_ 1p_ 2}} +\cdots+ {{2^{s-1} k_ s} \over {p_ 1\dots p_ s}} \right\}, \cdots,\\ \left\{ {k_ 1 \over p_ 1} + {{sk_ 2} \over {p_ 1p_ 2}} +\cdots+ {{s^{s-1} k_ s} \over {p_ 1\dots p_ s}} \right\} \Biggr);\end{split} \] \(k_ 1= 1,\dots, p_ 1;\;k_ 2= 1,\dots, p_ 1 p_ 2;\;\dots;\;k_ s=1,\dots,p_ 1 \cdots p_ s\); where \[ N=p_ 1^ s p_ 2^{s-1}\cdots p_ s\;\asymp\;p_ 1^{s(1+ 1/2+ \dots+ 1/s)}. \] Over the class of functions \(E_ s^ \alpha (C)\), that is, the functions \(f(x_ 1, \dots,x_ s)\) with period 1 in each variable whose Fourier coefficients satisfy \(| C(m_ 1,\dots, m_ s)|\leq C(\overline{m}_ 1 \dots \overline{m}_ s)^{-\alpha}\), where \(C>0\) and \(\alpha>1\), \(\overline{m}= \max (| m|,1)\), the error \(R_ N(f)\) of the quadrature formula \[ \int_ 0^ 1 \dots \int_ 0^ 1 f(x_ 1,\dots, x_ s)dx_ 1\dots dx_ s= {\textstyle{1\over N}} \sum_{\vec \xi\in M} f(\vec\xi)+ R_ N(f) \] satisfies \[ R_ N(f)= O(N^{-\alpha/ (1+ 1/2+ \dots+ 1/s)}). \] In comparison with N. M. Korobov’s parallelepiped meshes [Number- theoretical methods in approximate analysis (Fizmatgiz, Moskva 1963; Zbl 0115.117)], the author’s mesh is defined explicitly and has a simple construction, but the order of the error of the corresponding quadrature formula is worse, although it is better than that of the equidistribution mesh. Furthermore, its applicability in practice is quite limited, because the number \(N\) of mesh points increases too rapidly as the dimension \(s\) grows.