Construction of lattice rules with a trigonometric $d$-property on the basis of extreme lattices

Lattice rules with the trigonometric $d$-property that are optimal with respect to the number of points are constructed for the approximation of integrals over an $n$-dimensional unit cube. An extreme lattice for a hyperoctahedron at $n=4$ is used to construct lattice rules with the trigonometric $d$-property and the number of points
$$ 0.80822\ldots\cdot\Delta^4(1+o(1)),\quad\Delta\to\infty $$
($d=2\Delta-1\ge3$ is an arbitrary odd number). With few exceptions, the resulting lattice rules have the highest previously known effectiveness factor.