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)), \Delta \to \infty $
(
d = 2Δ ? 1 ≥ 3 is an arbitrary odd number). With few exceptions, the resulting lattice rules have the highest previously known effectiveness factor.