We report three main progresses we made in 1992: The magnetic structure of the lattice was modified in order to increase the following factor of the main dipoles in the regular cells. The tolerances on the field quality of the guiding magnet were investigated with long‐term computer tracking simulations as well as with experiments performed at the SPS. Additional studies were performed to speed up the simulations using truncated maps, to propose phenomenological descriptions of the diffusive behavior in the transverse phase spaces, and to obtain a deeper insight into the nonlinear motion in the case of dynamical systems as simple as the He´non map. (AIP)