We study the KPZ equation (in D = 2, 3 and 4 spatial dimensions) by using a restricted solid-on-solid discretization of the surface. We measure the critical exponents very precisely, and we show that the rational guess is not appropriate, and that D = 4 is not the upper critical dimension. We are also able to determine very precisely the exponent of the sub-leading scaling corrections, that turns out to be close to unity in all cases. We introduce and use a multi-surface coding technique, that allows a gain of the order of 30-fold over usual numerical simulations.