In this note, we analyze the discrete-time implementation of a second-order sliding mode control (2-SMC) scheme. The treatment is detailed for a simple class of feedback-linearizable nonlinear systems expressed in the Brunowsky normal form. First, it is shown that the direct discretization of a continuous-time 2-SMC scheme guarantees the finite-time attainment of a motion in an O(T²) boundary layer of the sliding manifold (T being the sampling period). Then, a suitable iterative learning procedure, that leads to the asymptotic reduction of the boundary layer to O(T³) is proposed. Simulation results are reported at the end of the paper.