  The standard method to calculate non-perturbatively the evolution of the
running coupling of a SU(N) gauge theory is based on the Schr\"odinger
functional (SF). In this paper we construct a family of boundary fields for
general values of N which enter the standard definition of the SF coupling. We
provide spatial boundary conditions for fermions in several representations
which reduce the condition number of the squared Dirac operator. In addition,
we calculate the O(a) improvement coefficients for N>3 needed to remove
boundary cutoff effects from the gauge action. After this, residual cutoff
effects on the step scaling function are shown to be very small even when
considering non-fundamental representations. We also calculate the ratio of
Lambda parameters between the MS-bar and SF schemes.
