Schr\"odinger functional boundary conditions and improvement for N>3

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.