  The leading method to study the running coupling constant of non-abelian
gauge theories is based on the Schr\"odinger functional scheme. However, the
boundary conditions and $\mathcal{O}(a)$ improvement have not been
systematically generalized for theories with more than three colors. These
theories have applications in BSM model building as well as in the large $N$
limit. We have studied the boundary conditions and improvement for the pure
Yang-Mills theory within the Schr\"odinger functional scheme. We have
determined for all values of $N$ the boundary fields which provide high
signal/noise ratio. Additionally, we have calculated the improvement
coefficient $c_t$ for the pure gauge to one loop order for SU($N$) gauge
theories with $N=2,\ldots,8$ from which $N\geq 4$ are previously unknown.
