  We show how to solve computationally a combinatorial problem about the
possible number of roots orthogonal to a vector of given length in $E_8$. We
show that the moduli space of K3 surfaces with polarisation of degree 2d is
also of general type for d=52. This case was omitted from the earlier work of
Gritsenko, Hulek and the second author. We also apply this method to some
related problems. In Appendix A, V. Gritsenko shows how to arrive at the case
d=52 and some others directly.
