  This paper explores a common class of diagonal-norm summation by parts (SBP)
operators found in the literature, which can be parameterized by an integer
triple $(s,t,r)$ representing the interior order of accuracy ($2s)$, the
boundary order of accuracy ($t$), and the dimension of the boundary closure
($r$). There is no simple formula for determining whether or not an SBP
operator exists for a given triple of parameters. Instead, one must check that
certain compatibility conditions are met: namely that a particular linear
system of equations has a positive solution. Partly because of the complexity
involved, not much is known about diagonal-norm SBP operators with $2s>10$.
  By utilizing a new algorithm for answering the question "Does an SBP operator
exist for the parameters $(s,t,r)$?", it is possible to explore the existence
of SBP operators with high order accuracy, and previously unknown SBP operators
with interior order of accuracy as large as $2s=30$ are found. Additionally, a
method for optimizing the spectral radius of the SBP derivative is introduced,
and the effectiveness of this method is explored through numerical experiment.
