  Determining the best attainable threshold for qudit magic state distillation
is directly related to the question of whether or not contextuality is
sufficient for universal quantum computation. We carry out a search for
high-threshold magic state distillation routines for a highly-symmetric qutrit
magic state known as the strange state. Our search covers a large class of
$[[n,1]]_3$ qutrit stabilizer codes with up to 23 qutrits, and is facilitated
by a theorem that relates the distillation performance of a qudit stabilizer
code to its weight-enumerators. We could not find any code with $n<23$ qutrits
that distills the strange state with better than linear noise suppression,
other than the 11-qutrit Golay code. However, for $n=23$, we find over 600 CSS
codes that can distill the qutrit strange state with cubic noise suppression.
While none of these codes surpass the threshold of the 11-qutrit Golay code,
their existence suggests that, for large codes, the ability to distill the
qutrit strange state is somewhat generic.
