Michael: could your program complete an exhaustive search for group G8? If it finds something larger than 50/4, maybe it will have an even larger symmetry group.
Great suggestion! Looks like the 50s are as good as that symmetry group gets. I just ran the exhaustive search for cap=4 codes invariant under your representation of the Quaternion units. There are 1,960,139 locally maximal codes, of which 144 are 50/4, and that's the largest that exists. (That 144 isn't reduced by symmetry at all, so probably there are fewer than that up to isomorphism; maybe I'll try to reduce this set to canonical forms later. Hmm, I'll need to figure out the normalizer of that quaternion representation...) For reference, that search took around 2 hours of compute time. --Michael -- Forewarned is worth an octopus in the bush.