Wow -- that was fast! wax & wane has found several more 50/4's, all with the same (quaternion unit) symmetry group. I'll try a deeper search to see if something else turns up. -Veit On Jan 6, 2013, at 3:13 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
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