Here's a symmetry update. Among the codes found by wax-&-wane for 48-50/4 only three groups have shown up: 1) The trivial group, G1. 2) The group of order 2, G2, generated by (1 3) (2 4) (5 7) (6 8) (9) (10) 3) The quaternion group of order 8, G8, generated by (1 2 3 4) (5 6 7 8) (9) (10) (1 6 3 8) (5 4 7 2) (9) (10) All three 50/4 codes found so far have symmetry G8. The four 49/4 codes I looked at (determining their symmetry is more work than generating them) all have symmetry G2. And three out of the four 48/4 codes I looked at all had no symmetry (G1); the fourth had symmetry G2. All of my codes are maximal, in the sense that additional codewords can't be added. This local optimality may account for some of the symmetry. On the other hand, Michael's Z2xZ2 group might be excessive, since there are 48/4 and 49/4 codes with symmetry groups that are proper subgroups of his group. 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. -Veit On Jan 3, 2013, at 11:01 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
Wow, super result! The spontaneous symmetry is fascinating, maybe there is something to that after all.
Would it be easy for you to modify your wax-and-wane code to enforce a symmetry constraint? Even a simple one, like adding and subtracting cyclic-rotation-by-5 pairs of points? Would be interesting to see how the codes that that produces differ from those produced by the unconstrained search.
(Hmm, I suppose during the wane step you'd need to figure out how to balance the removal of a single-point symmetry class like {1000010000} versus a two-point class {1000000000, 0000010000}.)
--Michael