Thanks for pointing out the rounding issue; it's gratifying to know I helped in a small way to salvage another bottle of wine! I spent a couple of hours turning my Mma program into C code, in case anyone wants to play with it. It generates 49/4 codes at a rate of about 1 per minute. Here's a brief description of the algorithm: There are two operations, wax and wane, in each cycle. As the names suggest, the code grows in size during wax and shrinks during wane. Wax is the simpler of the two: unused codewords are added as long as the "cap" on the code (4 in the case of 49/4) is not exceeded. This introduces randomness, because the algorithm tries the unused codewords in a random (not lexicographic) order. During wane, codewords are deleted that have an overabundant representation in the unions that are at their capped values. The rationale is that growth of the code is more likely when fewer unions are capped, and if the rule is to always delete the same number d of codewords, one should preferentially delete those that "un-cap" the greatest number of capped unions. So the algorithm assigns a cost to each codeword equal to the number of capped unions it contributes to and then deletes the d codewords having the highest cost. Starting from the empty code the algorithm performs many wax and wane cycles and outputs the code whenever an increased size is found. During wax there is no limit to the growth of the code, while the size always shrinks by d during wane. I get good results for the 49/4 code when d=5. -Veit On Jan 2, 2013, at 7:46 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
Oh hey, my Z2xZ2 symmetry search just turned up a 49/4 also:
5,160,640,20,6,192,384,12,10,320,33,528,34,65,272,520,40,257,80,514,51,609,816,537,60,897,240,519,66,264,77,418,712,278,90,834,360,267,102,195,408,780,132,145,548,169,293,596,658
(Note the presence of 132 = 0010000100, a symmetry orbit of size 1, allowing the code to be of odd size.)
Still eager to see Veit's approach, though; I remain afraid that this symmetry thing is ultimately a distraction.
--Michael