I have finally implemented the "missing link" for doing canonical augmentation right. This is the question of computing double-coset type problems. Suppose you have two properties P_L and P_R which form subgroups of a permutation group G (i.e. this is true of the elements for which these hold), and a third property P. Further suppose that if P holds for any permutation in G, then it forms both a left coset of P_L and a right coset of P_R. This is the kind of problem the double-coset approach tackles.

In particular, this is a more efficient method of isomorphism than the usual "canonical label times two" approach. Essentially, only one tree structure is traversed (instead of two), and worse case performance is when this one traversal takes about the same time. So the best case will be a little more than half the time of the other approach.

I have implemented randomized testing of the new code on graphs only thus far, but everything now seems to be working, and what I have is posted here.

This was the last obstacle before canonical augmentation itself could be tackled. Currently, in the code which implements augmentation for binary codes, the step which should be accomplished by a double-coset approach is farmed out to GAP, in an inefficient way. Essentially, GAP computes a group intersection and a coset traversal, in order to find whether (C,P) ~ (C,M). Instead, using refinements for each structure in the pair, we can adapt the (flexible, thanks to the generalized framework) new double coset program to compute this very efficiently. In practice, GAP can take up to 70% of cpu time when being used in this way, where the new approach is expected to be a much smaller fraction of computation time. This should speed up the classification under way here.

## Sunday, September 14, 2008

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