I'm currently working on refactoring the binary code part of sage into the new framework, and I'm trying to get my head clear on exactly what is to be done. I'll be rewriting a binary code class/struct where degree is arbitrary (the underlying implementation will use bitsets for the basis). In order to retain focus, I won't be tying this in to Sage's general linear codes until after my work for Google is done, keeping my work in the newly created sage/groups/perm_gps/partn_ref module.
To adapt incidence structures to the algorithm in general, the standard procedure is to produce a bipartite graph whose vertices are divided into two sets: points and lines of the incidence structure. However, the automorphism group of an incidence structure is a permutation group acting on the points of the structure, which induces an action on the lines. Binary codes are an example of incidence structures, with additional linear structure.
However, the bipartite graph approach can be more expensive-- this is in essence the solution to the riddle (item 4 here) of the projective planes of order 16 (which I will confirm experimentally later, once the machinery is there). Instead, the refinement tree only needs to be partitions of the points. Although it will be handy to keep the partition of the words of the code around, this need not be in the main algorithm, since it will only be used on refinement.