I suspect I’ve just made (for me) a breakthrough – relating the constructions of des with the need of the design to specify a coin flipping operator.
We have seen from quantum walks how different coin flippers work, in concert with a suitable starting state, to give an unbiased conditional operator. (And we saw the contrary effect too, related to cryptanalysis, wherein an asymmetrical biased walk can identify a “chi setting”.)
We see now how, from a comparison of the original feistel cipher and des, how the subkey generation process works. Evidently it’s job is to control, for 48 nominally-independent bit flows, whether the (bit) value calculated in the previous round does or does not appropriately flip the current one of 48 bits of plaintext. The net result then parameterizes the Hamiltonian (and the asymmetry inducing middle cycle, particular), which in turn controls which flow heads for which active sbox.
In general, we must think of the subkey-generation processes in des as a means to choose rows of two circulant matrices, where the “stepping sequence” ( of these pseudo tunny wheels) occasionally creates a row erasure (producing complexity, when guessing the mask on reversing the Rounds)
We must look at the middle Hamiltonian as a base matrix for generating a code of subjects whose valency works with the coin flipper, as above, to generate a second coin flipper – codifying an irregular ldpc graph (with different coin flipping operator for each degree encountered as one walks the varying dimensional space).