trellis coding | Peter’s ruminations


https://yorkporc.wordpress.com/2011/10/15/trellis-coding/

The evil cosets mentioned in the referenced page are less evil when (finally) we understand their relationship to “covers”.

From Turings manuscript we saw him discover the null space of the hamming cube, using permutation groups as representation theory. He showed how to search out wheel wiring so that the the underlying geometry of that null space emerged from wheel’s permutations cycles (and the langrangian, for 2 and 3 cycles, and invariant distances).

Later, he focusses on adjancency matrices, and powers thereof. But he also implicitely uses his group for the 7 point geometry as a cover for his quotient  group whose elements (the cosets themselves) are subject to the group operator, using the same generator set.

We see the notion of weight classes emerge, as the “level sets” into which the elements of the cover group are sorted.

The url adds phase shift keying/modulation to the picture.

We can so see how colossus method  is very quantum mechanical in the sense that the “certainty” that comes from a eigenfunction (the various weights attached to the nodes in the graph) for a given eigenvalue of the 7point/coset-based transform can be used to creAte a distance measure to non eigenfunction weightings – whose lessening guides the cryptographer as he guesses wheel bits (given a start).

We can also see how tension within the subsets of nodes kn “reliable” networks (the cayley graphs) allows them to characterise expander families, which ensure they do not converge (losing the remaining information eithin fhe correlations of the inner product space).

One sees how “uncertainty” relates to entropy in a very intuitive manner, as the weighting function that is at some (squared energy law) distance from the eigenfunction (certainty).

Perhaps one can look at the election of a colossus era run as selection of a sieve (of particular hole width) designed to sort the evidence (still in phase space) into  its weight “shells”.  Once sorted here, one can transform back to the 7point geometry world.

We can grasp now how turing/newman were thinking, about their nonlinear estimator (a numerical analysis exercise, indeed).

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About home_pw@msn.com

Computer Programmer who often does network administration with focus on security servers. Very strong in Microsoft Azure cloud!
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