## Klein Group, Tunny square, Lie algebra, decibannage

When we looked at the Tunny sqaure in the General Report (circa 1945), we quickly figured that it was related to the 4 –operator Klein Group. (It sticks out!…)

I see now how I need to improve the labels, so we accommodate the adjoint space, too. That is our picture (for the4-char group version of the algebra) SHOULD HAVE BEEN

Now the theory of repeats etc has a strong basis. Its really the commutator algebra in which the action of the normalizer compresses the original plaintext (down to) to the decibannage. Each commutator, being independent further compresses the length of the plaintext superpositions down, until there is nothing left to do which means the lengths in phase space are now associated with the radius of the normalizer line of the 7-point algebra – which have DEPENDENT terms (by definition). This means there is no further independence (or mutual information measure) to be obtained from the plaintext.

Fascinating to how these were little matrices, in a unique basis, all along.

We can assume that the still “secret” report from the Testery would show that folks were not ONLY doing mental arithmatic with 2 characters (having memorized their xor); but also had tables of larger and larger matrices, as above. Which leads to today….where its easy to do some algebras with even huge matrices, in fields Much larger than 2**5.

I suspect that element 1, in the picture I drew above is still wrong. It REALLY ought to be labeled S, not S squared. S squared is the group identity, is is a matrix of all \. But, near enough for me. We get the core idea!

Even the definition of S+ holds out, since xor is the same for subtraction or addition:

(this is bullshit, note. two matrices have constant product, the weight difference)

Now the theory of 2, 4, 8, 16 counts makes some theoretical sense, being an act of calculating at different levels of granularity/accuracy. One uses the 2-width matrix, the 16 width matrix… etc, which compares 2 and upto 16 of the source bits in a buffer , pairwise.

Putting this thought train together with that of the previous train (looking at amplitudes and intensity of light, detected by modern photodiodes rather than the 1945 era 5205 machine), let’s not forget our article that makes it all rather practical:

https://yorkporc.wordpress.com/2011/07/16/from-5205-to-balanced-homodyne-detection/

In some ways, the 2-,4-,8- counts are a little like a MRI machine, that slices cross-sectionally, thinking in terms of quantum “tomography”

We can show that product of two matrices do give a constant weight profile (the difference between each component). The profile is the diagonal of the matrix not used (ignoring the identity matrix). If one multiples with the identity matrix, the profile is that of the non-identity argument. Hmm!