Quantum Gravity 168.1: Hadamard Matrices in Cross-Phase Desingularizing/Decoding

From Osher Doctorow

Take a look at "Hadamard matrix" in Wikipedia, and then the new paper
"On Hadamard matrices at roots of unity," Teodor Banica and Jean-Marc
Schlenker, U. Toulouse III France, arXiv: 0707.4175 v1 [math.CO] 27
Jul 2007, 23 pp.

They prove:

1) if L = p1^a1 p2^a2...ps^as, then n has form p1N + ... + psN for s =
1, 2, or 3 (and conjectured for s > 3)

where L is the Lth root of unity and the Hadamard matrix has order n,
and p1N + ... + psN is a weighted sum of primes p1 through ps with
weights natural numbers (elements of N).

This result shows, among other things, that conversion of
multiplication to (weighted) addition is non-trivial, as I've been
arguing with regard to Probable Causation/Influence (PI).

With reference to physics, complex Hadamard matrices appear in
branches of coding theory and in quantum physics and have very non-
trivial quantum algebra invariants. Complex Hadamard matrices are
square matrices over the complex numbers with entries on the unit
circle and pairwise orthogonal rows.

Osher Doctorow

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