How are higher dimensional sphere packings (HDSPs) related to the error-correcting codes used by cell phones, space probes and the Internet for signal communication through noisy channels? HDSPs strike me as super abstract objects and I'm having a hard time even wrapping my head around their possible concrete applications.
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In what context did sphere packing come up? Please reference where you've read or heard about it. – Nick Alexeev Oct 09 '17 at 01:51
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Was reading an article published onQuanta on HDSPs and that's literally the context i.e. it's casually mentioned in a single sentence that HDSps are used thus. – user51309 Oct 09 '17 at 15:57
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That's not a reference to an article as references go, @user51309 . – Nick Alexeev Oct 09 '17 at 16:10
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https://www.quantamagazine.org/sphere-packing-solved-in-higher-dimensions-20160330/?utm_content=buffer15425&utm_medium=social&utm_source=facebook.com&utm_campaign=buffer – user51309 Oct 10 '17 at 13:06
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Does the sphere radius indicate the acceptable deviation away from nominal magnitude and phase? If the total energy (signal + noise + ISI + deterministic trash) moves outside that radius, the energy is interpreted as a different symbol.
analogsystemsrf
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The math is way over my head for the proof of \$E_8\$ and \$E_{24}\$ which defines how to arrange spheres to occupy the most space.
It uses d-dimensional Euclidean space, Fourier transforms, Schwartz function, holomorphic congruent subgroups, Eisenstein series, Fourier expansions, Eigenvalues, Modular Forms, Poincare´ series with Bessel Functions.
- Although some residuals were rounded off, the best computer tests could not find an error of more than \$10^{-28}\$ in \$E_{24}\$ .
Although based on the Leech Lattice, the inspiration for the proof used the \$G_{24}\$ 24 bit [Golay code] used for error correction such that 3 errors could be corrected in 12 bits of data in 24 bit words. 2
The symmetry of this lattice model may lead to more beautiful revelations in nature using this with Quantum Theory, not just how to stack oranges better in the supermarket.
Tony Stewart EE75
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