8 Sep
2016
8 Sep
'16
9:37 a.m.
In about a hundred seconds of assiduous research on the Internet, I wasn't able to find any accounts of people searching for an order-668 example. Is nobody looking? Or are the searchers being secretive, and not sharing search strategies in the hopes of being first? Or (most likely) is there an active, collegial search going on, and I simply failed to find it? I can think of a bunch of strategies involving hill-climbing and annealing-like processes. I'm guessing these all fail spectacularly. On Wed, Sep 7, 2016 at 10:14 PM, Dan Asimov <dasimov@earthlink.net> wrote: > Very interesting (as J.P. wrote). > > It may not help anything, but I like to think of an Hadamard matrix > geometrically. > > Consider the n-cube as [-1,1]^n. > > Then the rows (or columns) of an order-n Hadamard matrix are n vertices of > the n-cube whose directions from the origin are perpendicular. > > —Dan > > > > On Sep 7, 2016, at 7:57 AM, Veit Elser <ve10@cornell.edu> wrote: > > > > The Hadamard matrix conjecture holds that such matrices exist for all > orders that are divisible by 4. After surveying what’s been done on the > classification/enumeration of Hadamard matrices (e.g. > http://neilsloane.com/hadamard/) I’ve felt that what’s humbling about the > conjecture is that we lack the knowledge to prove the existence of even a > single Hadamard (at each possible order) when the evidence points to a very > rapid growth in their number. Now I’m not so sure. > > > > Let x = log_2(N), N = order of Hadamard (a multiple of 4), and y = > (1/N^2)log_2(num(N)), where num(N) is the number of Hadamard matrices of > order N. Sequence A206711 gives num(N) for N = 1, … , 32. If you plot y vs. > x you get a very straight line: y = 0.78785 - 0.09458 x. Taking this > literally, there should be a maximum in the number of Hadamard matrices at > order N = 196, and beyond that the number plummets, vanishing at around N = > 322. The available constructions (beyond this number) would then represent > isolated points. > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >