8 Sep
2016
8 Sep
'16
11:03 a.m.
Not specifically about n = 668, but you may want to contact Richard Brent (he is a very friendly guy and knows quite a bit about this topic). Our WDS has (IIRC) published something with Brent about Hadamard matrices. Best regards, jj * Allan Wechsler <acwacw@gmail.com> [Sep 08. 2016 19:05]: > 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 > > > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun